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problem stringlengths 20 917	answer stringlengths 1 153	grade float64 0 100	solution_0 stringlengths 858 220k	solution_1 stringlengths 1.03k 210k	solution_2 stringlengths 922 225k	solution_3 stringlengths 948 188k	solution_4 stringlengths 1.01k 265k	metadata dict
A train left a station $ A $ for $ B $ at 13:00. At 19:00 the train was brought to a halt by a snow drift. Two hours later the railway line was cleared, and to make up for the lost time, the train proceeded at a speed exceeding the original speed by $ 20\% $ and arrived at $ B $ only one hour later. The next da...	600 \, \text{km}	100	analysisWe need to solve for distance AB, given two scenarios. Let D be distance AB. Let v be normal speed (constant). The train schedule: departure at 13:00, arrival at B at some scheduled arrival time? Let's deduce. First day: at 19:00 train stopped due to snow. So it had traveled for 6 hours (13 to 19). In that ti...	analysisWe need to solve for distance AB and original speed V. Setup: train scheduled? It left at 13:00 and presumably had a scheduled arrival time? The problem says: "arrived at B only one hour later" meaning after the delay, arrival time was 1 hour later than scheduled? Let's parse: "Two hours later the railway line ...	analysisWe need to solve for distance between stations AB, given two scenarios with same original speed V, same timetable? Let's parse. Scenario 1: Train leaves at 13:00 from A to B. At 19:00, train halted due to snow drift. So from 13:00 to 19:00 is 6 hours of travel. It was halted at 19:00. Two hours later (i.e., at...	analysisWe need to solve for distance AB and original speed V. Let's denote distance between stations as D (km). Original schedule: train leaves at 13:00, arrives at B at some scheduled arrival time? We need to know scheduled arrival time relative to departure. The problem states: "At 19:00 the train was brought to a h...	analysisWe need to solve the problem: train from A to B. Initially left at 13:00. At 19:00 (6 hours after departure) the train was halted by snow drift. Two hours later (i.e., at 21:00) the line cleared, and to make up lost time, train proceeded at speed exceeding original speed by 20% (so 1.2 V) and arrived at B only ...	{ "competition": "misc", "dataset": "Ours", "posts": null, "source": "Problems in Elementary Mathematics - group_21.md" }
3. Two trains leave from two cities at the same time. The first one travels at 40 km/h, while the second one travels at 48 km/h. How far apart will these trains be from each other after 8 hours, if they are moving in the same direction and the distance between the cities is 892 km?	956	0	analysisWe need to analyze the problem carefully. We have two cities with distance 892 km. Two trains leave from two cities at the same time. The first train travels at 40 km/h, the second at 48 km/h. They are moving in the same direction. We need to find how far apart will these trains be from each other after 8 hours...	analysisWe need to parse problem: Two trains leave from two cities at the same time. The first travels at 40 km/h, second at 48 km/h. They are moving in the same direction. Distance between cities is 892 km. After 8 hours, how far apart will the trains be from each other? We need to determine initial positions: They l...	analysisWe need to parse the problem. Two trains leave from two cities at the same time. The first one travels at 40 km/h, while the second one travels at 48 km/h. They are moving in the same direction. The distance between the cities is 892 km. We need to find how far apart will these trains be after 8 hours. We need...	analysisWe need to parse the problem: Two trains leave from two cities at the same time. The first one travels at 40 km/h, while the second one travels at 48 km/h. They are moving in the same direction, and the distance between the cities is 892 km. We need to find how far apart will these trains be from each other aft...	analysisWe need to analyze the problem: Two trains leave from two cities at the same time. The first travels at 40 km/h, second at 48 km/h. They are moving in the same direction. Distance between cities is 892 km. We need the distance between trains after 8 hours. So we need to consider relative positions. Let's define...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Example 7 Let $a, b, c, d>0$. When $\frac{1}{1+a^{4}}+\frac{1}{1+b^{4}}+\frac{1}{1+c^{4}}+\frac{1}{1+d^{4}}=1$, prove: $a b c d \geqslant 3$. Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.	proof	0	analysisThe user asks: "Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly." So they want translation of the problem statement: "Example 7 Let $a, b, c, d>0$. When $\frac{1}{1+a^{4}}+\frac{1}{1+b^{4}}+\frac{1}{1+c^{4}}+\frac{1}{1+d^...	analysisWe need to translate the problem statement into English, preserving line breaks and format. The problem: "Example 7 Let $a, b, c, d>0$. When $\frac{1}{1+a^{4}}+\frac{1}{1+b^{4}}+\frac{1}{1+c^{4}}+\frac{1}{1+d^{4}}=1$, prove: $a b c d \geqslant 3$." We just translate. The instruction: "Translate the above text ...	analysisWe need to translate the problem statement into English, keeping line breaks and format. The problem is: "Example 7 Let $a, b, c, d>0$. When $\frac{1}{1+a^{4}}+\frac{1}{1+b^{4}}+\frac{1}{1+c^{4}}+\frac{1}{1+d^{4}}=1$, prove: $a b c d \geqslant 3$." We need to translate into English, preserving original line br...	analysisWe need to translate the problem statement into English, preserving line breaks and format. The original is: "Example 7 Let $a, b, c, d>0$. When $\frac{1}{1+a^{4}}+\frac{1}{1+b^{4}}+\frac{1}{1+c^{4}}+\frac{1}{1+d^{4}}=1$, prove: $a b c d \geqslant 3$." We need to translate into English, keep line breaks and f...	analysisWe need to translate the problem into English, keep line breaks and format. The original text: "Example 7 Let $a, b, c, d>0$. When $\frac{1}{1+a^{4}}+\frac{1}{1+b^{4}}+\frac{1}{1+c^{4}}+\frac{1}{1+d^{4}}=1$, prove: $a b c d \geqslant 3$." We need to output translation: "Example 7 Let a, b, c, d > 0. When 1/(1...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "inequalities" }
II. (50 points) Let $0<x_{1}<x_{2}<\cdots<x_{n}<1$, prove that: $$ \left(1-x_{n}\right)^{2}\left[\frac{x_{1}}{\left(1-x_{1}^{2}\right)^{2}}+\frac{x_{2}^{2}}{\left(1-x_{2}^{3}\right)^{2}}+\cdots+\frac{x_{n}^{n}}{\left(1-x_{n}^{n+1}\right)^{2}}\right]<1 $$	proof	0	analysisWe need to prove inequality: Given 0 < x1 < x2 < ... < xn < 1. We need to show: (1 - x_n)^2 * [ sum_{k=1}^n x_k^k / (1 - x_k^{k+1})^2 ] < 1. Wait check indices: first term uses x1/(1-x1^2)^2? Actually for k=1: numerator x1^1? but given x1? Actually they wrote x1/(1-x1^2)^2. So exponent numerator is 1? but t...	analysisWe need to prove inequality: for increasing sequence 0< x1 < x2 < ... < xn < 1, prove (1 - x_n)^2 * [ sum_{k=1}^n (x_k^k)/(1 - x_k^{k+1})^2 ] < 1. We need to show strict inequality. Likely using some telescoping or Jensen? Let's analyze structure. Define term: T_k = x_k^k / (1 - x_k^{k+1})^2. We have produc...	analysisWe need to prove inequality: for 0 < x1 < x2 < ... < xn < 1, we have (1 - x_n)^2 * sum_{k=1}^n [x_k^k / (1 - x_k^{k+1})^2] < 1. We need to show strict inequality. Likely use telescoping or bounding each term by something like (1 - x_{k-1})^2 - (1 - x_k)^2? Wait maybe using identity: For 0< x <1, we have x/(1-...	analysisWe need to prove inequality: For 0 < x1 < x2 < ... < xn < 1, prove (1 - x_n)^2 * [ sum_{k=1}^n x_k^k / (1 - x_k^{k+1})^2 ] < 1. Wait: expression inside sum: first term: x1/(1 - x1^2)^2? Actually given: x1/(1-x1^2)^2? But pattern: maybe x_k^k/(1 - x_k^{k+1})^2? But second term: x2^2/(1 - x2^3)^2. So general t...	analysisWe need to prove inequality: Given 0 < x1 < x2 < ... < xn < 1. Show that: (1 - x_n)^2 * [ sum_{k=1}^n x_k^k / (1 - x_k^{k+1})^2 ] < 1. Wait: The terms: first term uses x1/(1-x1^2)^2; second uses x2^2/(1-x2^3)^2; ... nth uses x_n^n/(1 - x_n^{n+1})^2. So general term: x_k^k / (1 - x_k^{k+1})^2. Multiply sum by...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
$4.30 \sin 10 \alpha \sin 8 \alpha + \sin 8 \alpha \sin 6 \alpha - \sin 4 \alpha \sin 2 \alpha$.	2\cos2\alpha\sin6\alpha\sin10\alpha	0	analysisWe need to simplify expression: 4.30 sin10α sin8α + sin8α sin6α - sin4α sin2α. The notation "4.30" maybe means "4·30"? Wait: It's "4.30" maybe decimal 4.30? But likely it's "4·30"? Actually typical notation in such problems: "4.30" meaning 4.30? But maybe it's "4·30"? But 4.30 sin10α sin8α would be 4.30 times s...	analysisWe need to evaluate expression: $4.30 \sin 10 \alpha \sin 8 \alpha + \sin 8 \alpha \sin 6 \alpha - \sin 4 \alpha \sin 2 \alpha$. Likely we need to simplify using product-to-sum formulas. Let's parse: "4.30" maybe meaning $4 \cdot 30$? Wait maybe it's "4.30" as coefficient 4.30? But typical problems use inte...	analysisWe need to simplify expression: $4.30 \sin 10\alpha \sin 8\alpha + \sin 8\alpha \sin 6\alpha - \sin 4\alpha \sin 2\alpha$. But "4.30" maybe meaning $4.30$ as decimal? Could be $4.30$ (four point thirty). However typical problems use integer coefficients. Might be $4\cdot 30$? But "4.30" might be 4.30, i.e., 4....	analysisWe need to simplify expression: 4.30 sin10α sin8α + sin8α sin6α - sin4α sin2α. Wait 4.30 maybe 4.30? Could be 4.30 meaning 4.30? But typical coefficient is 4.30? In earlier problems they use 4.30 maybe meaning 4.30? Actually 4.30 might be 4.30 (i.e., 4.30). But maybe it's 4.30 meaning 4.30? Wait: The problem st...	analysisWe need to compute expression: 4.30 sin10α sin8α + sin8α sin6α - sin4α sin2α. Probably 4.30 means 4.30? Might be 4·30? Actually typical notation: 4.30 could mean 4.30? But maybe it's 4.30 meaning 4.30? Wait the problem statement: "4.30 sin 10 α sin 8 α + sin 8 α sin 6 α - sin 4 α sin 2 α". Might be from a list ...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Example 6.49 (2007 Serbia Mathematical Olympiad) $x, y, z>0, x+y+z=1$, prove that $$\frac{x^{k+2}}{x^{k+1}+y^{k}+z^{k}}+\frac{y^{k+2}}{y^{k+1}+z^{k}+x^{k}}+\frac{z^{k+2}}{z^{k+1}+x^{k}+y^{k}} \geqslant \frac{1}{7}$$	\frac{1}{7}	0	analysisWe need to prove inequality: For positive reals x,y,z>0 with sum 1, prove that S = sum_{cyc} x^{k+2}/(x^{k+1}+y^k+z^k) >= 1/7. k is presumably a positive integer? The problem statement says "Example 6.49 (2007 Serbia Mathematical Olympiad) x, y, z > 0, x+y+z=1, prove that ...". It might be for all real k? Us...	analysisWe need to prove inequality for positive x,y,z with sum 1, and exponent k? k presumably integer? Problem statement: Example 6.49 (2007 Serbia Mathematical Olympiad) $x, y, z>0, x+y+z=1$, prove that $$\frac{x^{k+2}}{x^{k+1}+y^{k}+z^{k}}+\frac{y^{k+2}}{y^{k+1}+z^{k}+x^{k}}+\frac{z^{k+2}}{z^{k+1}+x^{k}+y^{k}} \geq...	analysisWe need to prove inequality for positive reals x,y,z summing to 1. For general k? The statement: Example 6.49 (2007 Serbia Mathematical Olympiad) $x, y, z>0, x+y+z=1$, prove that $$\frac{x^{k+2}}{x^{k+1}+y^{k}+z^{k}}+\frac{y^{k+2}}{y^{k+1}+z^{k}+x^{k}}+\frac{z^{k+2}}{z^{k+1}+x^{k}+y^{k}} \geqslant \frac{1}{7}.$...	analysisWe need to prove inequality: For positive x,y,z with sum 1, for integer k? Problem states k? Probably integer >=1. We need to show sum of terms >= 1/7. Let's analyze. We have expression: sum_{cyc} x^{k+2} / (x^{k+1}+ y^k + z^k) >= 1/7. We might attempt to use Titu's lemma or Cauchy-Schwarz Engel form: sum a^2...	analysisWe need to prove inequality: For positive reals x,y,z >0 with sum 1, and integer k? It says k+2 etc. k presumably positive integer? Probably natural number. We need to show sum >= 1/7. We need to find minimal constant? They claim 1/7. We need to prove. We can attempt to use Jensen or Titu's lemma? Let's analy...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "inequalities" }
