🧠:Okay, so I need to figure out the set of points (x, y) in the plane where y is equal to sin x divided by the absolute value of sin x. Let me start by understanding what this expression means. First, let's rewrite the equation: y = (sin x) / |sin x|. Hmm, that looks like the sign function applied to sin x. Because if I have any number divided by its absolute value, that gives me 1 if the number is positive and -1 if the number is negative. But what happens when sin x is zero? Because then the denominator would be zero, making the expression undefined. So, the first thing to note is that this equation is only defined when sin x is not zero. So, x cannot be an integer multiple of π, right? Because sin x is zero at x = 0, π, 2π, and so on, both positive and negative. So, the domain of this function is all real numbers except x = nπ where n is any integer.Now, let's think about the possible values of y. Since sin x is either positive, negative, or zero, but we've already excluded the zeros, y can only be 1 or -1. So, for each x in the domain, y will be 1 if sin x is positive and -1 if sin x is negative. Therefore, the graph of this equation should consist of horizontal lines at y = 1 and y = -1, but only where sin x is positive or negative, respectively.But where exactly is sin x positive or negative? Let's recall that sin x is positive in the intervals (0, π), (2π, 3π), (-2π, -π), etc., and negative in the intervals (π, 2π), (-π, 0), (3π, 4π), etc. So, between each multiple of π, the sign of sin x alternates. So, for example, between 0 and π (but not including 0 and π), sin x is positive, so y would be 1 there. Between π and 2π, sin x is negative, so y would be -1 there. This pattern repeats every 2π.Therefore, the graph of y = sin x / |sin x| should be a series of horizontal line segments at y = 1 and y = -1, each of length π, alternating between the two. However, at each multiple of π, there should be a discontinuity because the function is undefined there. So, the points (x, y) would lie on y = 1 and y = -1, but only in the intervals where sin x is positive or negative, respectively.Wait a second, but how exactly does this look? Let me visualize the sine curve. Sin x starts at 0, goes up to 1 at π/2, back to 0 at π, then down to -1 at 3π/2, back to 0 at 2π, and so on. So, in the interval (0, π), sin x is positive, so y would be 1. Then in (π, 2π), sin x is negative, so y would be -1. Similarly, in (-π, 0), sin x is negative, so y would be -1, and so on.Therefore, the set of points is two horizontal lines, one at y = 1 and one at y = -1, but each line is broken into intervals of length π, alternating between the upper and lower half of the plane. But wait, each horizontal line is continuous over each interval where sin x is positive or negative. So, for example, from 0 to π (excluding endpoints), y = 1; from π to 2π (excluding endpoints), y = -1; then from 2π to 3π, y = 1 again, etc. Similarly, going into the negative x-direction, from -π to 0, y = -1; from -2π to -π, y = 1, and so on.But how do we represent this as a set of points? The key is to note that for each x not equal to nπ, y is either 1 or -1. So, the set is the union of the lines y = 1 and y = -1, but with holes at every x where sin x is zero, which are the points x = nπ. Therefore, the graph is two horizontal lines with gaps at every integer multiple of π. Each line segment between nπ and (n+1)π is either y = 1 or y = -1, depending on whether n is even or odd. Let me check:Take n even: Suppose n = 0. Then from 0 to π, sin x is positive, so y = 1. If n = 2, then from 2π to 3π, sin x is positive again. Similarly, n = -2, from -2π to -π, sin x is positive. So, for even n, the interval (nπ, (n+1)π) has y = 1. For odd n, like n = 1, from π to 2π, sin x is negative, so y = -1. Similarly, n = -1, from -π to 0, sin x is negative. So, yes, if n is even, y = 1; if n is odd, y = -1. Therefore, the set is the union over all integers n of the intervals (nπ, (n+1)π) × { (-1)^n }.Wait, let's see: If n is even, then (-1)^n is 1, and if n is odd, it's -1. So, for each interval (nπ, (n+1)π), the y-coordinate is (-1)^n. Therefore, the set is the union for all integers n of the set { (x, (-1)^n) | x ∈ (nπ, (n+1)π) }.So, in summary, the set consists of horizontal lines at y = 1 and y = -1, each broken at every multiple of π, alternating between the two lines. Each segment of y = 1 is between x = 2kπ and x = (2k+1)π for integers k, and each segment of y = -1 is between x = (2k+1)π and x = (2k+2)π for integers k. Similarly in the negative direction.But perhaps it's simpler to just state that the set is all points (x, 1) where x is in the interval (2kπ, (2k+1)π) for some integer k, and all points (x, -1) where x is in the interval ((2k+1)π, (2k+2)π) for some integer k. So, each horizontal line at y = 1 and y = -1 is split into open intervals of length π, alternating between the two.Therefore, the final answer should describe these points as the union of these open intervals on the horizontal lines y=1 and y=-1, excluding the points where x is an integer multiple of π.Let me verify with specific examples. For x = π/2, sin x = 1, so y = 1/1 = 1. So, (π/2, 1) is in the set. For x = 3π/2, sin x = -1, so y = -1/1 = -1, so (3π/2, -1) is in the set. For x = π, sin x = 0, so it's excluded. Similarly, x = 0, 2π, etc., are all excluded. That makes sense.Another test: x = π/4, which is in (0, π), so y=1. Correct. x = 5π/4, which is in (π, 2π), so y=-1. Correct. x = 3π, which is excluded. So, the points at multiples of π are not included. So, the graph is two horizontal lines with holes at each multiple of π.Therefore, the set is the union of the open intervals along y=1 and y=-1 between each nπ and (n+1)π, with n even for y=1 and odd for y=-1.Alternatively, more formally, the set can be written as:{ (x, 1) | x ∈ ℝ { nπ | n ∈ ℤ } and sin x > 0 } ∪ { (x, -1) | x ∈ ℝ { nπ | n ∈ ℤ } and sin x < 0 }But since sin x > 0 exactly when x is in (2kπ, (2k+1)π) for some integer k, and sin x < 0 when x is in ((2k+1)π, (2k+2)π) for some integer k, we can write it as:⋃_{k ∈ ℤ} [ (2kπ, (2k+1)π) × {1} ] ∪ [ ((2k+1)π, (2k+2)π) × {-1} ]Yes, that seems precise. Each interval of length π where sin x is positive contributes a horizontal line segment at y=1, and each interval where sin x is negative contributes a segment at y=-1, with the endpoints excluded.So, putting it all together, the set is two horizontal lines at y=1 and y=-1, each consisting of open intervals of length π, alternating between the two lines at each multiple of π. The points where x is an integer multiple of π are excluded because the original expression is undefined there.

✅:The set of points ((x, y)) in the plane satisfying (y = frac{sin x}{|sin x|}) is characterized as follows:1. Expression Analysis: The equation simplifies to (y = text{sgn}(sin x)), where (text{sgn}) is the sign function. This means: - (y = 1) when (sin x > 0), - (y = -1) when (sin x < 0), - Undefined when (sin x = 0) (i.e., (x = npi) for (n in mathbb{Z})).2. Intervals of (sin x): - (sin x > 0) on intervals ((2kpi, (2k+1)pi)) for (k in mathbb{Z}), - (sin x < 0) on intervals (((2k+1)pi, (2k+2)pi)) for (k in mathbb{Z}).3. Resulting Set: The solution set comprises horizontal line segments at (y = 1) and (y = -1), excluding points where (x) is a multiple of (pi): - For (y = 1): All points ((x, 1)) where (x in (2kpi, (2k+1)pi)) for some integer (k). - For (y = -1): All points ((x, -1)) where (x in ((2k+1)pi, (2k+2)pi)) for some integer (k).Final Answer: The set is the union of horizontal lines at (y = 1) and (y = -1), partitioned into open intervals of length (pi) excluding multiples of (pi). Formally: [boxed{bigcup_{k in mathbb{Z}} left( left(2kpi, (2k+1)piright) times {1} cup left((2k+1)pi, (2k+2)piright) times {-1} right)}]

🔑:1. We start by analyzing the given function: [ y = frac{sin x}{|sin x|} ] This expression can be simplified by considering the properties of the absolute value function. Specifically, the fraction (frac{sin x}{|sin x|}) will depend solely on the sign of (sin x).2. Let us rewrite the function (y) by removing the absolute value in the denominator: [ y = begin{cases} 1 & text{ if } sin x > 0 -1 & text{ if } sin x < 0 end{cases} ]3. Next, determine the intervals where (sin x > 0). Recall that the sine function is positive in the intervals ((2npi, (2n+1)pi)), where (n in mathbb{Z}). Therefore: [ sin x > 0 text{ for } x in left(2npi, (2n+1)piright) ]4. Similarly, determine the intervals where (sin x < 0). The sine function is negative in the intervals (((2n+1)pi, (2n+2)pi)), where (n in mathbb{Z}). Hence: [ sin x < 0 text{ for } x in left((2n+1)pi, (2n+2)piright) ]5. Combining these observations, we can describe the intervals for (y): [ y = begin{cases} 1 & text{ for } x in left((2n-1)pi, 2npiright) -1 & text{ for } x in left(2npi, (2n+1)piright) end{cases} ] for (n in mathbb{Z}).6. Now, note that the function (y = frac{sin x}{|sin x|}) has discontinuities at points where (sin x = 0). These points occur at (x = npi), where (n in mathbb{Z}). Thus, the function (y) is not defined at these points. Conclusion:Therefore, the solution requires plotting intervals of constant values ((y = 1) and (y = -1)) with breaks at (x = npi) for all integers (n), forming a series of discontinuous steps parallel to the (x)-axis.[boxed{begin{cases}y = 1 & text { for } x in (pi(2n-1), 2npi) text { excluding } x = pi ny = -1 & text { for } x in (2npi, pi(2n+1)) text { excluding } x = 2npiend{cases}text{ for } n in mathbb{Z}}]

🧠:Alright, let's try to figure out this football tournament problem. So, there are five teams: A, B, C, D, E. Each plays exactly once against each other, so each team plays four matches. The points are 3 for a win, 1 for a draw, 0 for a loss. The key points given are:1. All teams have different points.2. Team A has the most points, but lost to Team B.3. Teams B and C did not lose any game.4. Team C has fewer points than Team D.We need to find the points each team earned. Let's break this down step by step.First, let's recall that in a round-robin tournament with five teams, each plays four matches. So, there are a total of C(5,2) = 10 matches. Each match contributes either 3 points (if someone wins) or 2 points (if it's a draw) to the total points. So, the total points in the tournament can range from 10*2=20 (all draws) to 10*3=30 (all wins). But since B and C didn't lose any games, their matches might involve more draws or wins.Starting with Teams B and C: they didn't lose any games. That means in all their matches, they either won or drew. Let's think about their results.Each team plays four matches. Team B didn't lose any game, so their results are either wins or draws. Similarly for Team C.But Team A lost to Team B. So, in the match between A and B, B won, which means B got 3 points from that game, and A got 0. But Team A still ended up with the highest total points. So, Team A must have won all their other matches except the one against B. Wait, but if A lost to B, then A's maximum possible points would be 3 wins and 1 loss: 3*3 + 0 = 9 points. But let's check that.Wait, no. If A lost to B, then they have three other matches: against C, D, E. If they won all three, that's 9 points. But Team B, since they didn't lose any games, their points would be at least 3 (from beating A) plus whatever they get from their other three matches (against C, D, E). Since B didn't lose, those three matches could be wins or draws. Similarly for Team C.But we also know that all teams have different points, and C has fewer points than D.Let me try to outline the matches each team played and their possible results.First, Team B didn't lose any games. So, their results: vs A (win), vs C, vs D, vs E (either wins or draws). Similarly, Team C didn't lose any games: vs A, vs B, vs D, vs E (either wins or draws). But since B and C played each other, their match must have been a draw because neither lost any game. Wait, because if they played each other, and neither lost, then the result must be a draw. Because if one had won, the other would have lost, contradicting the fact that neither B nor C lost any games. So, B vs C must have been a draw. That gives both B and C 1 point from that game.So, updating:Team B's matches: vs A (win, 3 points), vs C (draw, 1 point), vs D and vs E (either wins or draws).Team C's matches: vs A (unknown, but since C didn't lose, they either drew or won), vs B (draw, 1 point), vs D and vs E (either wins or draws).But Team A lost to B, so in A's matches: vs B (loss, 0 points), vs C, vs D, vs E (results unknown). However, Team A has the most points overall. So, after losing to B, A must have maximized their points in the remaining three games (against C, D, E). Since A needs to have the highest points, they probably won all three remaining matches. But wait, could they have drawn any? If they drew, they would get fewer points. Let's check.If Team A won all three remaining matches (against C, D, E), they would get 3*3 = 9 points from those, plus 0 from the loss to B, total 9 points. If they drew any of those, they would get 1 point instead of 3, which would lower their total. Since they need the highest points, they probably won all three. But we need to check if this is possible given the constraints on Teams B and C.But wait, Team C didn't lose any games. So, in the match between A and C, if A won, then C lost, which contradicts the fact that C didn't lose any games. Therefore, the match between A and C must have been a draw. So, Team A drew with C, giving both 1 point. But wait, that would mean Team A's results: lost to B, drew with C, and then played D and E. To maximize points, A would need to win against D and E. Let's see:If A drew with C, then against C: A gets 1, C gets 1. Then A plays D and E. If A wins both, that's 3+3=6 points. So total points for A: 0 (vs B) + 1 (vs C) + 6 (vs D, E) = 7 points. But is that enough for A to be the highest? Let's compare with Team B.Team B has beaten A (3 points), drawn with C (1 point), and plays D and E. If B wins those two, that's 3+3=6, total points for B: 3+1+6=10. Then B would have 10 points, which is more than A's 7. But the problem states that A has the most points. Therefore, this scenario is invalid. So, there must be a mistake here.Wait, this indicates that if A drew with C, then even if A wins against D and E, they only get 7 points, while B could get up to 10. But the problem states that A is the highest. Therefore, our assumption that A drew with C must be wrong. But Team C cannot lose, so the only possibility is that A vs C was a draw. Therefore, there's a contradiction here. Hmm.Wait, so maybe Team A didn't play against C? No, each team plays every other team. So, A must have played C. Since C didn't lose any games, the match between A and C must have been a draw. Therefore, A gets 1 point from that game, C gets 1 point. Then A's remaining games are against D and E. To maximize points, A must win both, giving 3+3=6, so total 0 (B) +1 (C) +6 (D,E) =7. But B, having beaten A and drawn with C, has 3 +1 from those two games, and then plays D and E. If B wins both, that's 3+3=6, total 3+1+6=10. But that's more than A's 7, which contradicts A being the highest. Therefore, our initial assumption that A drew with C is correct, but this leads to a contradiction. Therefore, our approach must be wrong.Wait, maybe Team B didn't win both remaining games against D and E? If B drew some of them, then their total points would be lower. Let's see. If B drew both D and E, then B would have 3 (vs A) +1 (vs C) +1+1=6 points. Then A has 7, which is higher. That works. But Team B didn't lose any games, so they can have draws. Similarly, Team C's points: they drew with A and B, so 1+1=2 from those, plus their games against D and E. If C drew both, they would have 2+1+1=4 points. But Team C must have fewer points than D. Let's see D's points. D plays against A, B, C, E. If D lost to A (since A beat D and E), but D's results against B and C: since B didn't lose, D could have drawn with B and C. Wait, but D plays B and C. If B didn't lose any games, then B vs D must be a draw or a B win. Similarly, C vs D: since C didn't lose, it's a draw or C win.But let's try to model this step by step.First, matches for each team:Team A: plays B, C, D, E. Lost to B, drew with C, won against D and E. So A's points: 0 +1 +3 +3 =7.Team B: plays A, C, D, E. Beat A, drew with C. Against D and E: let's say drew both. So B's points:3 +1 +1 +1=6.Team C: plays A, B, D, E. Drew with A and B. Against D and E: let's say drew both. So C's points:1+1+1+1=4.Team D: plays A, B, C, E. Lost to A (since A won against D). Against B: drew (since B drew all others). Against C: drew. Then plays E. If D beats E, then D's points:0 (vs A) +1 (vs B) +1 (vs C) +3 (vs E)=5. Then E's points: lost to A, lost to D, and their other matches. Wait, E plays A, B, C, D. Lost to A and D. Against B and C: since B and C didn't lose any games, E could have drawn with them. So E's points:0 (A) +1 (B) +1 (C) +0 (D)=2. So D has 5, C has 4, which satisfies C < D. Then Team B has 6, Team A has7. All points:7,6,5,4,2. All different, and ordered A(7), B(6), D(5), C(4), E(2). But wait, but the problem states that all teams have different points, which is satisfied here. Also, C has fewer than D (4<5). Team A has the most (7), even though they lost to B (who has 6). That seems to fit. But let's check if this is possible.Wait, but Team B's matches: they drew with C, D, and E. So B's results: W D D D. So 3+1+1+1=6. Team C drew with A, B, D, E. So four draws: 1*4=4. Team D: lost to A, drew with B, drew with C, beat E: 0+1+1+3=5. Team E: lost to A and D, drew with B and C: 0+0+1+1=2. Team A: lost to B, drew with C, beat D and E: 0+1+3+3=7. This seems to work. All points are different:7,6,5,4,2. C (4) < D (5). A (7) is highest. B didn't lose any games, C didn't lose any. But wait, Team C drew all their games, right? So they didn't lose any. Team B won one (against A) and drew the rest. Yes. But let's check if there are other possible distributions.Alternatively, maybe Team B didn't draw all three remaining games but won some. For example, if B wins one and draws two. Let's see. If B beats D and draws with C and E. Wait, but B already drew with C. So B's matches: vs A (W), vs C (D). Then vs D and E. Suppose B beats D and draws with E. Then B's points:3 +1 +3 +1=8. Then Team A has 7, which is less than B's 8. But the problem says A has the most points. So this is invalid. Therefore, B cannot win any other matches besides A. Therefore, B must have drawn all remaining matches after A and C. So B's points are 3 (A) +1 (C) +1 (D) +1 (E)=6. So this seems necessary to keep B's points below A's.Similarly, Team C's points: if they drew with A and B, and their matches against D and E. Suppose instead of drawing both, C wins one and draws one. But C didn't lose any games, so they can win. Wait, but if C wins against D, then D loses that game. But D could still have points from other matches. Let's see:If C vs D: C wins (3 points), and C vs E: draw (1 point). Then C's points:1 (A) +1 (B) +3 (D) +1 (E)=6. Then Team D's points: lost to A (0), drew with B (1), lost to C (0), and plays E. If D beats E, then D's points:0+1+0+3=4. Then C has 6, D has4, which contradicts C < D. So that's invalid. Therefore, C cannot win against D. Alternatively, if C draws with D and E, then C's points 4, D's points from drawing with C (1) plus others. As before, D would have 5. That works. So the initial scenario is the only valid one.Now, let's check Team E. E plays A (loss), B (draw), C (draw), D (loss). So E's points:0+1+1+0=2. Yes.So the points would be:A:7, B:6, D:5, C:4, E:2. All different, ordered A > B > D > C > E. But wait, the problem says all teams have different points, and C < D. This fits.But wait, the problem states that Team C earned fewer points than Team D. In this case, C has4, D has5, so that's correct. Also, B didn't lose any games, C didn't lose any. A lost to B but still has the highest points.But let's check total points. Total points in the tournament:7+6+5+4+2=24. Since there are 10 matches, each contributing 3 or 2 points. Let's see: 24 total points. Let's count the number of wins and draws.Each win gives 3 points, each draw gives 2 points. So 3*W + 2*D =24. Also, total matches W + D=10.So solving:3W +2D=24W + D=10 → D=10 -WSubstitute:3W +2(10 -W)=24 →3W +20 -2W=24→W +20=24→W=4So 4 wins and 6 draws. Let's check if this matches our scenario.In the scenario above:Wins:- B beat A (1)- A beat D (2)- A beat E (3)- D beat E (4)That's 4 wins. The rest are draws:- B vs C (D)- B vs D (D)- B vs E (D)- C vs A (D)- C vs D (D)- C vs E (D)Wait, no. Wait, let's list all matches:Matches:1. A vs B: B wins (W)2. A vs C: draw (D)3. A vs D: A wins (W)4. A vs E: A wins (W)5. B vs C: draw (D)6. B vs D: draw (D)7. B vs E: draw (D)8. C vs D: draw (D)9. C vs E: draw (D)10. D vs E: D wins (W)So total wins: 1 (B vs A), 3 (A vs D, A vs E, D vs E) → total 4 wins. Draws:6. That matches 4*3 +6*2=12+12=24. Correct.So this scenario is valid. Therefore, the points are:A:7B:6D:5C:4E:2But the question asks to list them as:Team A earned:Team B earned:Team C earned:Team D earned:Team E earned:So in order, A:7, B:6, C:4, D:5, E:2. But wait, the order of C and D is important. The problem says C earned fewer than D, so C=4, D=5. Correct.But let's check if there's another possible distribution. For example, could Team B have more points than in this scenario but still less than A? Suppose Team B drew two and won one of their remaining games (D and E). But if B wins one more game, say against D, then B's points would be 3 (A) +1 (C) +3 (D) +1 (E)=8. Then A has7, which is less, which is invalid. So B can't win any more games. Therefore, the only way is B draws all remaining matches.Thus, the only possible points are as above.Another check: Team C has four points from four draws (A, B, D, E). Team D has five points: drew with B and C, lost to A, beat E. Yes.Yes, this seems consistent. All constraints are satisfied.

✅:Team A earned: boxed{7} points Team B earned: boxed{6} points Team C earned: boxed{4} points Team D earned: boxed{5} points Team E earned: boxed{2} points Step-by-Step Explanation:1. Tournament Structure: Each of the 5 teams plays 4 matches (total 10 matches). Points: 3 for a win, 1 for a draw.2. Key Constraints: - All teams have distinct points. - Team A has the most points but lost to Team B. - Teams B and C never lost a game. - Team C’s points < Team D’s points.3. Analyzing Team B and C: - Team B beat Team A (3 points) and drew all other matches (vs C, D, E: 1 point each). Total: 3 + 1 + 1 + 1 = 6. - Team C drew all matches (vs A, B, D, E: 1 point each). Total: 1 times 4 = 4.4. Team A’s Performance: - Lost to B (0 points) but won against D and E (3 points each) and drew with C (1 point). Total: 0 + 3 + 3 + 1 = 7.5. Team D and E: - Team D lost to A (0), drew with B and C (1 each), and beat E (3). Total: 0 + 1 + 1 + 3 = 5. - Team E lost to A and D (0 each), drew with B and C (1 each). Total: 0 + 0 + 1 + 1 = 2.6. Validation: - All points are distinct: 7, 6, 5, 4, 2. - Team A (7) > Team B (6) > Team D (5) > Team C (4) > Team E (2). - Total points: 7 + 6 + 5 + 4 + 2 = 24 (consistent with 4 wins and 6 draws).

🔑:To determine how many points each team scored, we need to use the given conditions and logical deductions based on the structure of the tournament.1. Points Representation: - Let the points scored by teams ( A, B, C, D, ) and ( E ) be ( a, b, c, d, ) and ( e ), respectively.2. Total Matches and Points: - Each team plays each other exactly once, resulting in ( binom{5}{2} = 10 ) matches. - A total of 30 points can be gained if there are no draws as each match awards 3 points to the winning team. For each draw, 2 points are awarded in total.3. Conditions and Bounds: - Since all teams scored different points and points can be from 0 to 12 in steps of 3 (maximum points is 9, but possible stepwise values analyzed similarly). - Team ( A ) scored the most points, having lost to ( B ). - Teams ( B ) and ( C ) did not lose any matches. - Team ( C ) scored fewer points than ( D ).4. Analysis: - From the given conditions, ( C ) must have drawn all matches since they didn't lose any and didn't score very high. This implies ( c = 4 ) (each draw gives 1 point across 4 matches). - ( A ) must have the highest after ( C ), and as ( C ) scored 4 points through draws (no more than this), and it must fit the differential structure.5. Steps: - Since ( B ) also did not lose and scored more points than ( C ), and won against ( A ), thus by other deductions ( B ) has played conservatively to draw most except possibly the win against ( A ). This implies ( b = 6 ), hinting more draws. - From the hierarchy, since ( A ) won against ( D ) and ( E ), but drew one (and lost against B) hence for the highest: - ( A: 3(wins) + 1(draw) = 3 * 2 + 1 = 7 ) - Similarly, team ( D ) scored more than ( C ) so: - wins against ( E ) and draws against ( C ) and others might place - ( d = 5 ) - Remaining points infer ( E ) might be formed from least: - ( E = 2 )# Conclusion:Thus by analyzing each team's condition, points validated, each respectively scored points:[boxed{A = 7, B = 6, C = 4, D = 5, E = 2}]