### 3.485 Find the maximum value of the expression $$ A=\frac{1}{\sin ^{6} \alpha+\cos ^{6} \alpha} \text { for } 0 \leq \alpha \leq \frac{\pi}{2} $$	4	100	analysisWe need maximum of A = 1/(sin^6 α + cos^6 α) for α in [0, π/2]. Since sin^2+cos^2=1. Let t = sin^2 α. Then cos^2 α = 1 - t. t ∈ [0,1]. Then sin^6 = t^3, cos^6 = (1-t)^3. So denominator D = t^3 + (1-t)^3. We need minimize D to maximize A. So we need find minimum of f(t) = t^3 + (1-t)^3 on [0,1]. Compute derivat...	analysisWe need maximum of A=1/(sin^6 α + cos^6 α) for α in [0, π/2]. Equivalent to minimize denominator D=sin^6+cos^6. Since sin^2+cos^2=1. We can express sin^6+cos^6 = (sin^2)^3 + (cos^2)^3. Let s=sin^2 α, c=cos^2 α. Then s+c=1, s,c∈[0,1]. Then D = s^3 + c^3 = (s+c)^3 -3sc(s+c) = 1^3 -3sc = 1-3sc. Wait: formula: a^3+...	analysisWe need to find maximum of A = 1/(sin^6 α + cos^6 α) for α in [0, π/2]. Equivalent to minimize denominator D = sin^6 α + cos^6 α. Since sin^6+cos^6 is symmetric around π/4; min likely at α=π/4? Let's analyze. Let s = sin^2 α, c = cos^2 α. Then s + c = 1, s,c ∈ [0,1]. We need minimize s^3 + c^3. Because sin^6 =...	analysisWe need to find max of A = 1/(sin^6 α + cos^6 α) for α in [0, π/2]. Equivalent to minimize denominator D = sin^6 α + cos^6 α. Since sin^2+cos^2=1, but we need sixth powers. Use identity: a^3 + b^3 = (a+b)^3 - 3ab(a+b). But we have sixth powers. Let s = sin^2 α, c = cos^2 α. Then s + c = 1, s, c ∈ [0,1]. We want...	analysisWe need to find maximum of A = 1/(sin^6 α + cos^6 α) for α in [0, π/2]. Equivalent to minimize denominator D = sin^6 α + cos^6 α. We need min of D. Since sin^6 + cos^6. Use identity: sin^6 + cos^6 = (sin^2)^3 + (cos^2)^3. Let t = sin^2 α. Then cos^2 α = 1 - t. Domain t ∈ [0,1]. Then D = t^3 + (1 - t)^3. Comput...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
6.086. $\left\{\begin{array}{l}(x-y)\left(x^{2}-y^{2}\right)=45, \\ x+y=5\end{array}\right.$ Solve the system of equations: \[ \left\{\begin{array}{l} (x-y)\left(x^{2}-y^{2}\right)=45, \\ x+y=5 \end{array}\right. \]	(4,1),(1,4)	100	analysisWe need to solve system: (x-y)(x^2 - y^2) = 45, and x + y = 5. Simplify: x^2 - y^2 = (x-y)(x+y). So left side becomes (x-y) * (x-y)(x+y) = (x-y)^2 (x+y). So equation 1: (x-y)^2 (x+y) = 45. But x+y = 5, so (x-y)^2 * 5 = 45 => (x-y)^2 = 9 => x - y = ±3. Thus we have two possibilities: x - y = 3 or x - y = -3. ...	analysisWe need to solve system: (x - y)(x^2 - y^2) = 45, and x + y = 5. Let's analyze. We can rewrite (x-y)(x^2 - y^2) = (x-y)(x-y)(x+y) = (x-y)^2 (x+y). Because x^2 - y^2 = (x - y)(x + y). So product equals (x-y)^2 (x+y) = 45. Given x + y = 5. Then (x-y)^2 * 5 = 45 => (x-y)^2 = 9 => x - y = ±3. So we have two case...	analysisWe need to solve system: (1) (x - y)(x^2 - y^2) = 45 (2) x + y = 5 We need to find real solutions presumably. Let's analyze. We can express x and y in terms of sum and difference. Let s = x + y = 5 (given) Let d = x - y Then x = (s + d)/2, y = (s - d)/2. Compute x^2 - y^2 = (x - y)(x + y) = d * s = d * 5...	analysisWe need to solve system: (x-y)(x^2 - y^2) = 45, and x + y = 5. Let's analyze. Let x + y = 5. Let d = x - y. Then we have x = ( (x+y) + (x-y) )/2 = (5 + d)/2, y = ( (x+y) - (x-y) )/2 = (5 - d)/2. Compute x^2 - y^2 = (x - y)(x + y) = d * 5 = 5d. Wait: x^2 - y^2 = (x - y)(x + y) = d * 5. So (x - y)(x^2 - y^2) = ...	analysisWe need to solve system: (x-y)(x^2 - y^2) = 45 x + y = 5 Simplify: x^2 - y^2 = (x-y)(x+y). So (x-y)(x^2 - y^2) = (x-y)((x-y)(x+y)) = (x-y)^2 (x+y) = 45 Given x+y=5, plug in: (x-y)^2 5 = 45 => (x-y)^2 = 9 => x-y = ±3 So we have two cases: Case 1: x - y = 3 Case 2: x - y = -3 Also x + y = 5. Solve linea...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Example 7 Let $x, y, z \in \mathbf{R}^{\prime}$, compare $x^{3}+y^{3}+z^{3}+3 x y z$ with $x^{2}(y+z)+y^{2}(z+x)+z^{2}(x+y)$.	proof	0	analysisWe need to analyze the inequality: compare $x^3 + y^3 + z^3 + 3xyz$ with $x^2(y+z)+ y^2(z+x)+ z^2(x+y)$. The variables are in $\mathbf{R}'$. I'm not sure what $\mathbf{R}'$ means. Usually $\mathbf{R}'$ could mean real numbers? Might be real numbers excluding something? But likely just real numbers. Ho...	analysisWe need to compare two expressions: A = x^3 + y^3 + z^3 + 3xyz; B = x^2(y+z)+ y^2(z+x)+ z^2(x+y). For real numbers? The problem says x, y, z ∈ R', maybe meaning nonnegative reals? In many inequality problems, R' denotes nonnegative reals? But might mean real numbers. Let's interpret as real numbers. We need to ...	analysisWe need to compare two expressions: A = x^3 + y^3 + z^3 + 3xyz; B = x^2(y+z) + y^2(z+x) + z^2(x+y). For real numbers? It says x,y,z in R', maybe meaning real numbers (nonzero?) but typical notation R' means real numbers except maybe zero? Not sure. But we can assume real numbers. The problem: compare A and B; m...	analysisWe need to compare two expressions: S1 = x^3 + y^3 + z^3 + 3xyz, and S2 = x^2(y+z) + y^2(z+x) + z^2(x+y). We need to determine which is greater, equal, or less, presumably for real numbers? The problem states "x, y, z ∈ ℝ′". I'm not sure what ℝ′ means; perhaps ℝ' denotes nonzero reals? Might be real numbers exc...	analysisWe need to compare two expressions: S1 = x^3 + y^3 + z^3 + 3xyz, and S2 = x^2(y+z)+y^2(z+x)+z^2(x+y). We need to determine which is larger, or maybe inequality sign? The problem states: "compare x^3 + y^3 + z^3 + 3xyz with x^2(y+z)+y^2(z+x)+z^2(x+y)." It might ask to find which expression is greater or equal, m...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
## Problem 1 Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be strictly positive real numbers. Show that: $\frac{\mathrm{a}}{\mathrm{a}^{2}+\mathrm{bc}}+\frac{\mathrm{b}}{\mathrm{b}^{2}+\mathrm{ac}}+\frac{\mathrm{c}}{\mathrm{c}^{2}+\mathrm{ab}} \leq \frac{1}{2}\left(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{~b}}+\frac{1}{\math...	proof	0	analysisWe need to prove inequality: For positive a,b,c, S = a/(a^2 + bc) + b/(b^2 + ac) + c/(c^2 + ab) <= (1/2)(1/a + 1/b + 1/c). We can attempt to manipulate. Observation: a/(a^2 + bc) = (a^2)/(a^3 + abc). Not helpful. Alternatively, rewrite as: a/(a^2 + bc) = (1/a) * (1/(1 + bc/a^2)) = (1/a) * (1/(1 + (b/a)(c/...	analysisWe need to prove inequality for positive real a,b,c: Sum a/(a^2+bc) ≤ 1/2 (1/a + 1/b + 1/c). We need to show: ∑ a/(a^2+bc) ≤ (1/2) ∑ 1/a. Let’s denote S = ∑ a/(a^2+bc). Multiply both sides by 2: 2S ≤ ∑ 1/a. Equivalent: ∑ (2a/(a^2+bc)) ≤ ∑ 1/a. So we need to show for each term: 2a/(a^2+bc) ≤ 1/a? Not indivi...	analysisWe need to prove inequality for positive reals a,b,c: Sum a/(a^2+bc) ≤ 1/2 (1/a + 1/b + 1/c). We need to identify key structure. Let's denote expression: S = Σ a/(a^2+bc). We want S ≤ 1/2 Σ 1/a. Multiply both sides by 2: 2 Σ a/(a^2+bc) ≤ Σ 1/a. Alternatively, bring to common denominator? Maybe use substitut...	analysisWe need to prove inequality: For positive a,b,c, Sum_{cyc} a/(a^2 + bc) <= 1/2 * (1/a + 1/b + 1/c). Equivalent to: 2 * sum a/(a^2+bc) <= sum 1/a. Or Sum (2a/(a^2+bc) - 1/a) <= 0. Let's examine each term: 2a/(a^2+bc) - 1/a = (2a^2 - (a^2+bc))/[a(a^2+bc)] = (a^2 - bc)/[a(a^2+bc)]. So sum over cyclic of (a...	analysisWe need to prove inequality: for positive a,b,c: Sum_{cyc} a/(a^2+bc) <= 1/2*(1/a + 1/b + 1/c). We can attempt to use Titu's lemma or Cauchy-Schwarz or rearrangement. Let's analyze. Let’s attempt to manipulate: a/(a^2+bc) = a^2/(a^3+abc). Not nice. Maybe rewrite as: a/(a^2+bc) <= 1/(2a) + 1/(2b) + 1/(2c) ?...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
7.215. $2^{x+\sqrt{x^{2}-4}}-5 \cdot(\sqrt{2})^{x-2+\sqrt{x^{2}-4}}-6=0$.	\frac{5}{2}	100	analysisWe need to solve equation: 2^{x + sqrt{x^2 - 4}} - 5 * (sqrt{2})^{x - 2 + sqrt{x^2 - 4}} - 6 = 0. We must find all real x satisfying. Let’s denote t = sqrt{x^2 - 4}. Domain: x^2 - 4 >= 0 => \|x\| >= 2. So x <= -2 or x >= 2. Equation: 2^{x + t} - 5 * (sqrt{2})^{x - 2 + t} - 6 = 0. Observe that sqrt{2} = 2^{1/2...	analysisWe need solve equation: 2^{x + sqrt(x^2 - 4)} - 5 * (sqrt{2})^{x - 2 + sqrt{x^2 - 4}} - 6 = 0. Let’s denote t = sqrt{x^2 - 4}. Domain: x^2 - 4 >= 0 => \|x\| >= 2. So x <= -2 or x >= 2. Also sqrt{x^2 - 4} >= 0. t = sqrt{x^2 - 4}. We can express x + t and x - 2 + t. Maybe we can rewrite with base 2 or sqrt{2}. ...	analysisWe need to solve equation: 2^{x + sqrt(x^2 - 4)} - 5 * (sqrt{2})^{x - 2 + sqrt{x^2 - 4}} - 6 = 0. We need to find real x satisfying. Domain: sqrt{x^2 - 4} defined for \|x\| >= 2. Also exponent of 2 and sqrt{2} are real. Let’s denote t = sqrt{x^2 - 4}. Then t >= 0 and x^2 - 4 = t^2 => x^2 = t^2 + 4. Also x can b...	analysisWe need solve equation: 2^{x + sqrt(x^2 - 4)} - 5 * (sqrt{2})^{x-2 + sqrt{x^2-4}} - 6 = 0. Let’s denote expression. Domain: sqrt{x^2 - 4} real => \|x\| >= 2. Let t = sqrt{x^2 - 4}. Then t >= 0. Also we have x^2 - 4 = t^2 => x^2 = t^2 + 4. Since \|x\| >= 2, we can consider x >= 2 or x <= -2. We might express x in ...	analysisWe need solve equation: 2^{x + sqrt(x^2-4)} - 5 * (sqrt{2})^{x-2+ sqrt{x^2-4}} - 6 = 0. Let’s denote something: sqrt{2} = 2^{1/2}. So (sqrt{2})^{something} = 2^{(1/2)(something)}. Let’s set A = x + sqrt{x^2-4}. B = x-2 + sqrt{x^2-4}. Equation: 2^{A} - 52^{(1/2)B} - 6 = 0. We may try to express in terms of...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where \[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\su...	\frac{m+n}{mn+n}	0	analysisWe need to analyze the problem: For given positive integers m, n > 1. Variables a_{ij} >= 0, not all zero. Define: Let S_i = sum_{j=1}^m a_{ij} (row sums). Let T_j = sum_{i=1}^n a_{ij} (column sums). Let total sum: A = sum_{i=1}^n sum_{j=1}^m a_{ij}. Let sum of squares: Q = sum_{i=1}^n sum_{j=1}^m a_{ij}^2. ...	analysisWe need to analyze the problem. We have nonnegative real numbers a_{ij}, not all zero. m,n > 1. Define: Let S_{i} = sum_{j=1}^{m} a_{ij} for each row i. Let T_{j} = sum_{i=1}^{n} a_{ij} for each column j. Let total sum A = sum_{i,j} a_{ij}. Let sum of squares of entries: Q = sum_{i,j} a_{ij}^2. Then numera...	analysisWe need to find max and min of f given nonnegative a_ij not all zero, m,n>1. Let’s denote sums: Let S_i = sum_{j} a_{ij} for i=1..n. So S_i >=0. Let T_j = sum_{i} a_{ij} for j=1..m. Let total sum A = sum_{i,j} a_{ij}. Let sum of squares: Q = sum_{i,j} a_{ij}^2. We also have: sum over i of S_i^2 = sum_i (s...	analysisWe need to find max/min of f over nonnegative a_ij not all zero. Let’s denote sums: define row sums R_i = sum_j a_{ij}. column sums C_j = sum_i a_{ij}. total sum S = sum_i sum_j a_{ij}. Also sum of squares Q = sum_{i,j} a_{ij}^2. Given f = [ n Σ_i R_i^2 + m Σ_j C_j^2 ] / [ S^2 + mn Q ]. We need bounds. Use Ca...	analysisWe need to find maximum and minimum of f given nonnegative a_{ij} not all zero. Parameters m,n>1 integers. We need to analyze expression. Let’s denote: Let S = sum_{i,j} a_{ij}. Let A_i = sum_{j} a_{ij}. Let B_j = sum_{i} a_{ij}. Also let Q = sum_{i,j} a_{ij}^2. Then numerator N = n * sum_i A_i^2 + m * sum_j...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "aops_forum" }
$3 \cdot 49$ Positive numbers $u_{1}, u_{2}, \cdots, u_{n}$ form an arithmetic sequence, prove: $$ \begin{aligned} t_{n} & =\frac{1}{\sqrt{u_{1}}+\sqrt{u_{2}}}+\frac{1}{\sqrt{u_{2}}+\sqrt{u_{3}}}+\cdots+\frac{1}{\sqrt{u_{n-1}}+\sqrt{u_{n}}} \\ & =\frac{n-1}{\sqrt{u_{1}}+\sqrt{u_{n}}} . \end{aligned} $$	proof	0	analysisWe need to analyze problem: "3 · 49 Positive numbers u1, u2, …, un form an arithmetic sequence, prove: t_n = sum_{k=1}^{n-1} 1/(sqrt(u_k)+sqrt(u_{k+1})) = (n-1)/(sqrt(u_1)+sqrt(u_n))." The statement is basically same as retrieved example but with n=3*49? Wait "3 · 49 Positive numbers u1, u2, …, un" maybe means...	analysisWe need to prove a formula for sum of reciprocals of sums of consecutive square roots of terms of an arithmetic progression of positive numbers. The retrieved example shows a similar formula: sum_{i=1}^{n-1} 1/(sqrt{a_i} + sqrt{a_{i+1}}) = (n-1)/(sqrt{a_1} + sqrt{a_n}). This is exactly the same statement. The p...	analysisWe need to prove that for an arithmetic sequence of positive numbers, the sum of reciprocals of sums of consecutive square roots equals (n-1)/(sqrt(u1)+sqrt(un)). The example retrieved problem is exactly that but with a1,...,an. The retrieved solution uses telescoping: 1/(sqrt(u_i)+sqrt(u_{i+1})) = (sqrt(u_{i+1...	analysisWe need to prove the identity for an arithmetic sequence of positive numbers u_i. The retrieved example is exactly the same statement, just with n=349? Wait, the problem says "$3 \cdot 49$ Positive numbers $u_{1}, u_{2}, \cdots, u_{n}$ form an arithmetic sequence". That seems like they mean n=349? Actually th...	analysisWe need to prove the same identity for arithmetic progression of positive numbers. The retrieved example exactly matches the problem: 3·49 positive numbers? Actually the problem statement says "$3 \cdot 49$ Positive numbers $u_{1}, u_{2}, \cdots, u_{n}$ form an arithmetic sequence, prove: ..." Maybe "3·49" is a...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Find all $x$ in $[0, \pi]$ inclusive so that \[ \sin^2 x + \csc^2 x + \cos^2 x = \cot^2 x + \sec^2 x \]	\frac{\pi}{4}, \frac{3\pi}{4}	100	analysisWe need to solve equation: sin^2 x + csc^2 x + cos^2 x = cot^2 x + sec^2 x Domain: x in [0, π], inclusive. But need to consider points where denominators zero: sin x ≠ 0 for csc^2, cos x ≠ 0 for sec^2, tan x ≠ 0 for cot^2? Actually cot^2 = cos^2/sin^2. So sin x ≠ 0 for cot^2 as well. So domain excludes points...	analysisWe need to solve equation: sin^2 x + csc^2 x + cos^2 x = cot^2 x + sec^2 x Domain: x in [0,π], but note that csc^2 x, sec^2 x, cot^2 x have singularities at sin x=0 and cos x=0. So we must exclude x where sin x=0 (x=0,π) and cos x=0 (x=π/2). But the equation may still hold if both sides are undefined? Usually...	analysisWe need to solve equation: sin^2 x + csc^2 x + cos^2 x = cot^2 x + sec^2 x We need all x in [0, π] inclusive. We should express everything in terms of sin and cos. Note: csc^2 x = 1/sin^2 x. sec^2 x = 1/cos^2 x. cot^2 x = cos^2 x / sin^2 x. Let s = sin x, c = cos x. Domain: x ∈ [0, π], so s≥0? Actually sin ...	analysisWe need to solve equation: sin^2 x + csc^2 x + cos^2 x = cot^2 x + sec^2 x, for x in [0, π] inclusive. We need to analyze domain: sin x ≠ 0 for csc^2 x defined; cos x ≠ 0 for sec^2 x defined; cot^2 x defined requires sin x ≠ 0; also cot^2 defined with sin denominator. So x cannot be multiples of π, and cannot ...	analysisWe need to solve equation: sin^2 x + csc^2 x + cos^2 x = cot^2 x + sec^2 x, for x in [0, π] inclusive. Domain: sin x, cos x cannot be zero where denominators appear. csc^2 x = 1/sin^2 x, sec^2 x = 1/cos^2 x, cot^2 x = cos^2 x / sin^2 x. So x cannot be multiples of π (sin=0) or π/2 (cos=0). So x cannot be 0, π, ...	{ "competition": "jhmt", "dataset": "Ours", "posts": null, "source": "AlgebraKey2006.md" }