🧠:Okay, so I need to figure out how many 31-element subsets of the set S = {1, 2, ..., 1990} have a sum that's divisible by 5. These are called "good subsets." Hmm, let's start by understanding the problem better.First, the set S has elements from 1 to 1990. We need to consider all possible subsets of size 31 and count those whose elements add up to a multiple of 5. The total number of subsets of size 31 is C(1990, 31), which is a huge number. But obviously, not all of them will have sums divisible by 5. I need a smarter way than brute force.I remember that when dealing with problems involving sums modulo some number, generating functions or the principle of inclusion-exclusion might be useful. Alternatively, maybe using combinatorial methods with modulo arithmetic. Let me recall if there's a standard approach for counting subsets with a given sum modulo m.Ah, yes, generating functions can be helpful here. The idea is to construct a generating function where each element's contribution is a term in a polynomial, and the coefficient of x^k in the product gives the number of subsets that sum to k. Then, if we're interested in subsets where the sum is divisible by 5, we need the sum of coefficients where the exponent is congruent to 0 mod 5.But since we're dealing with large numbers (1990 elements), directly expanding such a generating function isn't feasible. However, maybe we can use the roots of unity filter to extract the coefficients corresponding to multiples of 5. That sounds like a plan.The generating function for the entire set S would be the product over all elements (1 + x^element). Then, the number of subsets of size 31 with sum divisible by 5 is the coefficient of x^{5k} times y^{31} in this generating function. Wait, but how do we handle both the subset size and the sum?Actually, the generating function needs to track both the number of elements and their sum. So, maybe a two-variable generating function where each term is (1 + y*x^element). Then, the coefficient of y^31 x^{5k} would give the number of 31-element subsets with sum 5k. Then, we need the sum over all k of these coefficients.To compute this, we can use the roots of unity method on both variables? Hmm, but handling two variables might complicate things. Alternatively, since the subset size is fixed at 31, maybe first fix the size and then apply the roots of unity filter for the sum modulo 5.Yes, that might work. Let me formalize this.Let’s denote by N the total number of 31-element subsets, and we want the number of those where the sum is ≡ 0 mod 5. Let’s recall that the number we want is (1/5) * Σ_{j=0}^4 ω^{-j*0} * N(ω^j), where ω is a primitive 5th root of unity, and N(ω) is the generating function evaluated at ω with subsets of size 31.Wait, more precisely, the generating function for subsets of size 31 is the coefficient of y^31 in the product_{k=1}^{1990} (1 + y*x^k). Then, to get the number of such subsets with sum ≡ 0 mod 5, we can evaluate the coefficient of y^31 in (1/5) * Σ_{j=0}^4 ω^{-j*0} * Product_{k=1}^{1990} (1 + y*ω^{j*k}).But this seems quite involved. Maybe there's another way.Alternatively, for each element in S, we can consider its residue mod 5. Since the sum modulo 5 depends only on the residues of the elements, we can partition the set S into residue classes mod 5, and then model the problem as a counting problem with these classes.Let's try this approach.First, partition S into 5 residue classes modulo 5: residues 0, 1, 2, 3, 4.Let’s count how many numbers are in each residue class.Numbers from 1 to 1990:For residue 0: numbers divisible by 5. The first is 5, the last is 1985 (since 1985 = 5*397). So, the number of elements in residue 0 is 1985 / 5 = 397. Wait, 5*397 = 1985, then 1985 + 5 = 1990, which is the next number, but 1990 is included in the set S. Wait, S is {1, ..., 1990}, so 1990 is included. Therefore, numbers divisible by 5 in S are 5, 10, ..., 1990. The number of such elements is 1990 / 5 = 398. So, residue 0 has 398 elements.Similarly, for residue 1: numbers congruent to 1 mod 5. These are 1, 6, 11, ..., up to the largest number ≤1990. Let's compute how many. The sequence is 1 + 5k, where 5k ≤ 1989. 1989 /5 = 397.8, so k goes from 0 to 397, giving 398 numbers. Wait, 1 + 5*397 = 1 + 1985 = 1986, then 1986 +5 =1991, which is over. So the last term is 1986, which is 1 mod 5. So, residue 1 has 398 elements.Wait, let me check again. The numbers congruent to 1 mod 5 in S: starting at 1, each subsequent number is 5 more. So, the nth term is 1 +5(n-1). The largest term ≤1990 is 1 +5(n-1) ≤1990 → 5(n-1) ≤1989 → n-1 ≤1989/5=397.8 → n-1=397 → n=398. So, 398 elements.Similarly, residues 2, 3, 4:For residue 2: starting at 2, next is 7, 12, ..., up to ≤1990. Similarly, 2 +5(n-1) ≤1990 → 5(n-1) ≤1988 → n-1=1988/5=397.6, so n=398. So residue 2: 398 elements.Residue 3: starting at 3, 8, ..., 3 +5(n-1) ≤1990 → 5(n-1) ≤1987 → n-1=397.4 → n=398. So residue 3: 398 elements.Residue 4: starting at 4, 9, ..., 4 +5(n-1) ≤1990 → 5(n-1) ≤1986 → n-1=397.2 → n=398. But wait, 4 +5*397 =4 +1985=1989. Then 1989 +5=1994>1990, so yes, residue 4 also has 398 elements.Wait, but 1990 divided by 5 is exactly 398. So 5*398=1990. Therefore, each residue class mod 5 has exactly 398 elements. That's nice, the set S is evenly distributed modulo 5. Wait, but 5*398=1990, so each residue from 0 to 4 has exactly 398 elements. That simplifies things!So, each residue class 0,1,2,3,4 has exactly 398 elements. Perfect. So when we choose a subset, we can choose some number of elements from each residue class, say, c_0 from residue 0, c_1 from residue 1, etc., such that c_0 + c_1 + c_2 + c_3 + c_4 =31. Then, the sum modulo 5 is (0*c_0 +1*c_1 +2*c_2 +3*c_3 +4*c_4) mod5. We need this to be congruent to 0 mod5.So the problem reduces to counting the number of solutions (c_0,c_1,c_2,c_3,c_4) with each c_i between 0 and 398 (since each residue class has 398 elements), c_0 + c_1 + c_2 + c_3 + c_4 =31, and 1*c_1 +2*c_2 +3*c_3 +4*c_4 ≡0 mod5.Therefore, we need to compute the number of such tuples.This is a classic problem in combinatorics with congruence constraints. The standard approach is to use generating functions with roots of unity to pick out the congruence condition.Let me formalize this.The number of ways to choose c_i elements from residue class i is C(398, c_i) for each i. So the total number of subsets is the product over i of C(398, c_i), multiplied over all valid c_i such that sum c_i =31 and sum i*c_i ≡0 mod5.Therefore, the total number is the coefficient of y^31 z^0 in the generating function:Product_{i=0}^4 [Sum_{c_i=0}^{398} C(398, c_i) y^{c_i} z^{i*c_i} } ]But since we need the coefficient of y^31 and z^{0 mod5}, we can use roots of unity for the z variable.Alternatively, note that we can write the generating function as:Product_{i=0}^4 (1 + y*z^i)^{398}Then, the coefficient of y^31 z^{5k} in this product is the number of 31-element subsets with sum 5k. To get the total number of such subsets for any k, we need to sum over all k, which corresponds to the coefficient of y^31 in the generating function evaluated at z=1, but only considering the exponents of z divisible by 5.Using the roots of unity filter, the number we want is (1/5) * Sum_{j=0}^4 G(ω^j), where ω is a primitive 5th root of unity, and G(z) is the generating function Product_{i=0}^4 (1 + y*z^i)^{398} evaluated at y^31.Wait, actually, the generating function for the sum is Product_{i=0}^4 (1 + y*z^i)^{398}, and we need the coefficient of y^31 z^{0 mod5}. So to extract the coefficient of z^{0 mod5}, we can use the roots of unity filter on z. Then, the number is (1/5) * Sum_{j=0}^4 G(ω^j), where G(z) = [Product_{i=0}^4 (1 + y*z^i)^{398}]_{y^31}.But actually, since we need the coefficient of y^31 as well, perhaps we need to handle both variables. Let me clarify.Let’s denote F(y, z) = Product_{i=0}^4 (1 + y*z^i)^{398}Then, the coefficient of y^31 z^{5m} in F(y, z) is the number of 31-element subsets with sum 5m. Therefore, the total number of good subsets is the sum over m of these coefficients, which is equal to the coefficient of y^31 in the sum over m of the coefficient of z^{5m} in F(y, z). To compute this, we can apply the roots of unity filter on z:Sum_{m} [Coefficient of z^{5m} in F(y, z)] = (1/5) * Sum_{j=0}^4 F(y, ω^j)Therefore, the number of good subsets is the coefficient of y^31 in (1/5) * Sum_{j=0}^4 F(y, ω^j) = (1/5) * Sum_{j=0}^4 [Product_{i=0}^4 (1 + y*ω^{i j})^{398} } ]Therefore, substituting, we get:Number of good subsets = (1/5) * Sum_{j=0}^4 [Product_{i=0}^4 (1 + y*ω^{i j})^{398} ] evaluated at y^31.Now, we need to compute this for each j from 0 to 4 and then take the average.Let me compute each term separately.First, for j=0:ω^{0} =1, so each term becomes (1 + y*1^{i*0})^{398} = (1 + y)^{398} for each i from 0 to 4. But wait, the product is over i=0 to 4. Wait, but for j=0, ω^{i*j} = ω^0 =1 for all i. Therefore, each factor in the product becomes (1 + y*1)^{398} = (1 + y)^{398}. Since we have 5 such factors (i=0 to 4), the entire product becomes [(1 + y)^{398}]^5 = (1 + y)^{1990}. Then, the coefficient of y^31 in this is C(1990, 31). Therefore, the term for j=0 is C(1990, 31).For j=1,2,3,4, we need to compute the product over i=0 to 4 of (1 + y*ω^{i j})^{398} and then take the coefficient of y^31.Let’s note that ω is a primitive 5th root of unity, so ω^5=1. Also, for j=1,2,3,4, the exponents i*j mod5 cycle through different residues.Let’s take j=1 first.For j=1, the exponents i*j mod5 are 0,1,2,3,4 for i=0,1,2,3,4. Therefore, the product becomes:(1 + y*ω^0)^{398} * (1 + y*ω^1)^{398} * (1 + y*ω^2)^{398} * (1 + y*ω^3)^{398} * (1 + y*ω^4)^{398}= [ (1 + y) * (1 + y*ω) * (1 + y*ω^2) * (1 + y*ω^3) * (1 + y*ω^4) ]^{398}But notice that (1 + y*ω^k) for k=0 to4 is the factorization of 1 + y^5, since the roots of 1 + y^5 are -ω^k for k=0 to4. Wait, actually, 1 + y^5 = Product_{k=0}^4 (1 + y*ω^k). Let me check:The roots of 1 + y^5 =0 are y = -ω^k for k=0 to4. Therefore, 1 + y^5 = Product_{k=0}^4 (y + ω^k). But we have Product_{k=0}^4 (1 + y*ω^k). Let’s substitute z = y, then Product_{k=0}^4 (1 + z*ω^k). Let’s compute this:Let’s take z = y, then Product_{k=0}^4 (1 + z*ω^k) = ?Using the formula for cyclotomic polynomials or maybe generating functions. Alternatively, note that if we set z = -1, the product becomes Product (1 - ω^k) = 1 -1 +1 -1 +1 -1? Wait, no. Wait, for z = -1, Product_{k=0}^4 (1 - ω^k). But ω^0=1, so the first term is (1 -1)=0, hence the product is 0. But actually, maybe expanding the product:Let’s multiply out the terms:(1 + z)(1 + zω)(1 + zω^2)(1 + zω^3)(1 + zω^4)First, multiply (1 + z)(1 + zω^4):=1 + z(1 + ω^4) + z^2 ω^4Then multiply (1 + zω)(1 + zω^3):=1 + zω(1 + ω^2) + z^2 ω^4Wait, maybe this approach is too tedious. Alternatively, note that the product over (1 + zω^k) is equal to 1 + z + z^2 + z^3 + z^4 + z^5. Wait, is that true?Wait, when z =1, the product is 2*2*2*2*2=32. But 1 +1 +1 +1 +1 +1=6. Not equal. So that’s incorrect.Wait, perhaps using generating functions. Let me think. The generating function Product_{k=0}^4 (1 + zω^k) is the same as evaluating the generating function Product_{k=0}^4 (1 + zω^k). Let’s note that ω is a primitive 5th root of unity, so ω^5 =1.Alternatively, log the product:ln(Product_{k=0}^4 (1 + zω^k)) = Sum_{k=0}^4 ln(1 + zω^k)But this might not be helpful.Wait, maybe expand the product step by step.First, multiply (1 + zω^0)(1 + zω^1)(1 + zω^2)(1 + zω^3)(1 + zω^4)Let’s note that the product is equal to the value of the polynomial f(z) = Product_{k=0}^4 (1 + zω^k). Let’s compute f(z):f(z) = Product_{k=0}^4 (1 + zω^k)Let’s note that if we set z = -1/ω^k, then each factor becomes 0. Therefore, the roots of f(z) are -1/ω^k for k=0 to4. Which is equivalent to -ω^{-k}. Since ω^5=1, ω^{-k} = ω^{5 -k}.Therefore, f(z) = Product_{k=0}^4 (1 + zω^k) = Product_{k=0}^4 (1 + zω^k)Alternatively, consider that if we let z = t, then f(t) = Product_{k=0}^4 (1 + tω^k)I recall that in complex analysis, such products can sometimes be expressed in terms of known polynomials. Let’s test for small degrees.Compute f(z) when z is small:f(z) = (1 + z)(1 + zω)(1 + zω^2)(1 + zω^3)(1 + zω^4)Multiply the first two factors: (1 + z)(1 + zω) =1 + z(1 + ω) + z^2 ωMultiply the next two factors: (1 + zω^2)(1 + zω^3) =1 + z(ω^2 + ω^3) + z^2 ω^5 =1 + z(ω^2 + ω^3) + z^2 ω^0 =1 + z(ω^2 + ω^3) + z^2Now, multiply these two results:[1 + z(1 + ω) + z^2 ω][1 + z(ω^2 + ω^3) + z^2]This seems complicated. Alternatively, note that multiplying all five factors may result in a symmetrical polynomial. Let me try expanding step by step.Alternatively, use the fact that ω^k are the 5th roots of unity, so we can use properties of roots of unity.Note that for any z, Product_{k=0}^4 (z - ω^k) = z^5 -1.But here we have Product_{k=0}^4 (1 + zω^k). Let’s make a substitution. Let’s set w = z, then:Product_{k=0}^4 (1 + wω^k) = ?Let’s note that if we substitute w = -u, then the product becomes Product_{k=0}^4 (1 - uω^k). Compare this to the generating function for cyclotomic polynomials. The 5th cyclotomic polynomial is Φ_5(x) = x^4 + x^3 + x^2 + x +1. But how does this relate?Alternatively, consider that Product_{k=0}^4 (x - ω^k) = x^5 -1. Therefore, Product_{k=0}^4 (x - ω^k) = x^5 -1. Let’s set x = -1/w, then:Product_{k=0}^4 (-1/w - ω^k) = (-1/w)^5 -1 = -1/w^5 -1But also, Product_{k=0}^4 (-1/w - ω^k) = Product_{k=0}^4 [ (-1 - wω^k)/w ] = (1/w^5) Product_{k=0}^4 (-1 - wω^k )Therefore,(1/w^5) Product_{k=0}^4 (-1 - wω^k ) = -1/w^5 -1Multiply both sides by w^5:Product_{k=0}^4 (-1 - wω^k ) = -1 - w^5Therefore,Product_{k=0}^4 (1 + wω^k ) = Product_{k=0}^4 (-1 - wω^k ) / (-1)^5 = (-1 - w^5)/(-1) =1 + w^5Ah! So Product_{k=0}^4 (1 + wω^k ) =1 + w^5Therefore, going back, f(w) =1 + w^5. So the product over k=0 to4 of (1 + wω^k) equals 1 + w^5. That's a key identity!Wow, that simplifies things. Therefore, for any variable w,Product_{k=0}^4 (1 + wω^k ) =1 + w^5Therefore, returning to our original product when j=1:Product_{i=0}^4 (1 + y*ω^{i*1})^{398} = [Product_{i=0}^4 (1 + y*ω^i)]^{398} = (1 + y^5)^{398}Similarly, for j=2,3,4, let's check if the same identity applies.For j=2:Product_{i=0}^4 (1 + y*ω^{i*2})^{398}But ω^{i*2} are ω^0, ω^2, ω^4, ω^6=ω^1, ω^8=ω^3. So the exponents are 0,2,4,1,3. But since multiplication is commutative, the product is the same as Product_{k=0}^4 (1 + y*ω^k)^{398} = (1 + y^5)^{398}Similarly, for j=3:Product_{i=0}^4 (1 + y*ω^{i*3})^{398}Exponents would be 0,3,6=1,9=4,12=2. Again, the exponents are 0,3,1,4,2. Which are the same as 0,1,2,3,4 in different order. Therefore, the product is again (1 + y^5)^{398}Similarly, for j=4:Exponents i*4 mod5: 0,4,3,2,1. Again, all residues, so product is (1 + y^5)^{398}Therefore, for all j=1,2,3,4, the product becomes (1 + y^5)^{398}Therefore, the term for j=1,2,3,4 is each [ (1 + y^5)^{398} ] and we need the coefficient of y^31 in each of these.But (1 + y^5)^{398} expands as Sum_{m=0}^{398} C(398, m) y^{5m}. Therefore, the coefficient of y^31 in (1 + y^5)^{398} is C(398, m) where 5m =31. But 31 is not divisible by5, so the coefficient is 0. Therefore, for j=1,2,3,4, the coefficient of y^31 is 0.Therefore, putting it all together:Number of good subsets = (1/5) [ C(1990, 31) + 0 +0 +0 +0 ] = C(1990,31)/5Wait, that can't be right. That would suggest that exactly 1/5 of all 31-element subsets have sum ≡0 mod5. But is this the case?Wait, but when the generating function for the sum modulo5 is symmetric, perhaps the counts are equally distributed among the residues. But in this case, because the elements are equally distributed among the residues mod5, and the subset size is fixed, but the problem is whether the sum is equally likely to be any residue.But is that true? For example, if the total number of subsets is C(1990,31), then if the sums are uniformly distributed modulo5, then each residue would have C(1990,31)/5 subsets. But maybe the symmetry here causes that.But let's verify the logic.We have for each j=0,1,2,3,4, the term G(ω^j) where G(z) is the generating function for the subsets of size 31. For j=0, it's C(1990,31). For j=1,2,3,4, it's the coefficient of y^31 in (1 + y^5)^{398}, which is zero because 31 isn't a multiple of5. Hence, the total number of good subsets is (1/5)(C(1990,31) +0+0+0+0)= C(1990,31)/5.But is there a mistake here? Let's think again.The critical step is recognizing that Product_{i=0}^4 (1 + y*ω^{i j})^{398} = (1 + y^5)^{398} for j=1,2,3,4. But wait, is that correct?Wait, when j=1, the product is over i=0 to4 of (1 + y*ω^{i*1})^{398} = [Product_{i=0}^4 (1 + y*ω^i)]^{398} = [1 + y^5]^{398}, as per the identity earlier. Therefore, yes, this is correct.Similarly, for other j, since multiplying the exponents just permutes the roots of unity, the product remains the same. Therefore, the conclusion is that for each j=1,2,3,4, the generating function becomes (1 + y^5)^{398}, which has no y^31 term, since 31 is not divisible by5. Hence, their contributions are zero.Therefore, the total number of good subsets is indeed C(1990,31)/5.But let me check this with a smaller case to see if it makes sense. Let's take a smaller set where calculations are manageable.Suppose S = {1,2,3,4,5}, and we want the number of 2-element subsets with sum divisible by5.Total subsets of size2: C(5,2)=10. The subsets with sum divisible by5 are {1,4}, {2,3}, and {5} but wait, {5} is size1. Wait, in S={1,2,3,4,5}, the 2-element subsets with sum divisible by5 are {1,4} (sum5), {2,3} (sum5). So there are 2 subsets. According to the formula, it would be C(5,2)/5=10/5=2. Which matches. So in this case, the formula works.Another test case: S={1,2,3,4,5,6}, which has 6 elements. Let's compute the number of 2-element subsets with sum divisible by3.Total subsets of size2: C(6,2)=15. The formula would predict 15/3=5.Let's count them:Possible sums modulo3:Sum=3: {1,2}, {3, any?} Wait, sum=3: 1+2=3, 3+0=3 but no zero. So only {1,2}.Sum=6: {1,5}, {2,4}, {3,3} but duplicates not allowed. So {1,5}, {2,4}, {3,3} invalid. So two subsets.Sum=9: Not possible with numbers up to6.Wait, but modulo3, the residues we want are sums ≡0 mod3.The possible pairs:1+2=3≡01+5=6≡02+4=6≡03+6=9≡04+5=9≡0So total of 5 subsets: {1,2}, {1,5}, {2,4}, {3,6}, {4,5}. That's 5 subsets, which is 15/3=5. So the formula works here too.Another example where the formula works. So this suggests that the formula is correct when the elements are uniformly distributed modulo m, and the subset size is arbitrary. Wait, but in the problem statement, the elements are uniformly distributed modulo5, each residue appearing exactly 398 times. Then, the number of subsets of size k with sum ≡0 mod5 is equal to C(398*5, k)/5 = C(1990, k)/5.But why is that?It seems like when the elements are equally distributed among the residues modulo m, then the number of subsets of size k with sum ≡0 mod m is just C(N, k)/m, where N is the total number of elements. But is this a general result?I need to verify if this is a known theorem. It seems related to the idea of uniform distribution and group actions. If the group (here, the additive group mod5) acts regularly on the elements, then the number of subsets with a given sum modulo m is equal for each residue class. But in this case, since each residue class has the same number of elements, the counts might indeed be equally distributed.However, in general, this isn't always true. For example, if the elements are not equally distributed, or if the subset size affects the possible combinations. But in our case, since each residue class is equally represented and the subsets are chosen without any bias except for the size, the number of subsets with sum ≡c mod5 should be the same for each c, hence each gets 1/5 of the total.But wait, in the earlier small example with S={1,2,3,4,5} (5 elements, each residue 0,1,2,3,4 once), choosing subsets of size2. Then total subsets 10, and two of them have sum ≡0 mod5. But 10/5=2, which matches. Similarly, the next example with 6 elements (but residues not equally distributed) actually gave a correct result. Wait, in the second example, S={1,2,3,4,5,6}, residues mod3:Residues:0: 3,6 → 2 elements1:1,4 → 2 elements2:2,5 →2 elementsSo equally distributed. Then subsets of size2: total 15, number with sum≡0 mod3 is 15/3=5, which worked. So when residues are equally distributed, the formula holds.Therefore, perhaps the general result is: If the set S is partitioned into m residue classes modm, each containing exactly k elements, then the number of n-element subsets of S with sum ≡c modm is equal to C(m*k, n)/m, provided that the generating function symmetry holds.But why does this happen? Because the generating function for each residue class is (1 + y)^k for residue0, (1 + y*ω)^k for residue1, etc., and when multiplied together, the generating function becomes [(1 + y)(1 + y*ω)...(1 + y*ω^{m-1})]^k. Then, using the identity that Product_{j=0}^{m-1} (1 + y*ω^j) =1 + y^m, which we proved earlier for m=5, but in general, this holds because the roots of 1 + (-y)^m are the -ω^j, so the product becomes 1 + y^m.Therefore, Product_{j=0}^{m-1} (1 + y*ω^j) =1 + y^m. Hence, the generating function becomes (1 + y^m)^k. Therefore, when expanding, the coefficient of y^n is C(k, n/m) if n is divisible bym, else 0. But wait, no:Wait, if we have (1 + y^m)^k, the expansion is Sum_{i=0}^k C(k, i) y^{m*i}. Therefore, the coefficient of y^n is C(k, floor(n/m)) if n is a multiple ofm, else 0. Wait, no, more precisely, the exponent is m*i, so for (1 + y^m)^k, the coefficient of y^t is C(k, t/m) if t is a multiple ofm, else 0.But in our problem, the generating function after applying the roots of unity filter for j≠0 gives us (1 + y^m)^{k}, and when we look for the coefficient of y^n, it's zero unless n is a multiple ofm, which in our original problem, n=31 and m=5, 31 isn't divisible by5, hence zero. Therefore, all terms except j=0 vanish, giving the result as C(N, n)/m, where N=m*k.Therefore, in general, when a set is equally distributed modulo m with each residue class having k elements, then the number of n-element subsets with sum ≡c modm is C(m*k, n)/m, due to the uniform distribution across residues.Therefore, applying this general result to our problem where m=5, k=398, N=5*398=1990, and n=31, the number of good subsets is C(1990, 31)/5.Therefore, the final answer is C(1990, 31)/5. Since the problem asks for the number, which should be an integer, we can express it as boxed{dfrac{1}{5} dbinom{1990}{31}}.But usually, in combinatorics problems, the answer is presented as a binomial coefficient divided by 5, but since 5 is a prime, and 1990 choose31 is divisible by5, this will indeed be an integer. To confirm that C(1990,31) is divisible by5, we can use Lucas' theorem or compute the exponent of5 in the prime factorization of C(1990,31). But given that the problem is posed, it's safe to assume that it's an integer, hence the answer is as above.Final AnswerThe number of good subsets of ( S ) is boxed{dfrac{1}{5} dbinom{1990}{31}}.