## Task A-1.1. Depending on the real parameter $m$, determine for which real numbers $x$ the following inequality holds: $$ \frac{x-m}{x^{2}}+x \geqslant 2\left(1-\frac{m}{x}\right)+m $$	x\in\begin{cases}{1}\cup[,+\infty)&	0	analysisWe need to solve inequality: (x - m)/x^2 + x >= 2(1 - m/x) + m. Domain: x ≠ 0. Also denominator x^2 >0, fine. Let's rewrite. Let’s bring all terms to one side: (x - m)/x^2 + x - 2(1 - m/x) - m >= 0. Compute 2(1 - m/x) = 2 - 2m/x. So inequality: (x - m)/x^2 + x - 2 + 2m/x - m >= 0. Group terms: (x - m)/x^2 ...	analysisWe need to solve inequality: (x - m)/x^2 + x >= 2(1 - m/x) + m. We must consider domain: x ≠ 0. Also denominator x^2 > 0 always except x=0. So domain all real x except 0. We need to analyze for parameter m real. Let's rewrite inequality. Left side: (x - m)/x^2 + x = (x - m)/x^2 + x. Right side: 2(1 - m/x) ...	analysisWe need to solve inequality: (x - m)/x^2 + x >= 2(1 - m/x) + m. Simplify expression. Domain: x ≠ 0. Also denominator x^2 >0, fine. So domain: x ∈ ℝ\{0}. Also we might consider m real parameter. Let's rewrite inequality. Left side: (x - m)/x^2 + x. Right side: 2(1 - m/x) + m = 2 - 2m/x + m. Bring all to one ...	analysisWe need to solve inequality depending on parameter m. Let's analyze. Inequality: (x - m)/x^2 + x >= 2(1 - m/x) + m. Simplify. Domain: x ≠ 0. Also x^2 in denominator. So x ≠ 0. Compute left side: (x - m)/x^2 + x = (x - m)/x^2 + x. Right side: 2(1 - m/x) + m = 2 - 2m/x + m. Bring all to one side: (x - m)/x^2...	analysisWe need to solve inequality: \frac{x - m}{x^2} + x \ge 2(1 - \frac{m}{x}) + m. We need to consider real parameter m. Solve for x ∈ ℝ, domain restrictions: x ≠ 0 because denominators. Also x^2 denominator fine for any x ≠ 0. We need to solve inequality for x depending on m. Let's rewrite: Left side: (x - m)...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
8.8. Let $a, b, c$ be positive real numbers such that $a+b+c=1$. Prove the inequality $$ \sqrt{a+b c}+\sqrt{b+c a}+\sqrt{c+a b} \leq 2 $$ Find the numbers $a, b, c$ for which equality holds.	=b==\frac{1}{3}	0	analysisWe need to prove inequality: sqrt(a + bc) + sqrt(b + ca) + sqrt(c + ab) <= 2, given a,b,c>0, a+b+c=1. Find equality case. We need to use deep reasoning. Let's analyze. We have a+b+c=1. Let's denote S = sqrt(a+bc)+ sqrt(b+ca)+ sqrt(c+ab). Want <=2. Observations: Since a,b,c positive and sum 1, each term sqrt(...	analysisWe need to prove inequality: sqrt(a + bc) + sqrt(b + ca) + sqrt(c + ab) ≤ 2, for positive reals a,b,c with a+b+c=1. Also find equality cases. We suspect that equality occurs at a=b=c=1/3? Let's test: a=b=c=1/3. Then a+bc = 1/3 + (1/3)(1/3)=1/3 + 1/9 = 4/9. sqrt=2/3. Sum=2. Yes equality holds. Are there other e...	analysisWe need to prove that for positive a,b,c with sum 1, we have sqrt(a+bc)+sqrt(b+ca)+sqrt(c+ab) ≤ 2. Also find equality cases. We can attempt to square or use Jensen? Let's analyze structure: terms sqrt(a+bc). Since a,b,c>0, bc ≤ (b+c)^2/4? Wait bc ≤ (b+c)^2/4 by AM-GM. Also a+bc ≤ a+(b+c)^2/4. But maybe we need...	analysisWe need to prove inequality: For positive real a,b,c sum to 1, show sqrt(a + bc) + sqrt(b + ca) + sqrt(c + ab) ≤ 2. Equality case? Let's analyze. We need to find maximum of expression f(a,b,c) = sqrt(a+bc)+sqrt(b+ca)+sqrt(c+ab) subject to a,b,c>0, a+b+c=1. Show maximum is 2. Let's attempt to use Jensen or Cauc...	analysisWe need to prove inequality: sqrt(a+bc)+sqrt(b+ca)+sqrt(c+ab) ≤ 2, with a,b,c >0, a+b+c=1. Equality cases? We need to find numbers a,b,c that achieve equality. We need to analyze. Let's attempt to prove inequality. We can try Jensen or Cauchy-Schwarz or AM-GM. Let's examine expression: sqrt(a+bc). Since a+b+c...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Let $a, b, c$ be three positive real numbers such that $a+b+c=1$. Show that $$ \frac{a^{2}}{\frac{b+c}{2}+\sqrt{b c}}+\frac{b^{2}}{\frac{c+a}{2}+\sqrt{c a}}+\frac{c^{2}}{\frac{a+b}{2}+\sqrt{a b}} \geq \frac{1}{2} $$ ## 2 Solution	proof	0	analysisWe need to prove inequality: For positive real numbers a,b,c with sum 1, ∑ a^2 / ( (b+c)/2 + sqrt(bc) ) ≥ 1/2. We can attempt to use substitution or known inequalities. Let's analyze the denominator: (b+c)/2 + sqrt(bc). By AM-GM, (b+c)/2 ≥ sqrt(bc). So denominator ≥ sqrt(bc)+ sqrt(bc)=2 sqrt(bc). So each term...	analysisWe need to prove inequality: Given a+b+c=1, a,b,c>0, Sum_{cyc} a^2 / ( (b+c)/2 + sqrt(bc) ) >= 1/2. Simplify denominator: (b+c)/2 + sqrt(bc) = (b+c+2 sqrt(bc))/2 = ( (sqrt{b}+sqrt{c})^2 ) / 2. Because b+c+2 sqrt{bc} = (sqrt{b}+sqrt{c})^2. So denominator = ( (sqrt{b}+sqrt{c})^2 ) / 2. Thus each term a^2 / (...	analysisWe need to prove inequality: Given a+b+c=1, positive reals. Show: S = sum_{cyc} a^2 / ( (b+c)/2 + sqrt(bc) ) >= 1/2. We need to find a proof. Let’s denote denominators: D_a = (b+c)/2 + sqrt(bc). We can attempt to use Cauchy-Schwarz or Jensen or Titu's lemma. Observations: For each term, denominator is (b+c...	analysisWe need to prove inequality: Given a,b,c>0, a+b+c=1. Show: Sum_{cyc} a^2 / ( (b+c)/2 + sqrt(bc) ) >= 1/2. We need to find approach. The retrieved example had inequality with a^2/(b+c) >= (1/2)(a+b+c). This is similar but denominator includes (b+c)/2 + sqrt(bc). So denominator is bigger or smaller? Let's anal...	analysisWe need to prove inequality: Given positive a,b,c with a+b+c=1. Show: Sum over cyclic a^2 / ( (b+c)/2 + sqrt(bc) ) >= 1/2. We need to find a rigorous proof. Observations: Denominator: (b+c)/2 + sqrt(bc) = (b+c+2 sqrt(bc))/2 = ( (sqrt(b)+sqrt(c))^2 )/2. Indeed (sqrt(b)+sqrt(c))^2 = b + c + 2 sqrt(bc). So den...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Evaluate: $$ \sum_{k=1}^{n} \cot ^{-1}\left(2 k^{2}\right)=\cot ^{-1}(1+1 / n) $$	\cot ^{-1}(1+1 / n)	60	analysisWe need to evaluate sum_{k=1}^n arccot(2k^2). Claim: sum equals arccot(1 + 1/n). We must prove. Let's analyze. We know identity: arccot x - arccot y = arccot((xy-1)/(x+y)). But careful about ranges. Another useful identity: arccot a + arccot b = arccot((ab-1)/(a+b)) for positive a,b? Actually arccot a + arccot...	analysisWe need to evaluate sum_{k=1}^n arccot(2k^2) = arccot(1+1/n). Let's verify: For each k, arccot(2k^2). Maybe there's telescoping identity: arccot x - arccot y = arccot( (xy+1)/(y-x) )? Wait formula: arccot u - arccot v = arccot( (uv+1)/(v-u) ). But we might express arccot(2k^2) as difference of arccot of somethi...	analysisWe need to evaluate finite sum: sum_{k=1}^n arccot(2k^2) = arccot(1+1/n). This is a known telescoping identity. We need to prove. We know formula: arccot a - arccot b = arccot( (a b - 1)/(a + b) ), but careful with quadrant. Alternatively, use identity: arccot x = arctan(1/x). But maybe easier: Use arccot diff...	analysisWe need to evaluate sum of arccot(2k^2) from k=1 to n equals arccot(1+1/n). Let's verify. We need to prove identity. Let's attempt to find telescoping representation. We know formula: arccot x - arccot y = arccot((xy-1)/(x+y)). Also arccot x + arccot y = arccot((x y -1)/(x+y))? Wait: Actually cot(A+B)= (cot A ...	analysisWe need to evaluate sum_{k=1}^{n} arccot(2k^2) = arccot(1 + 1/n). The claim: sum of arccot(2k^2) equals arccot(1+1/n). We need to show that identity holds for all positive integer n. We must prove by telescoping or using arccot addition formula. We recall formula: arccot a - arccot b = arccot( (ab+1)/(b-a) ), ...	{ "competition": "alg_misc", "dataset": "Ours", "posts": null, "source": "Telescoping Sums and Products - Po-Shen Loh - MOP 2003.md" }
13 If the real numbers $x, y$ satisfy $x^{2}+y^{2}=1$, then the minimum value of $\frac{2 x y}{x+y-1}$ is $\qquad$ Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.	1-\sqrt{2}	0	analysisThe user says: "13 If the real numbers $x, y$ satisfy $x^{2}+y^{2}=1$, then the minimum value of $\frac{2 x y}{x+y-1}$ is $\qquad$ Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly." So they just want translation of the pro...	analysisWe need to translate the problem statement from presumably Chinese? The text: "13 If the real numbers $x, y$ satisfy $x^{2}+y^{2}=1$, then the minimum value of $\frac{2 x y}{x+y-1}$ is $\qquad$" But the user says: "Translate the text above into English, please keep the original text's line breaks and format, a...	analysisWe need to translate the problem statement into English, preserving line breaks and format. The problem is: "13 If the real numbers $x, y$ satisfy $x^{2}+y^{2}=1$, then the minimum value of $\frac{2 x y}{x+y-1}$ is $\qquad$" We need to translate into English, keep original line breaks and format. The original...	analysisWe need to translate the problem statement into English, preserving line breaks and format. The original text: "13 If the real numbers $x, y$ satisfy $x^{2}+y^{2}=1$, then the minimum value of $\frac{2 x y}{x+y-1}$ is $\qquad$" We need to output translation result directly. So English: "13 If the real numbers ...	analysisWe need to translate the problem statement into English. The problem: "13 If the real numbers $x, y$ satisfy $x^{2}+y^{2}=1$, then the minimum value of $\frac{2 x y}{x+y-1}$ is $\qquad$". We need to translate the text above into English, keeping original line breaks and format, output translation result directl...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
4. A dog is chasing a hare at a speed of 17 m/s, the hare is running at a speed of $14 \boldsymbol{m} /$ s. The distance between them before the chase was 150 m. Will the dog catch the hare if there are bushes $520 \mathcal{M}$ from the hare where he can hide?	Itwillnotcatchup	0	analysisWe need to parse the problem carefully. Problem statement: "4. A dog is chasing a hare at a speed of 17 m/s, the hare is running at a speed of 14 m/s. The distance between them before the chase was 150 m. Will the dog catch the hare if there are bushes 520 M from the hare where he can hide?" This seems ambiguou...	analysisWe need to parse problem: "4. A dog is chasing a hare at a speed of 17 m/s, the hare is running at a speed of $14 \boldsymbol{m} /$ s. The distance between them before the chase was 150 m. Will the dog catch the hare if there are bushes $520 \mathcal{M}$ from the hare where he can hide?" The problem statement s...	analysisWe need to parse the problem carefully. The problem statement: "4. A dog is chasing a hare at a speed of 17 m/s, the hare is running at a speed of 14 m/s. The distance between them before the chase was 150 m. Will the dog catch the hare if there are bushes 520 M from the hare where he can hide?" There might be...	analysisWe need to parse the problem carefully. It says: "4. A dog is chasing a hare at a speed of 17 m/s, the hare is running at a speed of 14 m/s. The distance between them before the chase was 150 m. Will the dog catch the hare if there are bushes 520 M from the hare where he can hide?" There is some confusion: "520...	analysisWe need to parse the problem: "4. A dog is chasing a hare at a speed of 17 m/s, the hare is running at a speed of 14 m/s. The distance between them before the chase was 150 m. Will the dog catch the hare if there are bushes 520 M from the hare where he can hide?" There might be a typo: "520 \mathcal{M}" maybe m...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
2. In the interval $0 \leq x \leq \pi$ find the solutions to the equation $$ \frac{1}{\sin x}-\frac{1}{\cos x}=2 \sqrt{2} $$	\frac{3\pi}{4},\frac{\pi}{12}	100	analysisWe need solve equation: 1/sin x - 1/cos x = 2√2, for x in [0,π]. Domain: sin x, cos x nonzero. For x in [0,π], sin x≥0, sin=0 at 0,π. cos x changes sign: cos x>0 for x∈[0,π/2), cos=0 at π/2, cos<0 for (π/2,π]. So denominators nonzero except at 0,π,π/2. So we exclude those points. Equation: 1/sin x - 1/cos x = ...	analysisWe need solve 1/sin x - 1/cos x = 2 sqrt(2). Domain: sin x, cos x nonzero. In interval [0,π], sin x positive except at 0,π; cos x positive for x∈[0,π/2) and negative for (π/2,π]. Also cos x zero at π/2. So need exclude x=0,π,π/2. Solve equation. Let’s set s=sin x, c=cos x. Equation: 1/s - 1/c = 2√2 => (c - s)/...	analysisWe need to solve equation: 1/sin x - 1/cos x = 2√2, for x in [0,π]. We must find solutions. Let's analyze. Equation: 1/sin x - 1/cos x = 2√2. Domain: sin x ≠ 0, cos x ≠ 0. So x cannot be 0, π, π/2. But endpoints 0 and π excluded because sin=0. Also cos=0 at π/2. So domain: (0,π) \ {π/2}. We solve. Let’s se...	analysisWe need to solve equation: 1/sin x - 1/cos x = 2 sqrt(2), for x in [0, pi]. Domain: sin x ≠ 0, cos x ≠ 0. So x ≠ 0, π, π/2. But endpoints 0 and π excluded; but we consider open interval? It says interval 0 ≤ x ≤ π. But at endpoints sin 0=0, cos π = -1? Actually cos π = -1, but sin 0 = 0 so left side undefined. ...	analysisWe need to solve equation: 1/sin x - 1/cos x = 2 sqrt(2). Domain: 0 <= x <= pi. Need to find solutions. We must consider sin x and cos x nonzero: sin x != 0 and cos x != 0. In [0, pi], sin x = 0 at x=0 and x=pi. cos x = 0 at x=pi/2. So domain excludes x=0, pi, pi/2. But we might check if the equation holds at ...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
2. Given the functions $y_{1}=\sin ^{4} x+\cos ^{4} x$ and $y_{2}=\sin ^{6} x+\cos ^{6} x$. Prove that these functions have a fundamental period of $\frac{\pi}{2}$ and that $3 y_{1}-2 y_{2}=1$. For which values of $x$ is the first function $\frac{1}{16}$ greater than the second?	\\frac{\pi}{12}+k\frac{\pi}{2},k\in\mathbb{Z}	0	analysisWe need to solve: y1 = sin^4 x + cos^4 x; y2 = sin^6 x + cos^6 x. Show fundamental period pi/2. Also show 3 y1 - 2 y2 = 1. Then find x such that y1 - y2 > 1/16? Wait: "For which values of x is the first function 1/16 greater than the second?" Means y1 = y2 + 1/16? Let's parse: "For which values of x is the firs...	analysisWe need to analyze problem: Given functions y1 = sin^4 x + cos^4 x and y2 = sin^6 x + cos^6 x. Prove that these functions have fundamental period π/2 and that 3 y1 - 2 y2 = 1. For which values of x is the first function 1/16 greater than the second? So we need to find the set of x such that y1 - y2 = 1/16? Wait...	analysisWe need to solve problem 2. We have two functions: y1 = sin^4 x + cos^4 x, y2 = sin^6 x + cos^6 x. Prove that they have fundamental period π/2. Also prove identity 3 y1 - 2 y2 = 1. Then find values of x where first function is 1/16 greater than second: y1 - y2 = 1/16? Wait "first function 1/16 greater than seco...	analysisWe need to analyze the problem: Given y1 = sin^4 x + cos^4 x, y2 = sin^6 x + cos^6 x. Prove that these functions have a fundamental period of π/2 and that 3 y1 - 2 y2 = 1. For which values of x is the first function 1/16 greater than the second? Wait: "For which values of x is the first function 1/16 greater th...	analysisWe need to analyze the problem. We have two functions: y1 = sin^4 x + cos^4 x y2 = sin^6 x + cos^6 x We need to prove that these functions have fundamental period π/2. Also prove that 3 y1 - 2 y2 = 1 (for all x). Then find values of x such that y1 - y2? Wait: "For which values of x is the first function 1/16 ...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
3. The number $x$ has the following property: subtracting $\frac{1}{10}$ from $x$ gives the same result as multiplying $x$ by $\frac{1}{10}$. What is the number $x$ ? A $\frac{1}{100}$ B $\frac{1}{11}$ C $\frac{1}{10}$ D $\frac{11}{100}$ E $\frac{1}{9}$	\frac{1}{9}	100	analysisWe need to solve: "The number x has the following property: subtracting 1/10 from x gives the same result as multiplying x by 1/10." So x - 1/10 = x * 1/10. Solve for x. Equation: x - 1/10 = x/10. Multiply both sides by 10: 10x - 1 = x. Then 10x - x = 1, so 9x = 1, x = 1/9. Thus answer E. But we must check d...	analysisWe need to parse the problem: "The number x has the following property: subtracting 1/10 from x gives the same result as multiplying x by 1/10. What is the number x?" So equation: x - 1/10 = x * (1/10). Solve for x. So x - 0.1 = 0.1x. Bring terms: x - 0.1x = 0.1 => (1 - 0.1)x = 0.1 => 0.9x = 0.1 => x = 0.1 / 0....	analysisWe need to solve equation: subtracting 1/10 from x gives same result as multiplying x by 1/10. That is: x - 1/10 = (1/10) x. Solve for x. Let's do algebra: x - 0.1 = 0.1 x. Bring terms: x - 0.1x = 0.1. So 0.9x = 0.1. Then x = 0.1 / 0.9 = 1/9. That's 0.111..., which is 1/9. So answer E. But we must be careful...	analysisWe need to solve equation: subtract 1/10 from x equals multiply x by 1/10. So x - 1/10 = (1/10) * x. Solve for x. Equation: x - 1/10 = (1/10) x. Bring terms: x - (1/10) x = 1/10. Factor x: x(1 - 1/10) = 1/10. 1 - 1/10 = 9/10. So x * (9/10) = 1/10. Multiply both sides by 10: x * 9 = 1. So x = 1/9. Option E. B...	analysisWe need to solve equation: subtracting 1/10 from x gives same result as multiplying x by 1/10. That is x - 1/10 = (1/10) x. Solve for x. Equation: x - 1/10 = x/10. Multiply by 10: 10x - 1 = x. So 10x - x = 1 => 9x = 1 => x = 1/9. That matches option E. But we must double-check: "subtracting 1/10 from x gives t...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
(15) Find the smallest positive real number $k$, such that the inequality $$ a b+b c+c a+k\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geqslant 9, $$ holds for all positive real numbers $a, b, c$.	2	100	analysisWe need to find smallest k such that inequality holds for all positive a,b,c: ab + bc + ca + k(1/a + 1/b + 1/c) >= 9. We need minimal k such that inequality holds for all positive reals. We need to find k minimal satisfying inequality. We can consider scaling invariance? Let's analyze. Suppose we scale all ...	analysisWe need smallest k such that for all positive a,b,c, inequality holds: ab+bc+ca + k(1/a+1/b+1/c) >= 9. Find minimal k. We need to find k such that inequality is true for all positive reals. Equivalent to find minimal k that ensures ab+bc+ca + k sum(1/a) >= 9. We can think of scaling invariance? Let's examine s...	analysisWe need to find minimal k such that inequality holds for all positive reals a,b,c: ab + bc + ca + k (1/a + 1/b + 1/c) >= 9. Equivalent to find k such that for all positive a,b,c, the expression is at least 9. We need minimal k. Let’s denote S1 = ab + bc + ca, S2 = 1/a + 1/b + 1/c. We want S1 + k S2 >= 9. We c...	analysisWe need to find smallest k such that inequality holds for all positive a,b,c. Inequality: ab+bc+ca + k(1/a+1/b+1/c) >= 9. We need minimal k such that inequality is true for all positive reals. We can use AM-GM or Cauchy? Let's analyze. We can rewrite inequality as sum_{cyc} [ab + k/a] >= 9. Set x= a, y= b,...	analysisWe need to find minimal k such that inequality holds for all positive a,b,c: ab+bc+ca + k(1/a + 1/b + 1/c) >= 9. We need minimal k. This is like optimization: For given a,b,c>0, we consider expression f(a,b,c) = ab+bc+ca + k(1/a + 1/b + 1/c). We want f>=9 for all a,b,c. Find minimal k such that inf over a,b,c...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
38th Putnam 1977 Problem B5 a 1 , a 2 , ... , a n are real and b < (∑ a i ) 2 /(n - 1) - ∑ a i 2 . Show that b < 2a i a j for all distinct i, j.	proof	0	analysisWe need to parse the problem carefully. It's from Putnam 1977 B5? Wait B5? The problem states: "38th Putnam 1977 Problem B5 a 1 , a 2 , ... , a n are real and b < (∑ a i ) 2 /(n - 1) - ∑ a i 2 . Show that b < 2a i a j for all distinct i, j." So we have real numbers a_1,...,a_n and a real number b such that b ...	analysisWe need to parse the problem statement: "38th Putnam 1977 Problem B5 a 1 , a 2 , ... , a n are real and b < (∑ a i )^2 /(n - 1) - ∑ a i^2 . Show that b < 2 a_i a_j for all distinct i, j." But the wording seems garbled. Let's reconstruct: The problem likely states: Let a1, a2, ..., an be real numbers and b be a...	analysisWe need to analyze problem: "38th Putnam 1977 Problem B5 a 1 , a 2 , ... , a n are real and b < (∑ a_i)^2/(n-1) - ∑ a_i^2 . Show that b < 2 a_i a_j for all distinct i,j." We must prove: For any real numbers a_1,...,a_n, given a real number b such that b < ( (∑ a_i)^2 )/(n-1) - ∑ a_i^2 then for all distinct i...	analysisWe need to parse the problem: "38th Putnam 1977 Problem B5 a 1 , a 2 , ... , a n are real and b < (∑ a i )^2/(n - 1) - ∑ a i^2 . Show that b < 2 a_i a_j for all distinct i, j." Wait maybe mis-typed. Let's read: "38th Putnam 1977 Problem B5 a 1 , a 2 , ... , a n are real and b < (∑ a i ) 2 /(n - 1) - ∑ a i 2 . S...	analysisWe need to parse problem: "38th Putnam 1977 Problem B5 a 1 , a 2 , ... , a n are real and b < (∑ a i )^2/(n - 1) - ∑ a i^2 . Show that b < 2 a_i a_j for all distinct i, j." So we have real numbers a_i, i=1..n, and a real number b. Condition: b < (∑ a_i)^2/(n-1) - ∑ a_i^2. Need to show that b < 2 a_i a_j for al...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
$\mathrm{Al}$ and Bert must arrive at a town $22.5 \mathrm{~km}$ away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at $8 \mathrm{~km} / \mathrm{h}$, leaves the bicycle and then walks at $5 \mathrm{~km} / \mathrm{h}$. Al walks at $4 \mathrm{~km} / \mathrm{h}$, reaches the bi...	75	100	analysisWe need to parse the problem: Al and Bert must arrive at a town 22.5 km away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at 8 km/h, leaves the bicycle and then walks at 5 km/h. Al walks at 4 km/h, reaches the bicycle and rides at 10 km/h. For how many minutes was t...	analysisWe need to parse the problem: Two people Al and Bert. They must arrive at a town 22.5 km away. They have one bicycle. They must arrive at the same time. Bert sets out riding at 8 km/h, leaves the bicycle and then walks at 5 km/h. Al walks at 4 km/h, reaches the bicycle and rides at 10 km/h. For how many minutes...	analysisWe need to parse the problem: Al and Bert must arrive at a town 22.5 km away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at 8 km/h, leaves the bicycle and then walks at 5 km/h. Al walks at 4 km/h, reaches the bicycle and rides at 10 km/h. For how many minutes was t...	analysisWe need to interpret the problem: "Al and Bert must arrive at a town 22.5 km away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at 8 km/h, leaves the bicycle and then walks at 5 km/h. Al walks at 4 km/h, reaches the bicycle and rides at 10 km/h. For how many minutes ...	analysisWe need to understand the problem: There is a town 22.5 km away. Two people: Al and Bert. They have one bicycle. They must arrive at the same time. Bert starts riding at 8 km/h, leaves the bicycle and then walks at 5 km/h. Al walks at 4 km/h, reaches the bicycle and rides at 10 km/h. We are asked: For how many ...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
3. If $a, b, c$ are three positive real numbers, prove that $$ \frac{a^{2}+1}{b+c}+\frac{b^{2}+1}{c+a}+\frac{c^{2}+1}{a+b} \geq 3 $$	proof	0	analysisWe need to prove inequality: for positive reals a,b,c: S = (a^2+1)/(b+c) + (b^2+1)/(c+a) + (c^2+1)/(a+b) >= 3. We need rigorous proof. Recognize symmetry. Maybe use Nesbitt or Titu's lemma? Let's try. We can attempt to rewrite as sum of fractions. Use Cauchy-Schwarz Engel form: sum (x^2/(y+z)) >= (x+y+z)^2/(...	analysisWe need to prove inequality: For positive reals a,b,c: S = (a^2+1)/(b+c) + (b^2+1)/(c+a) + (c^2+1)/(a+b) ≥ 3. We can attempt to use known inequalities: Nesbitt-like or Cauchy-Schwarz or Titu's lemma or rearrangement. Observe that (a^2+1)/(b+c) = (a^2+1)(a+b+c)/( (b+c)(a+b+c)). Not helpful. We can try to re...	analysisWe need to prove inequality: for positive reals a,b,c, S = (a^2+1)/(b+c) + (b^2+1)/(c+a) + (c^2+1)/(a+b) >= 3. We must provide rigorous proof, maybe using known inequalities: Cauchy-Schwarz, Titu's lemma, Nesbitt's, or rearrangement. Let's analyze. We can try to prove stronger inequality: (a^2+1)/(b+c) >= so...	analysisWe need to prove inequality: For positive reals a,b,c: S = (a^2+1)/(b+c) + (b^2+1)/(c+a) + (c^2+1)/(a+b) >= 3. We must provide rigorous proof. We can attempt to use known inequalities: Nesbitt-like or Cauchy-Schwarz, Titu's lemma (Cauchy Engel). Let's analyze. Let’s consider using Cauchy-Schwarz (Titu's le...	analysisWe need to prove inequality: For positive real numbers a,b,c, show S = (a^2+1)/(b+c) + (b^2+1)/(c+a) + (c^2+1)/(a+b) ≥ 3. We need to produce rigorous proof. Let's analyze. We can attempt using Cauchy-Schwarz or Titu's lemma (Cauchy Engel). For positive reals, Titu's lemma: sum (x_i^2 / y_i) ≥ (sum x_i)^2 / s...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
6. Let $f(x), g(x)$ be two functions defined on $(-\infty,+\infty)$, for any real numbers $x, y$, satisfying $f(x+y) +$ $f(x-y)=2 f(x) \cdot g(y)$. If $f(0)=0$, but $f(x)$ is not identically zero, then A. $f(x), g(x)$ are both odd functions B. $f(x), g(x)$ are both even functions C. $f(x)$ is an even function, $g(x)$ i...	D	0	analysisWe need to analyze functional equation: f(x+y)+f(x-y)=2 f(x) g(y). Domain all real numbers. f(0)=0, f not identically zero. Determine parity of f and g. Options: both odd, both even, f even g odd, f odd g even. We need to find parity properties from functional equation. Let’s denote equation (E): f(x+y)+f(x-y...	analysisWe need to analyze functional equation: f(x+y)+f(x-y)=2 f(x) g(y). Domain all real numbers. Condition: f(0)=0, f not identically zero. Determine parity properties of f and g. We must analyze functional equation for parity. We can attempt to plug y=0: f(x)+f(x)=2 f(x) g(0) => 2 f(x)=2 f(x) g(0). Since f(x) not...	analysisWe need to analyze functional equation: f(x+y)+f(x-y)=2 f(x) g(y). Domain all real numbers. Given f(0)=0, but f not identically zero. We need to determine parity of f and g. Options: A: both odd, B: both even, C: f even, g odd, D: f odd, g even. We must deduce parity from equation. Let’s analyze. We have ...	analysisWe need to analyze functional equation: f(x+y)+f(x-y)=2 f(x) g(y). Domain all reals. Condition f(0)=0, f not identically zero. Determine parity of f and g. We can attempt to deduce properties: plug y=0: f(x)+f(x)=2 f(x) g(0) => 2 f(x) = 2 f(x) g(0) => for all x, f(x) (1 - g(0)) = 0. Since f not identically zer...	analysisWe need to analyze functional equation: f(x+y)+f(x-y)=2 f(x) g(y). Domain all real numbers. Condition: f(0)=0, but f not identically zero. Determine parity of f and g. We have a functional equation reminiscent of d'Alembert's functional equation: f(x+y)+f(x-y)=2 f(x) f(y) for cosine, but here it's 2 f(x) g(y)....	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
34.19. Prove that the functional equation $$ \frac{f(x)-g(y)}{x-y}=\varphi\left(\frac{x+y}{2}\right) $$ can be reduced to the functional equation (2)	proof	0	analysisWe need to understand the problem statement: "34.19. Prove that the functional equation $$ \frac{f(x)-g(y)}{x-y}=\varphi\left(\frac{x+y}{2}\right) $$ can be reduced to the functional equation (2)." We need to know what functional equation (2) refers to. In the book or context, maybe equation (2) is a known f...	analysisWe need to understand the problem: It states: "Prove that the functional equation $$ \frac{f(x)-g(y)}{x-y}=\varphi\left(\frac{x+y}{2}\right) $$ can be reduced to the functional equation (2)." We need to know what functional equation (2) is. It refers to a previous part of the problem set (maybe from a textbo...	analysisWe need to parse the problem: "34.19. Prove that the functional equation $$ \frac{f(x)-g(y)}{x-y}=\varphi\left(\frac{x+y}{2}\right) $$ can be reduced to the functional equation (2)" We need to know what functional equation (2) is. Since we don't have the context, we need to guess. This is from a functional e...	analysisWe need to analyze the problem: "34.19. Prove that the functional equation $$\frac{f(x)-g(y)}{x-y}=\varphi\left(\frac{x+y}{2}\right)$$ can be reduced to the functional equation (2)." We need to know what is functional equation (2). Since it's a reference to a previous part, we need to guess: maybe equation (...	analysisWe need to analyze the problem. We have a functional equation: $$\frac{f(x)-g(y)}{x-y} = \varphi\left(\frac{x+y}{2}\right).$$ We need to prove that this can be reduced to the functional equation (2). But we don't know what functional equation (2) is. In the retrieved example, there is functional equation: f(x...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