✅:Given the set ( S = {1, 2, ldots, 1990} ), we need to find the number of 31-element subsets of ( S ) whose sum of elements is divisible by 5. First, we note that the set ( S ) is evenly distributed modulo 5, with each residue class (0, 1, 2, 3, 4) containing exactly 398 elements. To solve this problem, we use generating functions and the roots of unity filter. The generating function for the entire set ( S ) is constructed as a product of generating functions for each residue class. Each residue class contributes a factor of ( (1 + y omega^{ij})^{398} ) where ( omega ) is a primitive 5th root of unity. Using the roots of unity filter, we evaluate the generating function at the 5th roots of unity and sum the results. For ( j = 0 ), the generating function simplifies to ( (1 + y)^{1990} ), and the coefficient of ( y^{31} ) gives the total number of 31-element subsets, ( binom{1990}{31} ). For ( j = 1, 2, 3, 4 ), the generating functions simplify to ( (1 + y^5)^{398} ), which have no term ( y^{31} ) since 31 is not divisible by 5. Thus, the number of good subsets is given by averaging the contributions from each root of unity, resulting in:[frac{1}{5} left( binom{1990}{31} + 0 + 0 + 0 + 0 right) = frac{1}{5} binom{1990}{31}]Therefore, the number of good subsets of ( S ) is (boxed{dfrac{1}{5} dbinom{1990}{31}}).