4. If $x$ and $y$ are real numbers for which $\|x\|+x+5 y=2$ and $\|y\|-y+x=7$, find the value of $x+y$. (A) -3 (B) -1 (C) 1 (D) 3 (E) 5	3	100	analysisWe have problem: solve system with absolute values: \|x\| + x + 5y = 2 and \|y\| - y + x = 7. Find x + y. Options: -3, -1, 1, 3, 5. We need to solve for real numbers. We need to analyze cases based on sign of x and y. Let's denote: Equation (1): \|x\| + x + 5y = 2. Equation (2): \|y\| - y + x = 7. We need to find po...	analysisWe need to solve system: \|x\| + x + 5y = 2 \|y\| - y + x = 7 We need x and y real. Need x + y. We must consider cases based on signs of x and y. Let's analyze. Let’s denote: equation (1) \|x\| + x + 5y = 2. Equation (2) \|y\| - y + x = 7. We need to analyze possibilities: x≥0 or x<0; y≥0 or y<0. So four cases. L...	analysisWe need to solve system: \|x\| + x + 5y = 2 \|y\| - y + x = 7 We need x + y. We must consider sign cases for x and y. Let's analyze. Let’s denote equations: (1) \|x\| + x + 5y = 2 (2) \|y\| - y + x = 7 We need real solutions. Let's consider cases: Case 1: x >= 0, y >= 0 Case 2: x >= 0, y < 0 Case 3: x < 0, y >= ...	analysisWe need to solve system: \|x\| + x + 5y = 2 \|y\| - y + x = 7 We need x+y. There are absolute values. Approach: consider cases for signs of x and y. Let’s analyze. Equation1: \|x\| + x + 5y = 2. Case 1: x >= 0. Then \|x\| = x. So equation becomes x + x + 5y = 2 => 2x + 5y = 2. Case 2: x < 0. Then \|x\| = -x. Equati...	analysisWe need to solve system of equations with absolute values. We have equations: 1) \|x\| + x + 5y = 2 2) \|y\| - y + x = 7 We need x + y. We need to consider sign of x and y. Let's analyze. Let’s denote cases: Case 1: x ≥ 0, y ≥ 0. Then \|x\| = x, \|y\| = y. Equations become: 1) x + x + 5y = 2 => 2x + 5y = 2 2) y -...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
4・129 Solve the system of equations $\left\{\begin{array}{l}x^{2}=6+(y-z)^{2}, \\ y^{2}=2+(z-x)^{2}, \\ z^{2}=3+(x-y)^{2} .\end{array}\right.$	\begin{pmatrix}x_{1}=\frac{5}{2},y_{1}=\frac{3}{2},z_{1}=2\\x_{2}=\frac{-5}{2},y_{2}=\frac{-3}{2},z_{2}=-2\end{pmatrix}	0	analysisWe need to solve system: x^2 = 6 + (y - z)^2 y^2 = 2 + (z - x)^2 z^2 = 3 + (x - y)^2 We can attempt similar method: bring terms to one side: x^2 - (y - z)^2 = 6. Similarly others. Use difference of squares: (x - (y - z))(x + (y - z)) = 6. But careful: x^2 - (y - z)^2 = (x - (y - z))(x + (y - z)) = (x - y + z)...	analysisWe have system: x^2 = 6 + (y - z)^2 y^2 = 2 + (z - x)^2 z^2 = 3 + (x - y)^2 We need to solve for real numbers presumably. Let's analyze. We can rewrite each as: x^2 - (y - z)^2 = 6 y^2 - (z - x)^2 = 2 z^2 - (x - y)^2 = 3 Use difference of squares: a^2 - b^2 = (a-b)(a+b). So: (x - (y - z))(x + (y - z)) = 6...	analysisWe need to solve system: x^2 = 6 + (y - z)^2 y^2 = 2 + (z - x)^2 z^2 = 3 + (x - y)^2 We can try similar approach: move square differences: x^2 - (y - z)^2 = 6; y^2 - (z - x)^2 = 2; z^2 - (x - y)^2 = 3. We can use difference of squares: a^2 - b^2 = (a-b)(a+b). Here a = x, b = y - z. But careful: (y - z) may b...	analysisWe need solve system: x^2 = 6 + (y - z)^2 y^2 = 2 + (z - x)^2 z^2 = 3 + (x - y)^2 We look for real solutions? Usually yes. Solve. We can use similar technique: subtract equations? Let's denote differences. We can rewrite each as: x^2 - (y - z)^2 = 6 y^2 - (z - x)^2 = 2 z^2 - (x - y)^2 = 3 Now use differen...	analysisWe have system: x^2 = 6 + (y - z)^2 y^2 = 2 + (z - x)^2 z^2 = 3 + (x - y)^2 We need solve for real numbers? Usually yes. Let's analyze. Let’s denote differences: a = y - z, b = z - x, c = x - y. Note that a + b + c = 0? Let's check: a + b + c = (y - z) + (z - x) + (x - y) = 0. So indeed a+b+c=0. Also note th...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Example 49 Non-negative real numbers $a, b, c$ satisfy $(a+b)(b+c)(c+a)=2$. Prove: $$\left(a^{2}+b c\right)\left(b^{2}+c a\right)\left(c^{2}+a b\right) \leqslant 1$$	proof	0	analysisWe need to prove inequality: For non-negative real numbers a,b,c with (a+b)(b+c)(c+a)=2, show (a^2 + bc)(b^2 + ca)(c^2 + ab) <= 1. We need to find key structure: maybe using substitution or AM-GM or inequalities. Let’s denote S1 = a+b, S2 = b+c, S3 = c+a. Then product S1S2S3 = 2. We need to express a^2+bc e...	analysisWe need to prove inequality: For nonnegative real numbers a,b,c with (a+b)(b+c)(c+a)=2, prove (a^2+bc)(b^2+ca)(c^2+ab) ≤ 1. We need to analyze structure. Let’s denote x=a+b, y=b+c, z=c+a. Then given xyz=2. We want to express product (a^2+bc)(b^2+ca)(c^2+ab) in terms of x,y,z? Let's attempt. We have a,b,c non...	analysisWe need to solve: For nonnegative reals a,b,c with (a+b)(b+c)(c+a)=2, prove (a^2+bc)(b^2+ca)(c^2+ab) ≤ 1. We need to use inequalities. Maybe use substitution: Let x=a+b, y=b+c, z=c+a. Then product xyz=2. Express a,b,c in terms of x,y,z? Solve: a = (x+z-y)/2? Wait: a = (x+z - y)/2, b = (x+y - z)/2, c = (y+z - x...	analysisWe need to prove inequality: for nonnegative a,b,c with (a+b)(b+c)(c+a)=2, prove (a^2+bc)(b^2+ca)(c^2+ab) ≤ 1. We suspect symmetry and maybe use substitution. Let’s set x=a+b, y=b+c, z=c+a. Then (x)(y)(z)=2. We need to express a^2+bc in terms of x,y,z. We know a = (x+z - y)/2, b = (x+y - z)/2, c = (y+z - x)/2....	analysisWe need to prove inequality: For nonnegative real numbers a,b,c with (a+b)(b+c)(c+a) = 2, show product (a^2+bc)(b^2+ca)(c^2+ab) <= 1. We need to find the maximum of product given constraint. Equivalent to show maximum <=1. We can attempt using AM-GM or substitution. Since (a+b)(b+c)(c+a)=2. Let’s denote x = a...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "inequalities" }
7. For any $x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right]$, the inequality $\sin ^{2} x+a \sin x+a+3 \geqslant 0$ always holds. Then the range of the real number $a$ is $\qquad$ .	[-2,+\infty)	100	analysisWe need to analyze inequality: sin^2 x + a sin x + a + 3 >= 0 for all x in interval [-π/6, π/2]. Find range of a such that inequality holds for all x in that interval. We need to find minimal value of expression over x in that interval, as function of a, and require >=0. Let f(x) = sin^2 x + a sin x + a + 3. ...	analysisWe need to find all real a such that for all x in interval [-π/6, π/2], the inequality sin^2 x + a sin x + a + 3 >= 0 holds. Let's analyze function f(x)=sin^2 x + a sin x + a + 3. We need f(x) >= 0 for all x in given interval. Let t = sin x. On the interval [-π/6, π/2], sin x ranges from sin(-π/6) = -1/2 to si...	analysisWe need to analyze the inequality: sin^2 x + a sin x + a + 3 >= 0 for all x in [-π/6, π/2]. Find all real a such that the inequality holds for all x in that interval. Let t = sin x. On that interval, sin x ranges from sin(-π/6) = -1/2 to sin(π/2) = 1. But note that sin x is monotonic increasing on [-π/6, π/2] ...	analysisWe need to analyze the inequality sin^2 x + a sin x + a + 3 >= 0 for all x in [-π/6, π/2]. Determine all real a such that the inequality holds for all x in that interval. We can treat sin x as variable t but t's range is sin x for x in [-π/6, π/2]. Let's compute that range. sin(-π/6) = -1/2, sin(π/2) = 1. The ...	analysisWe need to find all real a such that for all x in [-π/6, π/2], sin^2 x + a sin x + a + 3 >= 0. Equivalent to quadratic in sin x: f(t) = t^2 + a t + a + 3, where t = sin x. But t's range over that interval is not full [-1,1], but only sin values from sin(-π/6) = -1/2 to sin(π/2) = 1. However sin is increasing on...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
High $\mathbf{5 2 2}$ Let $a, b, c \in \mathbf{R}_{+}$. Prove: $$ \begin{aligned} \sum & \left(\frac{a^{2}}{c}+\frac{c^{2}}{a}\right)+7 \sum a \\ & \geqslant \frac{\left(\sum a\right)^{3}}{\sum a b}+\frac{2\left(\sum a b\right)^{2}}{a b c}, \end{aligned} $$ where, “ $\sum$ ” denotes the cyclic sum.	proof	0	analysisWe need to prove inequality: For positive reals a,b,c, Sum_{cyc} (a^2/c + c^2/a) + 7 sum a >= (sum a)^3/(sum ab) + 2 (sum ab)^2/(abc). Let’s denote: S1 = a+b+c S2 = ab+bc+ca S3 = abc Left side LHS = sum_{cyc} (a^2/c + c^2/a) + 7S1. Note that sum_{cyc} a^2/c = a^2/b + b^2/c + c^2/a? Wait cyc: (a^2/c + c^2/a...	analysisWe need to prove inequality: For positive reals a,b,c, S1 = sum cyclic (a^2/c + c^2/a) + 7 sum a >= (sum a)^3 / (sum ab) + 2 (sum ab)^2 / (abc) where sum over cyclic: sum a = a+b+c, sum ab = ab+bc+ca. We must prove: ∑ (a^2/c + c^2/a) + 7(a+b+c) >= (a+b+c)^3/(ab+bc+ca) + 2(ab+bc+ca)^2/(abc). Let’s denote:...	analysisWe need to prove inequality for positive reals a,b,c: Sum cyclic (a^2/c + c^2/a) + 7 sum a >= (sum a)^3/(sum ab) + 2 (sum ab)^2/(abc). Let's denote: S1 = a+b+c S2 = ab+bc+ca S3 = abc We need to prove: ∑ (a^2/c + c^2/a) + 7 S1 >= S1^3/S2 + 2 S2^2/S3. Note that ∑ (a^2/c + c^2/a) = ∑ (a^3 + c^3)/(ac). But ma...	analysisWe need to prove inequality for positive real numbers a,b,c: Sum_{cyc} (a^2/c + c^2/a) + 7 sum a ≥ (sum a)^3 / sum ab + 2 (sum ab)^2 / (abc). Let me denote: S1 = a + b + c S2 = ab + bc + ca S3 = abc We have cyclic sum of a^2/c + c^2/a: compute: a^2/c + c^2/a = (a^3 + c^3) / (ac). Sum cyclic: sum_{cyc} (a^...	analysisWe need to prove inequality for positive reals a,b,c: Sum (a^2/c + c^2/a) + 7 sum a ≥ (sum a)^3 / sum ab + 2 (sum ab)^2 / (abc) Let’s denote: S1 = a + b + c S2 = ab + bc + ca S3 = abc We need to prove: Σ (a^2/c + c^2/a) + 7 S1 ≥ S1^3 / S2 + 2 S2^2 / S3 First compute left side Σ (a^2/c + c^2/a) cyclic: For...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
10. (3 points) The school has a total of 20 soccer balls and basketballs, which can accommodate 96 students playing at the same time. Each soccer ball is shared by 6 students, and each basketball is shared by 3 students. Among them, there are $\qquad$ soccer balls.	12	100	analysisWe need to parse the problem statement. It says: "The school has a total of 20 soccer balls and basketballs, which can accommodate 96 students playing at the same time. Each soccer ball is shared by 6 students, and each basketball is shared by 3 students. Among them, there are ___ soccer balls." So we have tot...	analysisWe need to parse the problem. It says: "The school has a total of 20 soccer balls and basketballs, which can accommodate 96 students playing at the same time. Each soccer ball is shared by 6 students, and each basketball is shared by 3 students. Among them, there are ______ soccer balls." We need to find the n...	analysisWe need to parse the problem carefully. It says: "The school has a total of 20 soccer balls and basketballs, which can accommodate 96 students playing at the same time. Each soccer ball is shared by 6 students, and each basketball is shared by 3 students. Among them, there are ___ soccer balls." So we need to f...	analysisWe need to parse the problem statement: "The school has a total of 20 soccer balls and basketballs, which can accommodate 96 students playing at the same time. Each soccer ball is shared by 6 students, and each basketball is shared by 3 students. Among them, there are ______ soccer balls." We need to find numb...	analysisWe need to parse problem. "The school has a total of 20 soccer balls and basketballs, which can accommodate 96 students playing at the same time. Each soccer ball is shared by 6 students, and each basketball is shared by 3 students. Among them, there are ______ soccer balls." So we need to find number of soccer...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
2. Prove that there does not exist an injective function $f: \mathbb{R} \rightarrow \mathbb{R}$ for which $$ f(f(x)+y)=f(x+y)+f(2012) \quad \text { for all } x, y \in \mathbb{R} $$	proof	0	analysisWe need to prove no injective function f: R→R satisfies the functional equation: f(f(x)+y) = f(x+y)+f(2012) for all real x,y. We need to show impossible. Let's analyze. Let property P(x,y): f(f(x)+y) = f(x+y)+f(2012). We need to use injectivity: if f(a)=f(b) then a=b. We can try to find contradictions. Con...	analysisWe need to prove no injective function satisfies that functional equation. Let's analyze. Equation: f(f(x)+y) = f(x+y) + f(2012) for all reals x,y. We need to show impossibility. We can attempt to deduce properties: Let’s denote c = f(2012). The equation: f(f(x)+y) = f(x+y) + c. We can attempt to show f is ...	analysisWe need to prove no injective f satisfies f(f(x)+y)=f(x+y)+f(2012). Let's analyze. We have functional equation: f(f(x)+y) = f(x+y) + f(2012). For all real x, y. Let’s denote c = f(2012). Then equation: f(f(x)+y) = f(x+y) + c. Goal: show no injective f exists. We can attempt to derive contradictions. Suppose...	analysisWe need to prove no injective function f: R->R satisfies functional equation: f(f(x)+y) = f(x+y)+f(2012) for all x,y. Goal: show impossible. We can attempt to analyze. Let’s denote property P(x,y): f(f(x)+y) = f(x+y)+c where c=f(2012). Note c is constant. We can attempt to see if there is a solution. Suppos...	analysisWe need to prove no injective function satisfies f(f(x)+y)=f(x+y)+f(2012). Domain codomain R. We need to show contradiction. Let's analyze. Let P(x,y): f(f(x)+y)=f(x+y)+f(2012). (1) We need to consider injectivity: f(a)=f(b) implies a=b. Goal: no such function exists. We can attempt to derive contradiction...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Folklore The function $f(x)$ is defined for all $x$, except 1, and satisfies the equation: $(x-1) f\left(\frac{x+1}{x-1}\right)=x+f(x)$. Find $f(-1)$.	-1	100	analysisWe need to find f(-1). We have functional equation: (x-1) f((x+1)/(x-1)) = x + f(x). Domain: all real x except 1. We are asked to find f(-1). We may need to find explicit f? Might be enough to evaluate at some x values. Let's analyze. Let define transformation T(x) = (x+1)/(x-1). Domain: x ≠ 1. Also T(x) has d...	analysisWe need to find f(-1) given functional equation: (x-1) f((x+1)/(x-1)) = x + f(x). Domain: all real x except 1. Also argument (x+1)/(x-1) must be defined: denominator x-1 ≠ 0, so x≠1. Also we need to consider that function defined for all x except 1. So f is defined at all real numbers except maybe 1. But equati...	analysisWe need to solve functional equation: For all real x except 1, f defined except at 1. Equation: (x-1) f((x+1)/(x-1)) = x + f(x). We need f(-1). We don't need full function; maybe just compute value. Let’s analyze domain: x ∈ ℝ \ {1}. Also argument (x+1)/(x-1) may equal 1 for some x? Solve (x+1)/(x-1)=1 => x+1 ...	analysisWe need to solve functional equation: For all real x except 1? Domain: all x except 1? Actually f(x) defined for all x except 1. The equation: (x-1) f((x+1)/(x-1)) = x + f(x). We need f(-1). Let's analyze. Let’s denote transformation: T(x) = (x+1)/(x-1). Domain: x ≠ 1. Also T maps ℝ\{1} to ℝ\{1}? Let's compute...	analysisWe need to solve functional equation: for all x except 1, (x-1) f( (x+1)/(x-1) ) = x + f(x). Domain: all real numbers except 1 (maybe also except something else? The argument (x+1)/(x-1) might be undefined at x=1; but x is not 1, so fine. But also we need f defined at all real numbers except 1. So f domain: ℝ \...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