🔑:1. Define Subsets: For ( k = 0, 1, 2, 3, 4 ), define [ F_{k} = left{ A : A subseteq S, , |A| = 31, , text{sum of elements of } A equiv k pmod{5} right} ] where ( |A| ) indicates the number of elements in the set ( A ).2. Transformation and Mapping: Observe that ( 31 equiv 1 pmod{5} ). If the set ( {x_{1}, x_{2}, cdots, x_{31}} ) lies in ( F_{0} ), then the transformations ( {x_{1} + k, x_{2} + k, cdots, x_{31} + k} ) will be in ( F_{k} ). Here, if any ( x_i + k > 1990 ), we replace ( x_i + k ) with ( x_i + k - 1990 ).3. Bijection Between Sets: Clearly, this transformation creates a one-to-one correspondence from ( F_{0} ) to ( F_{k} ) for each ( k ). Thus, we have: [ |F_{0}| = |F_{1}| = |F_{2}| = |F_{3}| = |F_{4}| ]4. Total Subsets: Since ( F_{0}, F_{1}, F_{2}, F_{3}, ) and ( F_{4} ) are disjoint and cover all possible 31-element subsets, we know: [ sum_{k=0}^{4} |F_{k}| = binom{1990}{31} ]5. Solve for ( |F_{0}| ): Given there are 5 sets ( F_{0}, F_{1}, F_{2}, F_{3}, F_{4} ) and they are all equally sized, we find: [ |F_{0}| = frac{1}{5} binom{1990}{31} ] Conclusion:[boxed{frac{1}{5} binom{1990}{31}}]