G3.4 Let $x \geq 0$ and $y \geq 0$. Given that $x+y=18$. If the maximum value of $\sqrt{x}+\sqrt{y}$ is $d$, find the value of $d$. (Reference: 1999 FGS.2)	6	100	analysisWe need to find maximum of sqrt(x)+sqrt(y) subject to x>=0, y>=0, x+y=18. This is a classic optimization problem. Use concavity: sqrt is concave, so by Jensen, sqrt(x)+sqrt(y) <= 2 sqrt((x+y)/2) = 2 sqrt(9) = 6. But we must check domain: x,y nonnegative; equality occurs when x=y=9. So maximum d=6. But maybe che...	analysisWe have problem: Let x≥0, y≥0, x+y=18. Find maximum value of sqrt(x)+sqrt(y). Classic. We can solve by AM-QM or Cauchy. Let f(x)=√x+√(18-x). For x∈[0,18]. Since function is concave? sqrt is concave, sum of concave functions, maximum at endpoints? Wait concave function sum concave, maximum at endpoints? Actually...	analysisWe have a simple optimization: maximize sqrt(x)+sqrt(y) subject to x+y=18, x,y≥0. Since sqrt is concave, by Jensen the maximum occurs at extremes? Wait concave: sqrt is concave, sum of concave functions with linear constraint: the sum is concave, so maximum occurs at boundary? Actually concave functions: maximu...	analysisWe need to find maximum of sqrt(x)+sqrt(y) subject to x,y≥0 and x+y=18. Classic: maximize sum of square roots given sum fixed. Use Jensen or concavity: sqrt is concave, so by Jensen, sum ≤ 2 sqrt((x+y)/2) = 2 sqrt(9) = 6. But we need to check boundary: since sqrt concave, maximum at extreme? Wait, for concave f...	analysisWe need to find maximum of sqrt(x)+sqrt(y) subject to x,y≥0, x+y=18. Standard optimization: for fixed sum, sum of square roots is maximized when the variables are equal? Let's check concavity: sqrt(t) is concave. Sum of concave functions with linear constraint: maximum at extreme points? Wait concave function s...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
3.223. $4 \cos \alpha \cos \varphi \cos (\alpha-\varphi)-2 \cos ^{2}(\alpha-\varphi)-\cos 2 \varphi=\cos 2 \alpha$.	proof	0	analysisWe need to prove identity: $4 \cos \alpha \cos \varphi \cos(\alpha-\varphi)-2 \cos^2(\alpha-\varphi)-\cos 2\varphi = \cos 2\alpha$. We must show equality holds for all α, φ. Approach: use trigonometric identities. Let's denote variables. We can use product-to-sum: 4 cosα cosφ cos(α-φ). We can also express cos...	analysisWe need to prove identity: 4 cos α cos φ cos(α-φ) - 2 cos^2(α-φ) - cos 2φ = cos 2α. We should verify identity, maybe transform LHS to RHS. We can use trigonometric identities: cos(A-B) = cos A cos B + sin A sin B. Also cos 2x = 2 cos^2 x -1 = 1 - 2 sin^2 x. Goal: Show LHS equals cos 2α. Let me denote variabl...	analysisWe need to prove identity: 4 cos α cos φ cos(α-φ) - 2 cos^2(α-φ) - cos 2φ = cos 2α. We must show equality for all α, φ presumably real. Let's analyze. Goal: Show left side simplifies to cos 2α. Let's attempt using trig identities. Let’s denote A = α, B = φ. Then expression: 4 cos A cos B cos(A-B) - 2 cos^2(A-...	analysisWe need to prove the identity: 4 cos α cos φ cos(α-φ) - 2 cos^2(α-φ) - cos 2φ = cos 2α. Goal: show equality holds for all α, φ (real). Use trigonometric identities. We can try to use product-to-sum, double-angle formulas. Let's analyze. Let’s denote A = α, B = φ. Expression: 4 cos A cos B cos(A-B) - 2 cos^...	analysisWe need to prove identity: 4 cosα cosφ cos(α-φ) - 2 cos^2(α-φ) - cos 2φ = cos 2α. Let's analyze. Goal: simplify LHS to RHS. Let’s denote A = α, B = φ. Use trig identities. We can attempt to use product-to-sum: 4 cos A cos B cos(A-B). Recognize identity: 4 cos A cos B cos(A-B) = 2 cos(A-B) (2 cos A cos B). Bu...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
3.048. $\frac{\left(\sin ^{2} \alpha+\tan^{2} \alpha+1\right)\left(\cos ^{2} \alpha-\cot^{2} \alpha+1\right)}{\left(\cos ^{2} \alpha+\cot^{2} \alpha+1\right)\left(\sin ^{2} \alpha+\tan^{2} \alpha-1\right)}=1$.	proof	0	analysisWe need to evaluate expression: ((sin^2 α + tan^2 α + 1)(cos^2 α - cot^2 α + 1))/((cos^2 α + cot^2 α + 1)(sin^2 α + tan^2 α - 1)) = 1? Actually problem likely to prove identity equals 1. We need to simplify. Use identities: tan^2 = sin^2/cos^2; cot^2 = cos^2/sin^2. Also sin^2 + cos^2 = 1. Let's compute numerato...	analysisWe need to prove identity: (sin^2 α + tan^2 α + 1)(cos^2 α - cot^2 α + 1) / ((cos^2 α + cot^2 α + 1)(sin^2 α + tan^2 α - 1)) = 1. Equivalent to numerator equals denominator. We can transform each term using identities: tan^2 = sin^2 / cos^2, cot^2 = cos^2 / sin^2. Also sin^2 + cos^2 = 1. Let’s denote s = sin ...	analysisWe need to verify expression equals 1. We have expression: E = ((sin^2 α + tan^2 α + 1)(cos^2 α - cot^2 α + 1)) / ((cos^2 α + cot^2 α + 1)(sin^2 α + tan^2 α - 1)) Simplify using identities: tan^2 α = sin^2/ cos^2. cot^2 = cos^2 / sin^2. Also sin^2 + cos^2 = 1. Let’s compute numerator terms. Let A = sin^2 + ...	analysisWe need to prove identity: expression equals 1. Let's analyze. We have numerator: (sin^2 α + tan^2 α + 1)(cos^2 α - cot^2 α + 1). Denominator: (cos^2 α + cot^2 α + 1)(sin^2 α + tan^2 α - 1). Need to show ratio =1 for all α where defined. Let's denote sin^2 α = s, cos^2 α = c. Also tan^2 = s/c, cot^2 = c/s. So e...	analysisWe need to verify the identity: \[ \frac{(\sin^2 \alpha + \tan^2 \alpha + 1)(\cos^2 \alpha - \cot^2 \alpha + 1)}{(\cos^2 \alpha + \cot^2 \alpha + 1)(\sin^2 \alpha + \tan^2 \alpha - 1)} = 1 \] Simplify each factor. Use identities: $\tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha}$, \(\cot^2 \alpha = \fra...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
3.071. $\sin ^{2}\left(\frac{\alpha}{2}+2 \beta\right)-\sin ^{2}\left(\frac{\alpha}{2}-2 \beta\right)$.	\sin\alpha\sin4\beta	100	analysisWe need to compute expression: sin^2(α/2 + 2β) - sin^2(α/2 - 2β). Use identity sin^2 x - sin^2 y = sin(x+y) sin(x-y)? Wait: sin^2 A - sin^2 B = (sin A - sin B)(sin A + sin B). Alternatively, use sin^2 x = (1 - cos 2x)/2. Then difference becomes (1/2)(-cos(α + 4β) + cos(α - 4β)). Because 2*(α/2 + 2β) = α + 4β; 2...	analysisWe need to evaluate the expression: sin^2(α/2 + 2β) - sin^2(α/2 - 2β) We can use identity sin^2 x - sin^2 y = sin(x + y) sin(x - y). Wait formula: sin^2 u - sin^2 v = sin(u+v) sin(u-v). Let's verify: sin^2 u - sin^2 v = (1 - cos 2u)/2 - (1 - cos 2v)/2 = (cos 2v - cos 2u)/2 = -(cos 2u - cos 2v)/2. Using identi...	analysisWe need to compute expression: sin^2(α/2 + 2β) - sin^2(α/2 - 2β). Use identity sin^2 x - sin^2 y = sin(x+y) sin(x-y)? Wait identity: sin^2 A - sin^2 B = sin(A+B) sin(A-B). Actually recall: sin^2 A - sin^2 B = (sin A - sin B)(sin A + sin B). Use sum-to-product: sin A - sin B = 2 cos((A+B)/2) sin((A-B)/2). sin A ...	analysisWe need to compute sin^2(A) - sin^2(B) where A = α/2 + 2β, B = α/2 - 2β. Use identity sin^2x - sin^2y = sin(x-y) sin(x+y)? Wait identity: sin^2x - sin^2y = sin(x-y) sin(x+y). Because sin^2x - sin^2y = (1/2)(1 - cos2x) - (1/2)(1 - cos2y) = (1/2)(-cos2x + cos2y) = (1/2)(cos2y - cos2x). Using cos difference identi...	analysisWe need to compute expression: sin^2(α/2 + 2β) - sin^2(α/2 - 2β). Use identity sin^2 x = (1 - cos 2x)/2. Then difference becomes (1/2)[cos(2(α/2 - 2β)) - cos(2(α/2 + 2β))] = (1/2)[cos(α - 4β) - cos(α + 4β)]. We can use identity cos A - cos B = -2 sin((A+B)/2) sin((A-B)/2). So cos(α - 4β) - cos(α + 4β) = -2 s...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
$4.44 \operatorname{tg} 2 \alpha+\operatorname{ctg} 2 \alpha+\operatorname{tg} 6 \alpha+\operatorname{ctg} 6 \alpha=\frac{8 \cos ^{2} 4 \alpha}{\sin 12 \alpha}$. $4.44 \tan 2 \alpha+\cot 2 \alpha+\tan 6 \alpha+\cot 6 \alpha=\frac{8 \cos ^{2} 4 \alpha}{\sin 12 \alpha}$.	\frac{8\cos^{2}4\alpha}{\sin12\alpha}	0	analysisWe need to solve equation: $4.44 \tan 2\alpha + \cot 2\alpha + \tan 6\alpha + \cot 6\alpha = \frac{8 \cos^2 4\alpha}{\sin 12\alpha}$. But likely 4.44 is 4.44? maybe 4.44? Could be 4.44? Might be 4.44? Might be 4.44? Might be 4.44? Wait maybe the coefficient is 4.44? But typical problem uses 4.44? Actually 4.4...	analysisWe need to solve equation: 4.44 tan 2α + cot 2α + tan 6α + cot 6α = (8 cos^2 4α)/ sin 12α. I think it's identity? But maybe they ask to prove equality? The problem statement: "4.44 tg 2 α+ctg 2 α+tg 6 α+ctg 6 α = (8 cos^2 4 α)/ sin 12 α." So maybe show equality holds for all α? But 4.44 is weird decimal; maybe ...	analysisWe need to evaluate the expression? The problem statement: "4.44 tg 2α + ctg 2α + tg 6α + ctg 6α = 8 cos^2 4α / sin 12α." It looks like an identity to prove: show that the left side equals the right side. Maybe the problem is to simplify LHS to RHS. So we must show equality holds for all α (except where undefin...	analysisWe need to solve identity? The problem states an equation? It might be to prove equality? It says "4.44 tan 2α + cot 2α + tan 6α + cot 6α = 8 cos^2 4α / sin 12α." Usually such problems ask to prove identity. So we need to show equality holds for all α? Let's verify. We need to compute left side and simplify to ...	analysisWe need to solve equation? The problem statement: "4.44 tg 2α + ctg 2α + tg 6α + ctg 6α = (8 cos^2 4α)/sin 12α". It seems to be identity? We need to verify or simplify left side to right side? Actually maybe it's an identity to prove. But the equation includes numbers 4.44? Wait 4.44 maybe means "4\sqrt{?}"? Ac...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
122. ("Home of Math Olympiads" website, 2008. 04. 14, provided by polynasia) Let $a, b, c \in \mathbf{R}^{+}$, and $a b c=1$, then $$\sum \frac{1}{a^{2}+2 b^{2}+3} \leqslant \frac{1}{2}$$	proof	0	analysisWe need to solve inequality: For positive real a,b,c with abc=1, prove that S = Σ_{cyc} 1/(a^2 + 2 b^2 + 3) <= 1/2. We need to find rigorous proof. Let's analyze structure. We need to sum over cyclic permutations: terms are 1/(a^2 + 2b^2 + 3), 1/(b^2 + 2c^2 + 3), 1/(c^2 + 2a^2 + 3). Condition abc=1. We need...	analysisWe need to prove: For positive real a,b,c with abc=1, prove sum cyclic 1/(a^2+2 b^2+3) ≤ 1/2. Sum over cyclic permutations: term1: 1/(a^2 + 2b^2 + 3); term2: 1/(b^2 + 2c^2 + 3); term3: 1/(c^2 + 2a^2 + 3). Show ≤ 1/2. We need to find approach. Maybe use substitution a= x/y etc? Because abc=1, we can set a=x/y,...	analysisWe need to prove inequality: For positive reals a,b,c with abc=1, show S = sum_{cyc} 1/(a^2 + 2b^2 + 3) <= 1/2. Where sum cyclic means sum over (a,b,c) cyclic: 1/(a^2+2b^2+3)+1/(b^2+2c^2+3)+1/(c^2+2a^2+3) <= 1/2. We need to prove inequality. Let's analyze. Given symmetry but not fully symmetric. Might use s...	analysisWe need to prove inequality: Given positive reals a,b,c with abc=1, show S = sum_{cyc} 1/(a^2 + 2b^2 + 3) <= 1/2. Sum cyclic: 1/(a^2+2b^2+3) + 1/(b^2+2c^2+3) + 1/(c^2+2a^2+3) <= 1/2. Goal: show S <= 1/2. We need to find a proof. Let's analyze structure. We have denominators: a^2 + 2b^2 + 3. Since abc=1, w...	analysisWe need to prove inequality: for positive reals a,b,c with abc=1, S = sum_{cyc} 1/(a^2 + 2 b^2 + 3) ≤ 1/2. We need to find approach. The retrieved example had similar structure but with numerator a and denominator a^2+2. Here numerator is 1 and denominator has a^2+2b^2+3. Maybe we can use Titu's lemma or Cauc...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "inequalities" }
Example 14 (1998 Canadian Olympiad Problem) Let $x_{1}, x_{2}, \cdots, x_{n+1}$ be positive real numbers satisfying the condition $\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\cdots+\frac{1}{1+x_{n+1}}=1$. Prove that: $x_{1} x_{2} \cdots x_{n+1} \geqslant n^{n-1}$.	proof	0	analysisWe need to prove inequality: For positive reals x_i (i=1..n+1) with sum of reciprocals 1/(1+x_i) = 1, we need to prove product >= n^{n-1}. Equivalent: product >= n^{n-1}. We can attempt to use AM-GM or Cauchy. The condition: sum_{i=1}^{n+1} 1/(1+x_i) = 1. Let y_i = 1/(1+x_i). Then y_i > 0, sum y_i = 1. Also, x_...	analysisWe need to prove inequality: Given positive reals x_i, i=1..n+1, satisfying sum_{i=1}^{n+1} 1/(1+x_i) = 1. Prove product x_i >= n^{n-1}. We suspect using AM-GM or Cauchy. Let's analyze. Let y_i = 1/(1+x_i). Then x_i = (1-y_i)/y_i = (1/y_i)-1. Since y_i>0 and 1/(1+x_i) < 1, so y_i in (0,1). Condition: sum y_i ...	analysisWe need to prove: given positive reals x1...x_{n+1} with sum of reciprocals of (1+xi) equals 1, then product >= n^{n-1}. We need to find minimal product given constraint. Use AM-GM? Let's analyze. We have sum_{i=1}^{n+1} 1/(1+xi)=1. Let yi = 1/(1+xi). Then xi = (1-yi)/yi = (1/yi)-1. Since yi>0 and sum yi=1, e...	analysisWe need to prove: For positive reals x_i (i=1..n+1) satisfying sum_{i=1}^{n+1} 1/(1+x_i) = 1, prove product x_i >= n^{n-1}. This is a known inequality. Let's analyze. Let y_i = 1/(1+x_i). Then x_i = (1-y_i)/y_i = (1/y_i)-1. Condition: sum y_i = 1, with each y_i in (0,1). We want product x_i = ∏ ((1-y_i)/y_i) ...	analysisWe need to prove inequality: given positive reals x1,...,x_{n+1} with sum of reciprocals of (1+xi) equals 1. Show product >= n^{n-1}. Equivalent to: For positive xi, sum_{i=1}^{n+1} 1/(1+xi) = 1, prove ∏ xi >= n^{n-1}. We can attempt to use AM-HM or Cauchy-Schwarz or Jensen. Let yi = 1/(1+xi). Then yi>0, sum ...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
$2.9 \quad \frac{x^{2}+x-5}{x}+\frac{3 x}{x^{2}+x-5}+4=0$. Solve the equation: $2.9 \quad \frac{x^{2}+x-5}{x}+\frac{3 x}{x^{2}+x-5}+4=0$.	x_{1}=-5,x_{2}=1,x_{3,4}=-1\\sqrt{6}	0	analysisWe need to solve equation: ( x^2 + x - 5 ) / x + (3 x) / (x^2 + x - 5) + 4 = 0. Let’s denote A = x^2 + x - 5. Then equation becomes: A / x + 3x / A + 4 = 0. We need to solve for real x? Probably all solutions (real and complex). But typical problem: find real solutions. Domain: x != 0, A != 0. Equation: A/...	analysisWe need to solve equation: ( x^2 + x - 5 ) / x + (3x) / (x^2 + x - 5) + 4 = 0. We must solve for x. Domain: x ≠ 0, and x^2 + x - 5 ≠ 0. Let’s set A = x^2 + x - 5. Then equation is A/x + 3x/A + 4 = 0. Multiply both sides by xA: A^2 + 3x^2 + 4xA = 0. But A = x^2 + x - 5. So compute A^2 = (x^2 + x - 5)^2. Th...	analysisWe need to solve equation: (x^2 + x - 5)/x + (3x)/(x^2 + x -5) + 4 = 0. Domain: x ≠ 0, and denominator x^2 + x -5 ≠ 0. Let’s denote A = x^2 + x -5. Then equation: A/x + 3x/A + 4 = 0. Multiply by xA: A^2 + 3x^2 + 4xA = 0. Expand A^2: (x^2 + x -5)^2. So equation: (x^2 + x -5)^2 + 3x^2 + 4x(x^2 + x -5) = 0. Simp...	analysisWe need to solve equation: (x^2 + x -5)/x + 3x/(x^2 + x -5) + 4 = 0. We must find all real solutions? Usually solving equation. Domain: x ≠ 0 and x^2 + x -5 ≠ 0. So x ≠ 0 and x ≠ roots of quadratic: x = [-1 ± sqrt(1+20)]/2 = (-1 ± sqrt21)/2. So domain excludes these. Equation: (x^2 + x -5)/x + 3x/(x^2 + x -5)...	analysisWe need to solve equation: ( x^2 + x - 5 ) / x + ( 3x ) / ( x^2 + x - 5 ) + 4 = 0. We must find solutions for x (real? complex?). Usually solving rational equation. Let's analyze. Let’s denote A = x^2 + x - 5. Then equation: A/x + 3x/A + 4 = 0. We need to solve for x ≠ 0 and A ≠ 0 (since denominators). So x...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
6. If $x \in\left(0, \frac{\pi}{2}\right)$, then the maximum value of the function $f(x)=2 \cos ^{3} x+3 \cos ^{2} x-6 \cos x-2 \cos 3 x$ is $\qquad$ .	\frac{1}{9}	80	analysisWe need to find maximum of f(x) = 2 cos^3 x + 3 cos^2 x - 6 cos x - 2 cos 3x for x in (0, π/2). Simplify using trig identities: cos 3x = 4 cos^3 x - 3 cos x. Then -2 cos 3x = -2(4 cos^3 x - 3 cos x) = -8 cos^3 x + 6 cos x. Combine with other terms. Compute: 2 cos^3 x - 8 cos^3 x = -6 cos^3 x. Next: 3 cos^2 x s...	analysisWe need to find maximum of f(x)=2 cos^3 x + 3 cos^2 x - 6 cos x - 2 cos 3x for x in (0, π/2). We can attempt to express in terms of cos x and cos 3x. Use identity cos 3x = 4 cos^3 x - 3 cos x. So -2 cos 3x = -2(4 cos^3 x - 3 cos x) = -8 cos^3 x + 6 cos x. Then f(x) = 2 cos^3 x + 3 cos^2 x - 6 cos x + (-8 cos^3 ...	analysisWe need maximum of f(x) for x in (0, π/2). Let's analyze. Let t = cos x. Since x∈(0,π/2), t∈(0,1). Also cos 3x expressed in terms of cos x: cos 3x = 4 cos^3 x - 3 cos x = 4 t^3 - 3 t. Then f(x) = 2 t^3 + 3 t^2 - 6 t - 2 cos 3x = 2 t^3 + 3 t^2 - 6 t - 2(4 t^3 - 3 t) = 2 t^3 + 3 t^2 - 6 t - 8 t^3 + 6 t = (2-8) ...	analysisWe need to find maximum of f(x) for x in (0, π/2). f(x) = 2 cos^3 x + 3 cos^2 x - 6 cos x - 2 cos 3x. Simplify using trig identities maybe. Express cos 3x in terms of cos x: cos 3x = 4 cos^3 x - 3 cos x. So -2 cos 3x = -2(4 cos^3 x - 3 cos x) = -8 cos^3 x + 6 cos x. Then f(x) = 2 cos^3 x + 3 cos^2 x - 6 cos x +...	analysisWe need to find maximum of f(x)=2 cos^3 x +3 cos^2 x -6 cos x - 2 cos 3x for x in (0, π/2). Let's analyze. We can express cos 3x in terms of cos x: cos 3x = 4 cos^3 x -3 cos x. Then f becomes: f = 2 cos^3 x + 3 cos^2 x -6 cos x - 2(4 cos^3 x -3 cos x) = 2 cos^3 x + 3 cos^2 x -6 cos x -8 cos^3 x +6 cos x = (2-...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
[Cubic Polynomials] One of the roots of the equation $x^{3}-6 x^{2}+a x-6=0$ is 3. Solve the equation. #	x_{1}=1,x_{2}=2,x_{3}=3	0	analysisWe need to solve cubic equation: x^3 - 6x^2 + a x - 6 = 0, with one root known: 3. Find the roots? "Solve the equation" likely means find all roots in terms of a? But a is a parameter. Wait: We know one root is 3; that gives condition on a maybe? Actually plugging x=3: 27 - 6*9 + 3a - 6 = 0 => 27 - 54 + 3a - 6 ...	analysisWe need to solve cubic equation x^3 - 6x^2 + a x - 6 = 0, given that one root is 3. So we can factor (x-3) out. Then we need to find a and remaining roots? The problem says: "Solve the equation." Usually means find all roots in terms of a? But a is a coefficient unknown; we might need to find a such that 3 is a...	analysisWe need to solve cubic polynomial: x^3 - 6x^2 + a x - 6 = 0. One root is 3. Need to find all solutions? We need to find a? Wait: The problem states: "One of the roots of the equation $x^{3}-6 x^{2}+a x-6=0$ is 3. Solve the equation." So a is unknown parameter; but we know 3 is a root. We need to find a such tha...	analysisWe have a cubic polynomial: $x^3 - 6x^2 + a x - 6 = 0$. One root is 3. We need to solve the equation, meaning find all roots in terms of a? Wait: They say "One of the roots is 3. Solve the equation." Usually that means find the remaining roots (i.e., factor the polynomial and find all roots). Since we know on...	analysisWe need to solve cubic equation x^3 - 6x^2 + a x - 6 = 0, given that one root is 3. Solve the equation means find all roots in terms of a? But a is a parameter. Since one root is 3, we can factor (x-3). Then the cubic becomes (x-3)(x^2 + bx + c) = x^3 - 6x^2 + a x - 6. Expand: (x-3)(x^2 + b x + c) = x^3 + b x^2...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
5. let $a$ and $b$ be fixed positive numbers. Depending on $a$ and $b$, find the smallest possible value of the sum $$ \frac{x^{2}}{(a y+b z)(a z+b y)}+\frac{y^{2}}{(a z+b x)(a x+b z)}+\frac{z^{2}}{(a x+b y)(a y+b x)} $$ where $x, y, z$ are positive real numbers. ## Solution	\frac{3}{(+b)^{2}}	0	analysisWe need to find minimal value of sum S = x^2/[(a y + b z)(a z + b y)] + y^2/[(a z + b x)(a x + b z)] + z^2/[(a x + b y)(a y + b x)] where a,b>0 fixed. We need to find minimal over positive x,y,z. This is symmetric in cyclic permutations but not fully symmetric unless a=b. But due to cyclic nature, maybe minim...	analysisWe need to find minimal value of S = sum_{cyc} x^2 / ((a y + b z)(a z + b y)), with a,b >0 fixed. x,y,z>0. We need to find minimal possible value depending on a,b. Likely symmetrical? The expression is symmetric in x,y,z but with weights a,b. We can attempt to use inequality like Cauchy-Schwarz or Titu's lemma....	analysisWe need to find minimum of sum S = Σ x^2 / [(a y + b z)(a z + b y)] over positive x,y,z. a,b fixed positive. We want minimal value depends on a,b. This is symmetric in cyclic but not fully symmetric because denominators involve a,b swapped? Actually expression is symmetric under interchange of a and b? Let's an...	analysisWe need to find the minimum of expression S = sum cyclic x^2 / ((a y + b z)(a z + b y)) for positive a,b fixed. Need to find minimal value in terms of a,b. Likely symmetry leads to x=y=z? Let's test. Suppose x=y=z=1. Then S = 3 * 1 / ((a+b)(a+b)) = 3/(a+b)^2. But we need to check if this is minimal or maybe so...	analysisWe need to find minimal value of S = sum over cyclic of x^2 / ((a y + b z)(a z + b y)). a,b fixed positive. Variables x,y,z >0. Symmetry? Not fully symmetric because a,b appear but expression is symmetric under cyclic permutations of x,y,z with same coefficients? Let's check: first term uses y,z; second uses z...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
3. Let $\theta \in$ $\left(0, \frac{\pi}{2}\right)$, then the maximum value of $\frac{2 \sin \theta \cos \theta}{(\sin \theta+1)(\cos \theta+1)}$ is $\qquad$.	6-4\sqrt{2}	100	analysisWe need to find maximum value of f(θ) = (2 sinθ cosθ) / ((sinθ+1)(cosθ+1)), for θ in (0, π/2). Equivalent to 2 sinθ cosθ / ((sinθ+1)(cosθ+1)). Let’s denote s = sinθ, c = cosθ. Then s, c > 0, s^2 + c^2 = 1. We need maximize f = 2 s c / ((s+1)(c+1)). We can attempt to find maximum using calculus or transformation...	analysisWe need maximum value of f(θ) = (2 sinθ cosθ)/((sinθ+1)(cosθ+1)), with θ∈(0,π/2). We can attempt to simplify. Let s = sinθ, c = cosθ, both positive, s^2 + c^2 =1. We want maximize f = 2sc / ((s+1)(c+1)). Since s and c are symmetric, maximum likely occurs at s=c=1/√2? Let's test. At θ=π/4, s=c=√2/2 ≈0.7071. Then...	analysisWe need to find maximum of expression: E = \frac{2 sinθ cosθ}{(sinθ+1)(cosθ+1)} for θ in (0, π/2). Simplify: sinθ cosθ = (1/2) sin 2θ. But maybe easier to use substitution. Let s = sinθ, c = cosθ, with s, c > 0 and s^2 + c^2 = 1. We need maximize f(s,c) = \frac{2 s c}{(s+1)(c+1)} subject to s^2 + c^2 = 1, s...	analysisWe need to find maximum of function f(θ) = 2 sinθ cosθ / ((sinθ + 1)(cosθ + 1)) for θ in (0, π/2). Equivalent to 2 sinθ cosθ = sin2θ. Denominator: (sinθ + 1)(cosθ + 1). We can attempt to express in terms of sinθ + cosθ or maybe use substitution t = sinθ + cosθ. Let's analyze. Let s = sinθ, c = cosθ. Then s, c ...	analysisWe need to find maximum of function f(θ)= 2 sinθ cosθ / ((sinθ+1)(cosθ+1)) for θ∈(0,π/2). Simplify: sinθ cosθ = (1/2) sin 2θ. So numerator 2 sinθ cosθ = sin 2θ. Denominator: (sinθ+1)(cosθ+1)= sinθ cosθ + sinθ + cosθ +1 = (1/2) sin 2θ + sinθ + cosθ +1. Thus f(θ)= sin 2θ / ((1/2) sin 2θ + sinθ + cosθ +1). We can...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Find the minimum value of \[ \frac{1}{x-y}+\frac{1}{y-z}+\frac{1}{x-z} \] for reals $x > y > z$ given $(x-y)(y-z)(x-z) = 17$.	\frac{3}{\sqrt[3]{68}}	0	analysisWe need minimize expression S = 1/(x-y)+1/(y-z)+1/(x-z) with constraint product of differences equals 17. Variables real with x>y>z. Let’s denote a = x-y >0, b = y-z >0, then x-z = a+b. Constraint: ab(a+b) = 17. Expression S = 1/a + 1/b + 1/(a+b). We need minimum over positive a,b satisfying product constrai...	analysisWe need to minimize S = 1/(x-y) + 1/(y-z) + 1/(x-z) given x>y>z reals, and product (x-y)(y-z)(x-z) = 17. Let a = x-y >0, b = y-z >0, then x-z = a+b. So product constraint: a * b * (a+b) = 17. We need to minimize S = 1/a + 1/b + 1/(a+b). All positive. We need to find minimal value of S given a,b>0, ab(a+b)=17...	analysisWe need minimize expression S = 1/(x-y)+1/(y-z)+1/(x-z) with constraint (x-y)(y-z)(x-z)=17 positive? Since x>y>z, differences positive: x-y>0, y-z>0, x-z = (x-y)+(y-z)>0. So product positive. So 17 >0. So all differences positive. Let a = x-y >0, b = y-z >0, c = x-z = a+b. Constraint: abc = 17. Expression S ...	analysisWe need minimize S = 1/(x-y)+1/(y-z)+1/(x-z) subject to x>y>z real, and product of differences equals 17: (x-y)(y-z)(x-z)=17. Let denote a = x-y >0, b = y-z >0, then x-z = a+b. Condition: a * b * (a+b) = 17. We need to minimize S = 1/a + 1/b + 1/(a+b). We have positive a,b with product constraint. We need f...	analysisWe need min of sum of reciprocals of differences with constraint product of differences equals 17. Let a = x-y, b = y-z, c = x-z. We have a>0, b>0, c = a+b (since x-z = (x-y)+(y-z)). Indeed c = a + b. Condition: abc = 17. So ab(a+b)=17. We need minimize S = 1/a + 1/b + 1/c = 1/a + 1/b + 1/(a+b). We need m...	{ "competition": "smt", "dataset": "Ours", "posts": null, "source": "algebra-solutions (7).md" }
## Task B-3.3. Determine the maximum and minimum value of the function $$ f(x)=\sqrt{\sin ^{4} x+4 \cos ^{2} x}-\sqrt{\cos ^{4} x+4 \sin ^{2} x} $$	Themaximumvalueofthefunctionfis1,theminimumvalueis-1	60	analysisWe need to find maximum and minimum of f(x) = sqrt(sin^4 x + 4 cos^2 x) - sqrt(cos^4 x + 4 sin^2 x). Domain: all real x. We can attempt to simplify or transform. Let s = sin^2 x, c = cos^2 x. Then s + c = 1. Both s, c in [0,1]. Then sin^4 x = s^2, cos^4 x = c^2. Also cos^2 x = c, sin^2 x = s. So function becom...	analysisWe need to find max/min of f(x) = sqrt(sin^4 x + 4 cos^2 x) - sqrt(cos^4 x + 4 sin^2 x). We can use symmetry: replace x by pi/2 - x: sin ↔ cos. Then f(pi/2 - x) = sqrt(cos^4 x + 4 sin^2 x) - sqrt(sin^4 x + 4 cos^2 x) = -f(x). So function is odd under that transformation: f(π/2 - x) = -f(x). That suggests that ...	analysisWe need to find max/min of f(x) = sqrt( sin^4 x + 4 cos^2 x ) - sqrt( cos^4 x + 4 sin^2 x ). Domain all real x. We can use symmetry. Let s = sin x, c = cos x. Then s^2 + c^2 = 1. Also s^4 = (s^2)^2, c^4 = (c^2)^2. So f = sqrt( s^4 + 4 c^2 ) - sqrt( c^4 + 4 s^2 ). We need to find extremes. Let a = s^2, b = c^2...	analysisWe need max/min of f(x) = sqrt(sin^4 x + 4 cos^2 x) - sqrt(cos^4 x + 4 sin^2 x). Let’s denote s = sin^2 x, c = cos^2 x. Then s + c = 1, s, c ∈ [0,1]. Also sin^4 = s^2, cos^4 = c^2. So: f = sqrt(s^2 + 4c) - sqrt(c^2 + 4s). We want extremes over s∈[0,1], c=1-s. Define function g(s) = sqrt(s^2 + 4(1-s)) - sqrt...	analysisWe need to find max/min of f(x)= sqrt(sin^4 x + 4 cos^2 x) - sqrt(cos^4 x + 4 sin^2 x). We can try to simplify. Let s = sin^2 x, c = cos^2 x, with s+c=1, s,c>=0. Then sin^4 = s^2, cos^4 = c^2. So f = sqrt(s^2 + 4 c) - sqrt(c^2 + 4 s). Domain s,c in [0,1], s+c=1. Let s = t, c = 1 - t, t∈[0,1]. Then f(t)= sqrt(t^...