🧠:Okay, so I need to prove that 1/r = 1/R + 1/a, where r is the radius of a circle inscribed in a circular sector with radius R, and the chord AB of the sector is 2a. Hmm, let me try to visualize this first. A circular sector is like a slice of a circle, right? So, the sector has radius R, and there's a chord AB that's length 2a. The inscribed circle with radius r touches the two radii of the sector and the chord AB. Wait, how exactly is the circle inscribed here? Is it tangent to both sides of the sector and the chord? That makes sense. So the circle sits inside the sector, touching the two straight sides (the radii) and the curved side (the arc of the sector)? No, wait, the chord AB is a straight line, so maybe the sector is like a "V" shape with two radii and the chord AB connecting the endpoints? Then the inscribed circle is tangent to both radii and the chord AB. Okay, that seems right.Let me draw a diagram mentally. The sector has central angle θ, radius R, and chord AB with length 2a. The inscribed circle of radius r is tangent to OA, OB, and AB, where O is the center of the sector. So, the center of the inscribed circle must be somewhere inside the sector, equidistant from OA and OB (since it's tangent to both radii), so it lies along the bisector of angle θ. Also, it's at a distance r from each radius, and the distance from the center of the inscribed circle to chord AB is also r.Wait, maybe I should use coordinate geometry here. Let me set up a coordinate system. Let's place the vertex of the sector at the origin O(0,0), and let the sector be symmetric about the x-axis. So, the two radii OA and OB make angles of θ/2 above and below the x-axis. The chord AB is then the line connecting points A and B, which are located at (R cos(θ/2), R sin(θ/2)) and (R cos(θ/2), -R sin(θ/2)) respectively. The length of chord AB is 2a, which is given. So, the distance between A and B is 2a. Let me verify that. The coordinates of A and B are (R cos(θ/2), R sin(θ/2)) and (R cos(θ/2), -R sin(θ/2)), so the vertical distance between them is 2R sin(θ/2). Therefore, the length AB is 2R sin(θ/2) = 2a, so R sin(θ/2) = a. That gives sin(θ/2) = a/R. Okay, so θ is related to a and R by θ = 2 arcsin(a/R). I'll keep that in mind.Now, the inscribed circle with radius r is tangent to OA, OB, and AB. Let's find the coordinates of its center. Since it's tangent to both OA and OB, which are the two radii making angle θ, the center must lie along the angle bisector, which is the x-axis in this coordinate system. So the center of the inscribed circle is at (h, 0) for some h. The distance from this center to OA (or OB) must be equal to r. The distance from a point (h,0) to the line OA, which is at an angle θ/2 from the x-axis, can be calculated using the formula for the distance from a point to a line.The equation of OA can be written as y = tan(θ/2) x. Similarly, the equation of OB is y = -tan(θ/2) x. The distance from (h, 0) to OA is |tan(θ/2) h - 0| / sqrt(tan²(θ/2) + 1) ) = |h tan(θ/2)| / sqrt(tan²(θ/2) + 1). But sqrt(tan²(θ/2) + 1) is sec(θ/2), so the distance is |h tan(θ/2)| / sec(θ/2) = |h sin(θ/2)|. Since h is positive (the center is inside the sector), the distance is h sin(θ/2) = r. Therefore, h = r / sin(θ/2).But we also know that the center (h,0) must be at a distance r from the chord AB. The chord AB is the line connecting points A and B. Let me find the equation of chord AB. Points A and B are (R cos(θ/2), R sin(θ/2)) and (R cos(θ/2), -R sin(θ/2)), so chord AB is a vertical line at x = R cos(θ/2). Wait, no. Wait, if both points have the same x-coordinate, R cos(θ/2), but different y-coordinates, then chord AB is indeed a vertical line x = R cos(θ/2). But that can't be, unless θ is 180 degrees. Wait, no. Let me check again. If the sector is symmetric about the x-axis, then points A and B are mirror images over the x-axis. So, their coordinates would be (R cos(θ/2), R sin(θ/2)) and (R cos(θ/2), -R sin(θ/2)), so the chord AB is indeed a vertical line at x = R cos(θ/2), stretching from (R cos(θ/2), R sin(θ/2)) to (R cos(θ/2), -R sin(θ/2)). Therefore, the chord AB is vertical, and its equation is x = R cos(θ/2). Therefore, the distance from the center (h, 0) to the chord AB is |R cos(θ/2) - h|. Since the inscribed circle is tangent to AB, this distance must equal the radius r. Therefore:|R cos(θ/2) - h| = rBut h was already found to be h = r / sin(θ/2). So substituting:|R cos(θ/2) - r / sin(θ/2)| = rSince the center is inside the sector, h must be less than R cos(θ/2), because the chord AB is at x = R cos(θ/2), and the center is to the left of it. Therefore, R cos(θ/2) - h = rSubstituting h:R cos(θ/2) - r / sin(θ/2) = rRearranging:R cos(θ/2) = r + r / sin(θ/2) = r (1 + 1 / sin(θ/2)) = r ( (sin(θ/2) + 1 ) / sin(θ/2) )Wait, maybe I miscalculated. Let's do the algebra step by step.Starting with:R cos(θ/2) - h = rBut h = r / sin(θ/2), so:R cos(θ/2) - r / sin(θ/2) = rBring all terms to one side:R cos(θ/2) = r + r / sin(θ/2)Factor out r:R cos(θ/2) = r (1 + 1 / sin(θ/2))Then:r = R cos(θ/2) / (1 + 1 / sin(θ/2)) = R cos(θ/2) / ( (sin(θ/2) + 1 ) / sin(θ/2) ) = R cos(θ/2) * sin(θ/2) / (sin(θ/2) + 1 )Hmm. Not sure if this helps. Maybe we need to find another relation.Earlier, we had that AB = 2a, which gave us:AB = 2 R sin(θ/2) = 2a => sin(θ/2) = a / RSo θ/2 = arcsin(a/R), and cos(θ/2) = sqrt(1 - (a/R)^2 )But maybe instead of dealing with θ, we can express everything in terms of a and R.Given that sin(θ/2) = a/R, then cos(θ/2) = sqrt(1 - (a²/R²)) = sqrt( (R² - a²)/R² ) ) = sqrt(R² - a²)/RTherefore, cos(θ/2) = sqrt(R² - a²)/RSo plugging back into the equation for r:r = R * [ sqrt(R² - a²)/R ] * [ a/R ] / ( (a/R) + 1 )Wait, let's step back. Let me substitute cos(θ/2) and sin(θ/2) into the expression for r.We have:r = R cos(θ/2) * sin(θ/2) / ( sin(θ/2) + 1 )But sin(θ/2) = a/R, and cos(θ/2) = sqrt(1 - a²/R²). So:r = R * sqrt(1 - a²/R²) * (a/R) / ( (a/R) + 1 )Simplify numerator and denominator:Numerator: R * sqrt( (R² - a²)/R² ) * (a/R ) = R * ( sqrt(R² - a²)/R ) * (a/R ) = ( sqrt(R² - a²) * a ) / RDenominator: (a/R + 1 ) = (a + R)/RTherefore:r = [ ( sqrt(R² - a²) * a ) / R ] / ( (a + R)/R ) ) = [ ( sqrt(R² - a²) * a ) / R ] * [ R / (a + R) ) ] = ( sqrt(R² - a²) * a ) / (a + R )So r = (a sqrt(R² - a²)) / (a + R )Hmm, but we need to show that 1/r = 1/R + 1/a. Let's compute 1/r:1/r = (a + R) / (a sqrt(R² - a²)) )Hmm, not sure if that's equal to 1/R + 1/a. Let's compute 1/R + 1/a:1/R + 1/a = (a + R)/(a R)So unless (a + R)/(a R) equals (a + R)/(a sqrt(R² - a²)), which would require sqrt(R² - a²) = R, which is only true if a = 0, which isn't the case. So clearly, there's a mistake in my reasoning somewhere. Maybe my assumption about the distance from the center to chord AB is incorrect?Wait, chord AB is vertical at x = R cos(θ/2). The center of the inscribed circle is at (h, 0). The distance from (h,0) to the line x = R cos(θ/2) is | R cos(θ/2) - h |. Since the circle is tangent to AB, this distance should equal r. So R cos(θ/2) - h = r. Then h = R cos(θ/2) - r.But earlier, we found h = r / sin(θ/2). So:R cos(θ/2) - r = r / sin(θ/2)Therefore:R cos(θ/2) = r + r / sin(θ/2) = r ( 1 + 1 / sin(θ/2) )So:r = R cos(θ/2) / (1 + 1 / sin(θ/2) ) = R cos(θ/2) sin(θ/2) / ( sin(θ/2) + 1 )But since sin(θ/2) = a/R, then:r = R * cos(θ/2) * (a/R) / ( (a/R) + 1 ) = a cos(θ/2) / ( (a + R)/R ) ) = a R cos(θ/2) / (a + R )So r = (a R cos(θ/2)) / (a + R )But cos(θ/2) = sqrt(1 - (a/R)^2 ) = sqrt(R² - a²)/R, so:r = (a R * sqrt(R² - a²)/R ) / (a + R ) = (a sqrt(R² - a²) ) / (a + R )So 1/r = (a + R ) / (a sqrt(R² - a²) )Hmm, but according to the equation we need to prove, 1/r should be 1/R + 1/a = (a + R)/(a R). So unless sqrt(R² - a²) = R, which is not true unless a=0. Therefore, something is wrong here. Either my approach is incorrect or I made a mistake in the calculations.Wait, maybe my initial assumption about the position of the chord AB is wrong. The problem states that the chord AB has length 2a. In my coordinate system, chord AB is vertical with length 2R sin(θ/2) = 2a. Therefore, sin(θ/2) = a/R, which is correct. Then θ/2 = arcsin(a/R). So θ = 2 arcsin(a/R). But maybe there's another way to relate these variables.Alternatively, maybe instead of coordinate geometry, I should consider triangle geometry. The center of the sector is O, and the inscribed circle touches OA, OB, and AB. Let's denote the center of the inscribed circle as C. Then, the distance from C to OA and OB is r, and the distance from C to AB is also r. Since OA and OB are radii of the sector, and AB is the chord.The center C lies along the angle bisector of angle AOB, which is the x-axis in my coordinate system. So the distance from C to O is h, which I previously calculated as h = r / sin(θ/2). But also, the distance from C to AB is r, which is equal to the distance from C to the chord AB. The chord AB is at a distance of R cos(θ/2) from the origin O. Wait, because in the coordinate system, the chord AB is at x = R cos(θ/2), so the distance from O to AB is R cos(θ/2). But the center C is at (h, 0), so the distance from C to AB is R cos(θ/2) - h = r. Therefore, h = R cos(θ/2) - r. But h is also equal to r / sin(θ/2). Therefore:R cos(θ/2) - r = r / sin(θ/2)So rearranged:R cos(θ/2) = r ( 1 + 1 / sin(θ/2) )Now, from the chord length AB = 2a = 2 R sin(θ/2), so sin(θ/2) = a/R. Let's substitute sin(θ/2) = a/R and cos(θ/2) = sqrt(1 - (a/R)^2). Then:R * sqrt(1 - (a/R)^2 ) = r ( 1 + R/a )Multiply both sides by a:R a sqrt(1 - (a/R)^2 ) = r ( a + R )Divide both sides by (a + R):r = [ R a sqrt(1 - (a/R)^2 ) ] / ( a + R )Hmm, but how do we relate this to 1/r = 1/R + 1/a?Let me compute 1/r:1/r = ( a + R ) / [ R a sqrt(1 - (a/R)^2 ) ]But we need to show that 1/r = 1/R + 1/a = (a + R)/(R a )Comparing the two expressions, they would be equal only if sqrt(1 - (a/R)^2 ) = 1, which would mean a = 0, which is not possible. Therefore, there must be an error in my approach.Wait, maybe the chord AB is not the one I considered. Perhaps AB is not the chord subtended by the angle θ, but another chord. Wait, the problem says "the length of the chord AB is equal to 2a". In a circular sector, the chord AB typically refers to the chord connecting the two endpoints of the arc. So in that case, the chord length is 2 R sin(θ/2), which is 2a, so θ = 2 arcsin(a/R). But maybe the inscribed circle is tangent to the arc of the sector as well? Wait, no, the problem says it's inscribed in the sector, which usually means tangent to the two radii and the arc. But the problem statement says "inscribed in a circular sector", which might mean tangent to the two radii and the chord. Wait, the problem says "a circumference with radius r is inscribed in a circular sector of radius R. The length of the chord AB is equal to 2a." So the inscribed circle is tangent to the two radii and the chord AB, which is of length 2a. So maybe my previous analysis is correct, but somehow the relation 1/r = 1/R + 1/a must come out of this. But according to my calculations, 1/r = (a + R)/(a R sqrt(1 - (a/R)^2 )). Which is not equal to 1/R + 1/a unless sqrt(1 - (a/R)^2 ) =1, which is not true. Therefore, perhaps my entire approach is wrong.Alternative approach: Maybe using similar triangles or some geometric relations.Let me consider the center of the inscribed circle, point C. The distances from C to OA and OB are both r, and the distance from C to AB is also r. The center C lies along the angle bisector of angle AOB. Let’s denote the distance from O to C as d. Then, in the triangle OCP, where P is the foot of the perpendicular from C to OA, we have a right triangle with hypotenuse d, one leg r, and the angle at O equal to θ/2. Therefore, sin(θ/2) = r / d => d = r / sin(θ/2).Similarly, the distance from C to AB is r. The distance from O to AB is the length of the altitude from O to AB. In the sector, AB is a chord of the circle of radius R. The distance from O to AB is R cos(θ/2). Therefore, the distance from C to AB is equal to the distance from O to AB minus the distance from O to C along the bisector. Wait, no. If AB is a chord, then the distance from O to AB is R cos(θ/2). The center C is along the bisector (which is the same line as the altitude from O to AB in this symmetric case). Therefore, the distance from C to AB is equal to the distance from O to AB minus the distance from O to C. Wait, yes. Since O is at distance R cos(θ/2) from AB, and C is along that same line, then the distance from C to AB is R cos(θ/2) - d, where d is the distance from O to C. But that distance is equal to r. So:R cos(θ/2) - d = rBut earlier, d = r / sin(θ/2). Therefore:R cos(θ/2) - r / sin(θ/2) = rWhich is the same equation as before. So we get:R cos(θ/2) = r (1 + 1 / sin(θ/2) )Again, substituting sin(θ/2) = a/R, cos(θ/2) = sqrt(R² - a²)/R:R * sqrt(R² - a²)/R = r (1 + R/a )Simplify left side: sqrt(R² - a²) = r ( (a + R)/a )Therefore:r = sqrt(R² - a²) * a / (a + R )Thus, 1/r = (a + R) / (a sqrt(R² - a²) )But 1/R + 1/a = (a + R)/(a R )So unless sqrt(R² - a²) = R, which would require a =0, which is invalid, these two expressions are not equal. Therefore, there's a contradiction. Which suggests that either the problem statement is incorrect, or my approach is fundamentally wrong.Wait, let's check if I misinterpreted the problem. The problem says a circumference (which is a circle) with radius r is inscribed in a circular sector of radius R. The length of chord AB is equal to 2a. We need to prove 1/r = 1/R + 1/a.Alternatively, maybe the sector is not the one I'm thinking of. Perhaps the sector is a different configuration. Wait, in some contexts, a circular sector can refer to the area between two radii and an arc, but sometimes "inscribed circle" might refer to a circle tangent to both radii and the arc, not the chord. But the problem mentions the chord AB, so maybe the circle is tangent to the two radii and the chord AB, which is different from the arc.Wait, if the circle is tangent to the two radii and the arc, then the chord AB would be different. But the problem states the chord AB has length 2a. Hmm. Maybe I need to clarify.Alternatively, perhaps the chord AB is the one that the inscribed circle is tangent to, so AB is the chord which is the base of the sector, and the inscribed circle touches the two radii and the chord AB. Then, with AB = 2a, sector radius R, inscribed circle radius r. Then the formula 1/r = 1/R + 1/a should hold. But according to my calculations, that is not the case, unless there is a specific relation between R and a.Wait, maybe I need to approach this using trigonometry identities. Let's square both sides of the equation we need to prove. Wait, perhaps not. Let's see:Given that 1/r = 1/R + 1/a, then cross-multiplying:a R = r (a + R )So r = (a R)/(a + R )But according to my earlier result, r = (a sqrt(R² - a²) ) / (a + R )Therefore, if sqrt(R² - a² ) = R, then r = (a R)/(a + R ), which would imply 1/r = 1/R + 1/a. But sqrt(R² - a² ) = R implies a=0, which is impossible. Therefore, the only way my result matches the desired formula is if sqrt(R² - a² ) = R, which is not true. Therefore, my result contradicts the desired formula, which suggests that my approach is wrong.Alternatively, maybe the formula to be proven is approximate under certain conditions, but the problem states to prove it as an exact formula. Therefore, I must have made a mistake.Wait, let's try to consider specific values. Suppose R is much larger than a, so that a << R. Then sqrt(R² - a²) ≈ R - a²/(2R). Therefore, r ≈ [ a ( R - a²/(2R) ) ] / (a + R ) ≈ [ a R - a³/(2R) ] / (R + a ) ≈ [ a R / (R + a ) ] - [ a³ / (2 R (R + a )) ]If a << R, then R + a ≈ R, so first term ≈ a R / R = a, second term ≈ a³ / (2 R² ). So r ≈ a - a³/(2 R² ). But according to the formula 1/r = 1/R + 1/a, then r = (a R)/(a + R ). If a << R, then r ≈ a R / R = a. Which matches the first term. So for small a, the two expressions approximate each other. But they are not equal unless a=0. Hence, the formula seems incorrect unless there's a different configuration.Alternatively, maybe the chord AB is not the chord subtending the angle θ but another chord. Wait, in the problem statement, it's just mentioned that AB is a chord with length 2a. Maybe AB is not the chord that forms the arc of the sector, but another chord. For example, the inscribed circle touches the two radii and the arc of the sector, and there's a chord AB of length 2a related to this. But the problem states "a circumference with radius r is inscribed in a circular sector of radius R. The length of the chord AB is equal to 2a." So perhaps AB is the chord where the inscribed circle is tangent to the arc? Wait, but a circle inscribed in a sector touching the two radii and the arc would have its center along the bisector, and the radius r related to R and the angle θ. But the chord AB is mentioned, so maybe AB is the chord that is tangent to the inscribed circle. Hmm.Wait, if the inscribed circle is tangent to the two radii and the arc of the sector, then the points of tangency on the arc would form a chord. Wait, but the inscribed circle touches the arc at one point, not forming a chord. Wait, no. If the circle is tangent to the arc of the sector, it's tangent at a single point along the arc. Then AB might be another chord. This is getting confusing. Maybe I need to look for another method.Alternative approach: Use inversion or other geometric transformations, but that might be too complicated.Wait, let's try to use the formula for the radius of an inscribed circle in a sector. I recall that for a circle inscribed in a sector with radius R and angle θ, the radius r is given by r = R sin(θ/2) / (1 + sin(θ/2) ). Let me verify this formula.If the center of the inscribed circle is at distance d from the vertex O, then d = r / sin(θ/2). The distance from the center to the arc is R - d = R - r / sin(θ/2). But since the circle is tangent to the arc, this distance must equal r. Therefore:R - r / sin(θ/2) = r => R = r ( 1 + 1 / sin(θ/2) ) => r = R sin(θ/2) / (1 + sin(θ/2) )Yes, this is a known formula. So in this case, the radius of the inscribed circle tangent to the two radii and the arc is r = R sin(θ/2)/(1 + sin(θ/2) ). However, in our problem, the circle is tangent to the two radii and the chord AB, which has length 2a. In this case, the chord AB is related to the angle θ by AB = 2 R sin(θ/2) = 2a => a = R sin(θ/2). Therefore, sin(θ/2) = a/R. Substituting into the formula for r:r = R * (a/R) / (1 + a/R ) = a / (1 + a/R ) = (a R ) / ( R + a )Thus, r = (a R ) / (a + R )Therefore, 1/r = (a + R ) / (a R ) = 1/a + 1/RWhich is the desired result. So this suggests that my previous approach was considering the circle tangent to the arc instead of the chord, which led to the confusion. The correct approach is to realize that when the circle is inscribed in the sector and tangent to the two radii and the chord AB (rather than the arc), then the standard formula gives r = (a R)/(a + R ), hence 1/r = 1/a + 1/R.Therefore, the mistake in my earlier reasoning was assuming the circle was tangent to the arc, but in the problem, it's tangent to the chord AB. When the circle is tangent to the chord AB, the distance from the center of the sector O to the center of the inscribed circle C is d = r / sin(θ/2). The distance from C to AB is r, which must equal the distance from O to AB minus d. The distance from O to AB is R cos(θ/2). Therefore:R cos(θ/2) - d = r => R cos(θ/2) - r / sin(θ/2) = rBut since AB = 2a = 2 R sin(θ/2), we have a = R sin(θ/2). Therefore, sin(θ/2) = a/R, and cos(θ/2) = sqrt(1 - (a/R)^2 ). Substituting into the equation:R sqrt(1 - (a/R)^2 ) - r / (a/R) = r => R sqrt(1 - (a/R)^2 ) - (R r)/a = rMultiply both sides by a:R a sqrt(1 - (a/R)^2 ) - R r = a rBring all terms to one side:R a sqrt(1 - (a/R)^2 ) = r (a + R )But sqrt(1 - (a/R)^2 ) = sqrt( (R² - a²)/R² ) = sqrt(R² - a²)/RTherefore:R a * sqrt(R² - a²)/R = r (a + R )Simplify:a sqrt(R² - a² ) = r (a + R )But if we solve for r:r = (a sqrt(R² - a² )) / (a + R )But according to the previous correct formula, when the circle is tangent to the chord, we should have r = (a R)/(a + R ). This inconsistency suggests that there's a different configuration. Wait, maybe in the standard formula, the circle is tangent to the two radii and the arc, not the chord, and when it's tangent to the chord, the formula changes.But wait, the problem says the circle is inscribed in the sector, which typically means tangent to all three boundaries: two radii and the arc. However, in this problem, it's mentioned that the chord AB has length 2a. If the circle is tangent to the arc, then AB would be the chord of the arc, which has length 2 R sin(θ/2). Then, if we set that equal to 2a, we have a = R sin(θ/2). Then using the standard formula for the inscribed circle tangent to two radii and the arc, r = R sin(θ/2)/(1 + sin(θ/2)) = a/(1 + a/R ) = (a R)/(a + R ), which gives 1/r = (a + R )/(a R ) = 1/a + 1/R. Hence, the desired result. Therefore, the chord AB in the problem is the chord of the arc, and the inscribed circle is tangent to the arc (not the chord AB). Therefore, my initial misunderstanding was whether the circle is tangent to the arc or the chord. The problem states the circle is inscribed in the sector, which usually implies tangent to the two radii and the arc. However, the chord AB with length 2a is the chord of the arc, so the inscribed circle's radius relates to R and a via the formula 1/r = 1/R + 1/a.Therefore, the correct approach is:1. Given a circular sector with radius R, central angle θ, and chord AB (of the arc) with length 2a.2. Therefore, 2a = 2 R sin(θ/2) => a = R sin(θ/2).3. The radius r of the circle inscribed in the sector (tangent to the two radii and the arc) is given by r = R sin(θ/2)/(1 + sin(θ/2)) = a/(1 + a/R) = (a R)/(a + R).4. Therefore, 1/r = (a + R)/(a R) = 1/R + 1/a.Hence, proved.The confusion arose because initially, I considered the circle tangent to the chord AB, but the problem's wording implies that the circle is inscribed in the sector, which standardly means tangent to the two radii and the arc. The chord AB is the chord subtended by the arc, and its length relates to θ, allowing the elimination of θ and deriving the relation between r, R, and a.

✅:To prove that (frac{1}{r} = frac{1}{R} + frac{1}{a}) for a circle of radius (r) inscribed in a circular sector of radius (R) with chord (AB = 2a), follow these steps:1. Relate Chord Length to Central Angle: The chord (AB) subtends the central angle (theta) of the sector. The length of (AB) is given by: [ 2a = 2R sinleft(frac{theta}{2}right) implies sinleft(frac{theta}{2}right) = frac{a}{R}. ]2. Express the Inscribed Circle Radius: The radius (r) of the circle inscribed in the sector (tangent to both radii and the arc) is: [ r = frac{R sinleft(frac{theta}{2}right)}{1 + sinleft(frac{theta}{2}right)}. ]3. Substitute (sinleft(frac{theta}{2}right)): Using (sinleft(frac{theta}{2}right) = frac{a}{R}) from Step 1: [ r = frac{R cdot frac{a}{R}}{1 + frac{a}{R}} = frac{a}{1 + frac{a}{R}}. ]4. Simplify to Find (1/r): [ r = frac{aR}{R + a} implies frac{1}{r} = frac{R + a}{aR} = frac{1}{R} + frac{1}{a}. ]Conclusion: The relationship (frac{1}{r} = frac{1}{R} + frac{1}{a}) is derived by relating the chord length to the central angle and using the formula for the radius of an inscribed circle in a sector. [boxed{frac{1}{r} = frac{1}{R} + frac{1}{a}}]

🔑:We are given a circle of radius ( r ) inscribed in a sector of a circle with radius ( R ). The length of the chord ( AB ) is ( 2a ).1. Identify and denote the relevant points and segments: - Let ( O ) be the center of the larger circle. - Let ( O_1 ) be the center of the inscribed circle. - ( D ) is the point of tangency between ( AO ) and the inscribed circle.2. Establish the right angles and equal segments: - Since ( D ) is the point of tangency, ( OD ) is perpendicular to the tangent line at ( D ). Therefore, (angle OD O_1 = 90^circ). - The chord ( AB ) intersects at ( C ) such that ( AC = frac{AB}{2} = a). - It is also true that (angle OCA = 90^circ) where ( C ) is the midpoint of the chord ( AB ).3. Identify similar triangles: - Triangles ( triangle ODO_1 ) and ( triangle OCA ) are similar by AA criterion (since they both have a right angle and share (angle OD O_1 )).4. Set up the ratio of corresponding sides: - From the similarity of ( triangle ODO_1 ) and ( triangle OCA ), we have: frac{text{OO}_1}{text{OA}} = frac{text{O}_1text{D}}{text{AC}}. Substitute the known values: - (text{OO}_1 = R - r) - (text{OA} = R) - (text{O}_1D = r) - (text{AC} = a) Thus, the ratio becomes: frac{R - r}{R} = frac{r}{a}. 5. Solve for the relationship between ( r, R, ) and ( a ): - Cross-multiplying the proportion, we get: (R - r)a = rR. - Expanding and rearranging terms, we have: Ra - ra = rR. - Rearrange further to separate the fractions involving ( r ): Ra = rR + ra. - Divide through by ( rRa ): frac{R}{Rr} = frac{r}{Rr} + frac{a}{Rr}. - Simplify each term: frac{1}{r} = frac{1}{R} + frac{1}{a}. Conclusion:We have shown that:boxed{frac{1}{r} = frac{1}{R} + frac{1}{a}}.

🧠:Okay, let me try to figure out this geometry problem. So, we have a rectangular parallelepiped, which is like a 3D shape with all faces being rectangles. The points are labeled A B C D A1 B1 C1 D1. I need to visualize this. Maybe it's a box where ABCD is the base, and A1 B1 C1 D1 is the top face. So edges like AA1, BB1, etc., connect the base to the top.The problem mentions points K, P, H, and E. Let me note their positions:- K is the midpoint of BB1. So if BB1 is an edge from B to B1, then K is halfway up that edge.- P is the midpoint of A1 D1. Since A1 D1 is an edge on the top face, connecting A1 to D1, so P is halfway between those two points.- H is the midpoint of CC1. CC1 is the edge from C to C1, so H is halfway up that.- E is on edge B1 C1, and the ratio B1 E : E C1 is 1:3. So starting from B1, moving towards C1, E divides the edge into parts where B1E is 1 part and EC1 is 3 parts. That means E is closer to B1 than to C1. Since the entire length of B1C1 is divided into 4 parts, E is 1/4 of the way from B1 to C1.The question is whether the line KP intersects the lines AE and A1H. So, we need to check if line KP intersects both AE and A1H. The wording says "intersects the lines AE and A1H," so does that mean it intersects both? The question is asking if it's true that KP intersects both AE and A1H. So we need to verify both intersections.First, maybe I should assign coordinates to all the points to make this easier. Let's set up a coordinate system. Let me assume that the parallelepiped is a rectangular prism (so all angles are right angles). If it's a general parallelepiped, it might be more complicated, but since it's rectangular, edges are perpendicular.Let me assign coordinates:Let’s place point A at the origin (0, 0, 0). Then:- B would be along the x-axis: (a, 0, 0) for some a > 0.- D along the y-axis: (0, b, 0) for some b > 0.- A1 along the z-axis: (0, 0, c) for some c > 0.Then:- Point B1 is (a, 0, c)- Point C is (a, b, 0)- Point D1 is (0, b, c)- Point C1 is (a, b, c)Now, let's find coordinates for points K, P, H, E.- K is the midpoint of BB1. B is (a, 0, 0), B1 is (a, 0, c). Midpoint K: ((a + a)/2, (0 + 0)/2, (0 + c)/2) = (a, 0, c/2)- P is the midpoint of A1 D1. A1 is (0, 0, c), D1 is (0, b, c). Midpoint P: ((0 + 0)/2, (0 + b)/2, (c + c)/2) = (0, b/2, c)- H is the midpoint of CC1. C is (a, b, 0), C1 is (a, b, c). Midpoint H: ((a + a)/2, (b + b)/2, (0 + c)/2) = (a, b, c/2)- E is on B1C1 with ratio B1E:EC1 = 1:3. B1 is (a, 0, c), C1 is (a, b, c). The vector from B1 to C1 is (0, b, 0). So moving 1/4 of the way from B1 to C1: E = B1 + (1/4)(C1 - B1) = (a, 0, c) + (0, b/4, 0) = (a, b/4, c)So coordinates:- K: (a, 0, c/2)- P: (0, b/2, c)- H: (a, b, c/2)- E: (a, b/4, c)- A: (0, 0, 0)- A1: (0, 0, c)Now, need to check if line KP intersects lines AE and A1H.First, let's parametrize line KP. Points K and P are (a, 0, c/2) and (0, b/2, c). Let's find parametric equations for KP.Parametric equations for a line can be written as starting at point K and moving towards P. Let parameter t go from 0 to 1.So direction vector from K to P is P - K = (-a, b/2, c/2)So parametric equations:x = a - a*ty = 0 + (b/2)*tz = c/2 + (c/2)*tfor t ∈ [0,1]Similarly, parametrize line AE. Points A (0,0,0) and E (a, b/4, c). Direction vector E - A = (a, b/4, c). So parametric equations:x = 0 + a*sy = 0 + (b/4)*sz = 0 + c*sfor s ∈ [0,1]Similarly, parametrize line A1H. Points A1 (0,0,c) and H (a, b, c/2). Direction vector H - A1 = (a, b, -c/2). Parametric equations:x = 0 + a*ry = 0 + b*rz = c + (-c/2)*r = c - (c/2)*rfor r ∈ [0,1]Now, we need to check if line KP intersects line AE. So we need to find parameters t and s such that:a - a*t = a*s(b/2)*t = (b/4)*sc/2 + (c/2)*t = c*sSimilarly, check if there exists t and s in [0,1] satisfying these equations.Let’s solve the system:1) a - a*t = a*s ⇒ 1 - t = s2) (b/2)*t = (b/4)*s ⇒ (t) = (1/2)s3) c/2 + (c/2)*t = c*s ⇒ (1/2) + (1/2)t = sFrom equation 1: s = 1 - tFrom equation 2: t = (1/2)s ⇒ t = (1/2)(1 - t) ⇒ t = 1/2 - t/2 ⇒ t + t/2 = 1/2 ⇒ (3/2)t = 1/2 ⇒ t = (1/2)/(3/2) = 1/3Then s = 1 - t = 1 - 1/3 = 2/3Check equation 3:(1/2) + (1/2)(1/3) = 1/2 + 1/6 = 2/3 = s = 2/3. Yes, it holds.So t = 1/3, s = 2/3, both within [0,1]. Therefore, line KP intersects line AE at the point when t=1/3, which is:x = a - a*(1/3) = (2a/3)y = (b/2)*(1/3) = b/6z = c/2 + (c/2)*(1/3) = c/2 + c/6 = 2c/3So intersection point is (2a/3, b/6, 2c/3)Now, check if line KP intersects line A1H.Parametrize KP as before:x = a - a*ty = (b/2)*tz = c/2 + (c/2)*tParametrize A1H:x = a*ry = b*rz = c - (c/2)*rSet them equal:a - a*t = a*r ⇒ 1 - t = r(b/2)*t = b*r ⇒ (t)/2 = rc/2 + (c/2)*t = c - (c/2)*rFrom first equation: r = 1 - tFrom second equation: r = t/2Therefore:1 - t = t/2 ⇒ 1 = (3t)/2 ⇒ t = 2/3Then r = t/2 = (2/3)/2 = 1/3Check third equation:Left side (z from KP): c/2 + (c/2)*(2/3) = c/2 + c/3 = (3c + 2c)/6 = 5c/6Right side (z from A1H): c - (c/2)*(1/3) = c - c/6 = 5c/6So they match. So t = 2/3, r = 1/3, both within [0,1]. Therefore, line KP also intersects line A1H at:x = a - a*(2/3) = a/3y = (b/2)*(2/3) = b/3z = c/2 + (c/2)*(2/3) = c/2 + c/3 = 5c/6So intersection point is (a/3, b/3, 5c/6)Hence, line KP intersects both lines AE and A1H. Therefore, the statement is true.But wait, let me verify once more to make sure I didn't make any mistakes.First intersection with AE:Solved equations and found t=1/3, s=2/3. Plugging back into coordinates, they match. So that's correct.Second intersection with A1H:Found t=2/3, r=1/3. Coordinates also match. So that's correct too.Therefore, yes, the line KP intersects both lines AE and A1H. So the answer is yes, it is true.Final Answerboxed{True}

✅:Given a rectangular parallelepiped (ABCDA_1B_1C_1D_1), we need to determine if the line (KP) intersects the lines (AE) and (A_1H). Coordinates Assignment:- (A = (0, 0, 0))- (B = (a, 0, 0))- (D = (0, b, 0))- (A_1 = (0, 0, c))- (B_1 = (a, 0, c))- (C = (a, b, 0))- (D_1 = (0, b, c))- (C_1 = (a, b, c))Midpoints and Points:- (K) (midpoint of (BB_1)): ((a, 0, c/2))- (P) (midpoint of (A_1D_1)): ((0, b/2, c))- (H) (midpoint of (CC_1)): ((a, b, c/2))- (E) (divides (B_1C_1) in ratio 1:3): ((a, b/4, c))Parametric Equations:- Line (KP): [ begin{cases} x = a - at y = frac{b}{2}t z = frac{c}{2} + frac{c}{2}t end{cases} ]- Line (AE): [ begin{cases} x = as y = frac{b}{4}s z = cs end{cases} ]- Line (A_1H): [ begin{cases} x = ar y = br z = c - frac{c}{2}r end{cases} ]Intersection Check with (AE):1. Solve (a - at = as), (frac{b}{2}t = frac{b}{4}s), (frac{c}{2} + frac{c}{2}t = cs)2. Solutions: (t = frac{1}{3}), (s = frac{2}{3}) (valid within [0, 1])Intersection Check with (A_1H):1. Solve (a - at = ar), (frac{b}{2}t = br), (frac{c}{2} + frac{c}{2}t = c - frac{c}{2}r)2. Solutions: (t = frac{2}{3}), (r = frac{1}{3}) (valid within [0, 1])Both intersections are valid, confirming that line (KP) intersects both lines (AE) and (A_1H).[boxed{True}]

🔑:1. Identify the geometric elements and positions: Given a rectangular parallelepiped (A B C D A_1 B_1 C_1 D_1), we note the following positions and properties: - Point (K) is the midpoint of edge (B B_1). - Point (P) is the midpoint of edge (A_1 D_1). - Point (H) divides edge (C C_1) into two equal parts. - Point (E) lies on edge (B_1 C_1) and divides it such that ( B_1 E : E C_1 = 1 : 3 ).2. Analyze the projection plane and directions: Assume that the projection plane is parallel to the face (B B_1 C_1 C) of the parallelepiped (A B C D A_1 B_1 C_1 D_1). In this case, the line (KP) will appear a point in the projection along the direction perpendicular to this face.3. Visualize points and lines in the projection: On this projected plane: - Line (A_1 H) aligns with the edge (A_1 D_1) because they share this parallel direction. - Segment (A K) intersects segment (T B) at its midpoint (M).4. Establish the relationship between points (A), (K), and (E): Since triangles (M B K) and (E B_1 K) are congruent by the given ratios and midpoints, point (M) is determined to lie on the line that includes points (A), (K), and (E).5. Determine intersections: Given that points lie on these described segments and the congruence derived from the midpoints, we infer: - The image of line (A_1 H) goes through point (K). - Line (A E) also intersects at (K) or lines up with it. Thus, on the projected plane, the images of the lines (A_1 H) and (A E) intersect the point corresponding to (K=P), implying these lines genuinely intersect with line (K P).# Conclusion:Therefore, line (K P) does intersect both lines (A E) and (A_1 H).Thus, the claim is true.[boxed{text{True}}]

🧠:Okay, so I need to find the maximum value of M, where M is the minimum of four expressions: 1/a, 2/b, 4/c, and the cube root of abc. All a, b, c are positive numbers. Hmm, let me think about how to approach this.First, since M is the minimum of those four terms, we want each of those terms to be as large as possible. But since we're taking the minimum, the maximum M will be the highest value such that all four expressions are at least M. So, effectively, we need to set each of the four expressions equal to M and find the a, b, c that satisfy all those equalities. That way, M is the minimum, and we can maximize it.So, let me write that down. To have M as the minimum of those four, we need:1/a ≥ M,2/b ≥ M,4/c ≥ M,and ∛(abc) ≥ M.But since we're trying to maximize M, the maximum possible M will be when all these inequalities become equalities. Because if even one of them is greater than M, we might be able to increase M a bit more. So, assuming equality:1/a = M,2/b = M,4/c = M,and ∛(abc) = M.Now, solving these equations for a, b, c in terms of M.From the first equation: a = 1/M.Second equation: b = 2/M.Third equation: c = 4/M.Fourth equation: ∛(abc) = M. Let's plug a, b, c from above into this.Compute abc: (1/M) * (2/M) * (4/M) = (8)/(M^3).Then, ∛(abc) = ∛(8/M^3) = 2/M.So, set that equal to M: 2/M = M.Multiply both sides by M: 2 = M^2.So, M^2 = 2, which means M = √2. Since M is positive.Wait, so M is √2. Let me check if this works.So, if M = √2, then:a = 1/√2,b = 2/√2 = √2,c = 4/√2 = 2√2.Then, abc = (1/√2)(√2)(2√2) = (1/√2 * √2) = 1, then 1 * 2√2 = 2√2.Wait, wait, let me compute that again:a = 1/√2,b = 2/√2 = √2,c = 4/√2 = 2√2.So abc = (1/√2) * √2 * 2√2 = (1/√2 * √2) = 1, then 1 * 2√2 = 2√2. Then ∛(abc) = ∛(2√2).But 2√2 is 2 * 2^(1/2) = 2^(3/2). So the cube root of 2^(3/2) is 2^( (3/2) * (1/3) ) = 2^(1/2) = √2. So ∛(abc) = √2, which matches M. So all four expressions equal √2. Therefore, M = √2 is achievable.But wait, is this the maximum possible? Let me see.Is there a way to make M larger than √2? Let's suppose M is larger, say M = √2 + ε for some ε > 0. Then, according to the equations:From 1/a = M, a = 1/M, which would be smaller than 1/√2.Similarly, b = 2/M < √2,c = 4/M < 2√2.Then, abc = (1/M)(2/M)(4/M) = 8/M^3.So ∛(abc) = 2/M. If M is larger than √2, then 2/M would be smaller than 2/√2 = √2. Therefore, ∛(abc) would be less than M, which contradicts the requirement that ∛(abc) ≥ M. Hence, we can't have M larger than √2. Therefore, √2 is indeed the maximum.Wait, but let me verify again. Let me try with specific numbers. Suppose M is √2. Then, as we saw, all four expressions equal √2. If we try to increase M a bit, say M = 1.5 (which is greater than √2 ≈ 1.4142), then:a = 1/1.5 ≈ 0.6667,b = 2/1.5 ≈ 1.3333,c = 4/1.5 ≈ 2.6667,Then abc ≈ 0.6667 * 1.3333 * 2.6667 ≈ 0.6667 * 3.5555 ≈ 2.37.Then ∛(abc) ≈ ∛(2.37) ≈ 1.33, which is less than M=1.5. Therefore, the cube root term becomes smaller than M, which violates the condition that all four terms must be at least M. Thus, M cannot be 1.5.Alternatively, if we tried to set M somewhere between √2 and 1.5, say M = 1.5, as above, the cube root term would be the limiting factor. So the maximum M is indeed √2.Therefore, the maximum value of M is √2.But let me check again my initial approach. I set all four expressions equal to M and solved for a, b, c. Then, substituted back into the cube root expression. Got M^2 = 2, so M=√2. That seems consistent.Alternatively, maybe there's another approach where not all four expressions are equal, but maybe three of them are equal and the fourth is higher? But since we are taking the minimum, if one of them is higher, then the minimum would still be the lowest one, which would be among those set to M. So to maximize M, all four must be equal. Otherwise, if one is higher, you could potentially increase M until the others catch up. Hmm, perhaps.Wait, suppose three of the expressions are equal to M, and the fourth is greater than M. Then, the minimum would still be M. But in that case, maybe we can increase M slightly, which would lower the first three expressions (since a, b, c would be smaller), but the fourth expression would decrease as well. Wait, but if we set three expressions equal to M and let the fourth be larger, then increasing M would require decreasing a, b, or c, but this might affect the fourth term. Let me think.Suppose we set 1/a = M, 2/b = M, 4/c = M, so a=1/M, b=2/M, c=4/M. Then, the fourth term is ∛(abc) = ∛(8/M^3) = 2/M. So if we set 2/M = M, we get M=√2. But if instead, we set 1/a = M1, 2/b = M2, 4/c = M3, and ∛(abc) = M4, and set all Mi ≥ M, then the maximum M would be when all Mi are equal. Otherwise, if one is higher, you could increase M. So the conclusion is that equality is required for all four terms.Therefore, the maximum M is √2. So the final answer is √2. So boxed, that's boxed{sqrt{2}}.But wait, let me confirm once again. Let's take a=1/√2, b=√2, c=2√2. Then compute each term:1/a = √2,2/b = 2/√2 = √2,4/c = 4/(2√2) = 2/√2 = √2,and ∛(abc) = ∛( (1/√2)*(√2)*(2√2) ) = ∛( (1/√2 * √2) * 2√2 ) = ∛(1 * 2√2) = ∛(2√2) = (2√2)^(1/3) = 2^(1/3) * (2^(1/2))^(1/3) = 2^(1/3 + 1/6) = 2^(1/2) = √2. Yep, all four terms equal √2. So that's correct. Therefore, the maximum value of M is √2.Final AnswerThe maximum value of ( M ) is boxed{sqrt{2}}.

✅:To determine the maximum value of ( M = min left{frac{1}{a}, frac{2}{b}, frac{4}{c}, sqrt[3]{abc}right} ) where (a), (b), and (c) are positive numbers, we need to ensure that each of the four expressions is at least (M). To maximize (M), we set each expression equal to (M):1. (frac{1}{a} = M implies a = frac{1}{M})2. (frac{2}{b} = M implies b = frac{2}{M})3. (frac{4}{c} = M implies c = frac{4}{M})4. (sqrt[3]{abc} = M)Substituting (a), (b), and (c) into the fourth expression, we get:[sqrt[3]{left(frac{1}{M}right) left(frac{2}{M}right) left(frac{4}{M}right)} = sqrt[3]{frac{8}{M^3}} = frac{2}{M}]Setting this equal to (M) gives:[frac{2}{M} = M implies M^2 = 2 implies M = sqrt{2}]Verification shows that when (M = sqrt{2}), the values (a = frac{1}{sqrt{2}}), (b = sqrt{2}), and (c = 2sqrt{2}) make all four expressions equal to (sqrt{2}). Thus, the maximum value of (M) is indeed (sqrt{2}).[boxed{sqrt{2}}]

🔑:Given the problem, we need to find the maximum value ofM = min left{frac{1}{a}, frac{2}{b}, frac{4}{c}, sqrt[3]{a b c}right}where ( a, b, c ) are all positive numbers.1. From the definition of ( M ), we have: [ M leq frac{1}{a}, quad M leq frac{2}{b}, quad M leq frac{4}{c}, quad M leq sqrt[3]{a b c} ]2. Rewriting these inequalities in terms of ( M ), we get: [ a leq frac{1}{M}, quad b leq frac{2}{M}, quad c leq frac{4}{M} ]3. Using ( M leq sqrt[3]{a b c} ), we substitute the upper bounds of ( a, b, ) and ( c ): [ a leq frac{1}{M}, quad b leq frac{2}{M}, quad c leq frac{4}{M} ] Hence, [ abc leq left(frac{1}{M}right) left(frac{2}{M}right) left(frac{4}{M}right) = frac{8}{M^3} ]4. Since ( M leq sqrt[3]{a b c} ), substituting the inequality obtained for ( abc ): [ M leq sqrt[3]{frac{8}{M^3}} ]5. Simplifying the expression: [ M leq frac{2}{M} ]6. To solve for ( M ), multiply both sides of the inequality by ( M ): [ M^2 leq 2 ]7. Taking the positive square root of both sides, as ( M ) is positive: [ M leq sqrt{2} ]Therefore, the maximum possible value for ( M ) occurs when ( M = sqrt{2} ).8. Verify that ( M = sqrt{2} ) can be achieved when specific values for ( a, b, ) and ( c ) are chosen. For instance, if ( c = 2 ), ( b = 2sqrt{2} ) and ( a = 2sqrt{2} ), then: [ frac{1}{a} = frac{1}{2sqrt{2}}, quad frac{2}{b} = frac{2}{2sqrt{2}}, quad frac{4}{c} = frac{4}{2}, quad sqrt[3]{abc} = sqrt[3]{(2sqrt{2})^2 cdot 2} = sqrt[3]{8} = 2 ]This gives the minimum value in the set as ( sqrt{2} ), satisfying our conditions.Conclusion. [boxed{sqrt{2}}]