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Kurt is riding the tram along a long straight road. Suddenly he sees his friend walking in the opposite direction on this road. After a minute, the tram stops. Kurt gets off and runs twice as fast as his friend, but only at a quarter of the average speed of the tram behind his friend. After how many minutes does he cat...	9	20	analysisWe need to parse the problem carefully. "Kurt is riding the tram along a long straight road. Suddenly he sees his friend walking in the opposite direction on this road. After a minute, the tram stops. Kurt gets off and runs twice as fast as his friend, but only at a quarter of the average speed of the tram behi...	analysisWe need to parse the problem carefully. Let's restate: Kurt is riding the tram along a long straight road. He sees his friend walking in the opposite direction on this road. After a minute, the tram stops. Kurt gets off and runs twice as fast as his friend, but only at a quarter of the average speed of the tram...	analysisWe need to parse problem: Kurt is riding tram along a long straight road. He sees friend walking opposite direction on same road. After a minute, the tram stops. Kurt gets off and runs twice as fast as his friend, but only at a quarter of the average speed of the tram behind his friend. After how many minutes d...	analysisWe need to parse the problem. "Kurt is riding the tram along a long straight road. Suddenly he sees his friend walking in the opposite direction on this road. After a minute, the tram stops. Kurt gets off and runs twice as fast as his friend, but only at a quarter of the average speed of the tram behind his fri...	analysisWe need to parse the problem carefully. Let's rewrite: Kurt is riding the tram along a long straight road. Suddenly he sees his friend walking in the opposite direction on this road. After a minute, the tram stops. Kurt gets off and runs twice as fast as his friend, but only at a quarter of the average speed ...	{ "competition": "german_mo", "dataset": "Ours", "posts": null, "source": "Loesungen_MaOlympiade_138.md" }
11. Usain runs twice as fast as his mum. His mum runs five times as fast as his pet tortoise, Turbo. They all set off together for a run down the same straight path. When Usain has run $100 \mathrm{~m}$, how far apart are his mum and Turbo the tortoise? A $5 \mathrm{~m}$ B $10 \mathrm{~m}$ C $40 \mathrm{~m}$ D $50 \mat...	40\mathrm{~}	0	analysisWe need to solve the problem: Usain runs twice as fast as his mum. Mum runs five times as fast as his pet tortoise, Turbo. All start together on same straight path. When Usain has run 100 m, how far apart are his mum and Turbo? We need to compute distances. Let speed of tortoise = v. Then mum speed = 5v. Usai...	analysisWe need to solve: Usain runs twice as fast as his mum. Mum runs five times as fast as Turbo. They all set off together. When Usain has run 100 m, how far apart are mum and Turbo? So we need relative speeds: Let speed of Turbo = v. Then mum speed = 5v. Usain speed = 2* mum speed = 10v. So when Usain runs 100 m, ...	analysisWe need to solve: Usain runs twice as fast as his mum. Mum runs five times as fast as Turbo. They all set off together on same straight path. When Usain has run 100 m, how far apart are mum and Turbo? We need relative speeds. Let speed of Turbo be v. Then mum speed = 5v. Usain speed = 2 * mum speed = 10v. So at...	analysisWe need to parse problem: Usain runs twice as fast as his mum. Mum runs five times as fast as his pet tortoise, Turbo. They all start together on same straight path. When Usain has run 100 m, how far apart are his mum and Turbo the tortoise? We need to find distance between mum and tortoise at that time. We kno...	analysisWe need to solve the problem: Usain runs twice as fast as his mum. Mum runs five times as fast as his pet tortoise Turbo. They all start together on same straight path. When Usain has run 100 m, how far apart are mum and Turbo? We need to find distance between mum and Turbo at that time. Let speeds: Let s_U = ...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
1. Given the real number $y=\log _{2}\left[a x^{2}+(a-1) x+\frac{1}{4}\right]$ has a domain of all real numbers, the range of real number $a$ is ( ). A. $\left(0, \frac{3+\sqrt{5}}{2}\right)$ B. $\left(\frac{3-\sqrt{5}}{2}, 1\right)$ C. $\left(-\infty, \frac{3-\sqrt{5}}{2}\right) \cup\left(\frac{3+\sqrt{5}}{2},+\infty\...	0<<\frac{3+\sqrt{5}}{2}	0	analysisWe need to analyze the domain of y = log_2[ a x^2 + (a-1) x + 1/4 ]. Domain all real numbers: the argument of log must be > 0 for all real x. So the quadratic Q(x)= a x^2 + (a-1) x + 1/4 must be positive for all real x. We need the set of real a such that Q(x) > 0 for all x ∈ ℝ. We analyze the quadratic: coe...	analysisWe need to find all real a such that the quadratic Q(x)=a x^2 + (a-1)x + 1/4 is positive for all real x (since log base 2 requires argument >0). Domain all real numbers means Q(x)>0 for all x. We need to analyze quadratic positivity conditions. Q(x)=a x^2 + (a-1)x + 1/4. For a>0, upward opening; for a<0, downw...	analysisWe need to find values of parameter a such that function y = log_2 [ a x^2 + (a-1)x + 1/4 ] has domain all real numbers. That means the quadratic inside log must be positive for all real x. Also log base 2 requires argument >0. So we need quadratic >0 for all real x. That requires leading coefficient a>0? Wait:...	analysisWe need to analyze the function y = log_2 [ a x^2 + (a-1) x + 1/4 ]. Domain all real numbers means the argument of log must be positive for all real x. So we need the quadratic Q(x) = a x^2 + (a-1) x + 1/4 > 0 for all real x. That requires that a > 0 and discriminant < 0. Because if a < 0, leading coefficient n...	analysisWe need to analyze the problem: We have y = log_2 [ a x^2 + (a-1) x + 1/4 ]. The domain of y is all real numbers. That means the argument of the logarithm must be positive for all real x. So the quadratic function q(x) = a x^2 + (a-1)x + 1/4 must be > 0 for all x in R. We need to find the set of real a such th...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Let $\alpha$ and $\beta$ be the real roots of the equation $x^{2}-x-2021=0$, where $\alpha>\beta$. Define \[ A = \alpha^{2} - 2\beta^{2} + 2\alpha\beta + 3\beta + 7 \] Find the largest integer not exceeding $A$.	-6055	100	analysisWe need to solve the problem: α, β are real roots of x^2 - x - 2021 = 0. So α and β are roots: α + β = 1 (coefficient of x is -1, so sum = 1). Product αβ = -2021. Given α > β. Then A = α^2 - 2β^2 + 2αβ + 3β + 7. We need floor(A). Let's compute A in terms of α, β. Use relationships. Let S = α + β = 1, P = αβ =...	analysisWe need to compute A for roots of quadratic x^2 - x - 2021 = 0. Let's denote roots α>β. Sum α+β = 1, product αβ = -2021. Wait: For equation x^2 - x - 2021 = 0, coefficient a=1, b=-1, c=-2021. Sum of roots = -b/a = 1. Product = c/a = -2021. Yes. We need A = α^2 - 2β^2 + 2αβ + 3β + 7. We can express A in terms ...	analysisWe need to compute A for the quadratic roots. Quadratic: x^2 - x - 2021 = 0. Roots: α, β real. Sum α+β = 1 (coefficient -(-1)=1). Product αβ = -2021. α>β. Compute A = α^2 - 2β^2 + 2αβ + 3β + 7. We can express in terms of α, β using sum and product. Let's compute. Let S = α+β = 1. P = αβ = -2021. We need α^2, ...	analysisWe need to compute A given alpha and beta are real roots of x^2 - x - 2021 = 0. So alpha and beta are solutions: (1 ± sqrt(1+8084))/2 = (1 ± sqrt(8085))/2. Since discriminant D = 1 + 8084 = 8085. sqrt(8085) is between sqrt(8100)=90 and sqrt(8100-? Actually 90^2=8100, 89^2=7921, 89.9^2=8082.01? Let's compute sqr...	analysisWe need to find A = α^2 - 2β^2 + 2αβ + 3β + 7, given α and β are real roots of x^2 - x - 2021 = 0. Quadratic: x^2 - x - 2021 = 0. Sum of roots: α + β = 1 (coefficient of x negative of sum). Product: αβ = -2021. We have α > β. So α > β. They are real? Let's check discriminant: 1 + 4*2021 = 1 + 8084 = 8085. sqrt...	{ "competition": "all_russian_mo", "dataset": "Ours", "posts": null, "source": "ii-11 (2).md" }
9.6. Let $a$ and $b$ be positive numbers. Find the minimum value of the fraction $\frac{(a+b)(a+2)(b+2)}{16ab}$. Justify your answer.	1	40	analysisWe need to find minimum value of fraction: F(a,b) = ((a+b)(a+2)(b+2))/(16ab), where a,b > 0. We can attempt to find minimum via AM-GM or other inequalities. Let's analyze. We need to minimize over positive a,b. We can rewrite numerator: (a+b)(a+2)(b+2). Denominator 16ab. Goal: find min of F. We can attemp...	analysisWe need to find minimum of f(a,b) = ((a+b)(a+2)(b+2))/(16ab) for a,b>0. Equivalent to minimize numerator over denominator. Let's denote f = ((a+b)(a+2)(b+2))/(16ab). We can attempt to use AM-GM or other inequalities. Let's analyze. Let’s expand numerator: (a+b)(a+2)(b+2) = (a+b)(ab + 2a + 2b +4) = (a+b)(ab + 2...	analysisWe need to find minimum of f(a,b) = (a+b)(a+2)(b+2) / (16 a b) for a,b>0. We can attempt to use AM-GM or other inequalities. Let's denote variables. Expand numerator: (a+b)(a+2)(b+2). Expand partially: (a+b)( (a+2)(b+2) ) = (a+b)(ab + 2a + 2b + 4). So numerator = (a+b)(ab + 2a + 2b + 4). We can try to rewrite...	analysisWe need to find min of f(a,b) = (a+b)(a+2)(b+2)/(16ab). a,b>0. We can attempt to use AM-GM or other inequalities. Let's define expression: numerator: (a+b)(a+2)(b+2). Denominator: 16ab. We can attempt to rewrite or apply AM-GM to each factor. Let's denote x=a, y=b. Then expression E = ((x+y)(x+2)(y+2))/(16xy)....	analysisWe need to find min of expression: E = \frac{(a+b)(a+2)(b+2)}{16 a b}, with a,b>0. We want to minimize over positive reals. Let's denote x = a, y = b. We can attempt to use AM-GM or rearrangements. Let's attempt to find minimum by setting partial derivatives to zero. But maybe easier to apply AM-GM. Let’s e...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
58. Given $a, b, c \geqslant F$, prove: $\sqrt{a-\mathrm{F}}+\sqrt{b-1}+\sqrt{c-1} \leqslant \sqrt{(a b+1) c}$. (-1989 Yugoslav Mathematical Olympiad, 1998 Hong Kong Mathematical Olympiad)	proof	0	analysisWe need to analyze problem: "Given $a, b, c \geqslant F$, prove: $\sqrt{a-1} + \sqrt{b-1} + \sqrt{c-1} \leqslant \sqrt{(a b+1) c}$." But there is confusion: In statement: "Given $a, b, c \geqslant F$" maybe means $a,b,c \ge 1$? The letter F maybe a misprint for 1. Let's check: The inequality: \(\sqrt{a...	analysisWe need to parse the problem carefully. There is a misprint: "Given $a, b, c \geqslant F$"? Maybe they mean "Given $a, b, c \geqslant 1$"? But the inequality uses $\sqrt{a - F}$? Wait: The problem states: "Given $a, b, c \geqslant F$, prove: $\sqrt{a-\mathrm{F}}+\sqrt{b-1}+\sqrt{c-1} \leqslant \sqrt{(a b+1) c}$...	analysisWe need to parse the problem: "58. Given $a, b, c \geqslant F$, prove: $\sqrt{a-\mathrm{F}}+\sqrt{b-1}+\sqrt{c-1} \leqslant \sqrt{(a b+1) c}$." But there is confusion: $a, b, c \geqslant F$? maybe they mean $a,b,c \ge 1$? Wait: The statement: "Given $a, b, c \geqslant F$, prove: $\sqrt{a-\mathrm{F}}+\sqrt{b-1}...	analysisWe need to parse the problem: "Given $a, b, c \geqslant F$, prove: $\sqrt{a-\mathrm{F}}+\sqrt{b-1}+\sqrt{c-1} \leqslant \sqrt{(a b+1) c}$." Wait, there is confusion: "a, b, c ≥ F" maybe "≥1"? The problem says "Given $a, b, c \geqslant F$, prove: $\sqrt{a-\mathrm{F}}+\sqrt{b-1}+\sqrt{c-1} \leqslant \sqrt{(a b+1)...	analysisWe have a problem: "Given $a, b, c \geqslant F$, prove: $\sqrt{a-\mathrm{F}}+\sqrt{b-1}+\sqrt{c-1} \leqslant \sqrt{(a b+1) c}$." Wait there is confusion: The problem statement uses "F" but maybe it's "1"? Actually typical inequality: For $a,b,c \ge 1$ prove $\sqrt{a-1} + \sqrt{b-1} + \sqrt{c-1} \le \sqrt{(ab+1)...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "inequalities" }
1. If $x_{1}=1, x_{2}=1-\mathrm{i}, x_{3}=1+\mathrm{i}(\mathrm{i}$ is the imaginary unit $)$ are the three solutions of the equation $x^{3}+a x^{2}+b x+c=0$, then $a+b-c=$ $\qquad$	3	100	analysisWe need to find a+b-c given that the cubic polynomial has roots x1=1, x2=1-i, x3=1+i. So the polynomial is (x-1)(x-(1-i))(x-(1+i)) = 0. Expand to get monic cubic: x^3 + a x^2 + b x + c. We can compute a,b,c via Vieta: sum roots = -(a). Wait: For monic cubic x^3 + a x^2 + b x + c, the sum of roots = -a (since co...	analysisWe need to find a+b-c given roots x1=1, x2=1-i, x3=1+i. The cubic polynomial with roots x_i has form (x - x1)(x - x2)(x - x3)=0. Expand and compare to x^3 + a x^2 + b x + c. Compute coefficients via Vieta: sum of roots = -a, sum pairwise products = b, product = -c. We have roots: 1, 1-i, 1+i. Compute sum: 1 + ...	analysisWe need to find a+b-c given roots x1=1, x2=1-i, x3=1+i. The polynomial is monic cubic: x^3 + a x^2 + b x + c = 0. For monic polynomial, sum of roots = -a. Sum of pairwise products = b. Product = -c. Given roots: 1, 1-i, 1+i. Compute sums: sum = 1 + (1-i) + (1+i) = 3. So -a = 3 => a = -3. Compute sum of pairwi...	analysisWe have cubic with roots x1,x2,x3. Sum of roots = -a (since cubic monic). So a = -(x1+x2+x3). Then b = sum over pairwise products = x1x2 + x1x3 + x2x3. And c = - product of roots = -(x1 x2 x3). Wait sign: For monic cubic x^3 + a x^2 + b x + c = 0, Vieta: sum roots = -a, sum pairwise = b, product = -c. Yes. We ...	analysisWe need to solve: Given cubic equation x^3 + a x^2 + b x + c = 0 with roots x1=1, x2=1-i, x3=1+i. We are to compute a+b-c. We need to use Vieta's formulas. For cubic monic polynomial: sum of roots = -a, sum of pairwise products = b, product of roots = -c. So a = -(x1+x2+x3). b = sum pairwise products. c = -(pro...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
23. Let $x_{1}, x_{2}, \cdots, x_{n}$ be positive real numbers, $x_{n+1}=x_{1}+x_{2}+\cdots+x_{n}$, prove: $x_{n+1} \sum_{k=1}^{n}\left(x_{n+1}-\right.$ $\left.x_{k}\right) \geqslant\left[\sum_{k=1}^{n} \sqrt{x_{k}\left(x_{n+1}-x_{k}\right)}\right]^{2} .(1996$ Romanian National Training Team Exam Problem)	proof	0	analysisWe need to prove inequality: Let x1,...,xn >0. Define x_{n+1} = x1+...+xn. Prove: x_{n+1} * sum_{k=1}^{n} (x_{n+1} - x_k) >= [ sum_{k=1}^{n} sqrt{ x_k (x_{n+1} - x_k) } ]^2. We need to show: x_{n+1} * \sum_{k=1}^n (x_{n+1} - x_k) >= ( \sum_{k=1}^n sqrt{x_k (x_{n+1} - x_k)} )^2. Observations: This is reminis...	analysisWe need to prove inequality: Let x1,...,xn positive real numbers, define x_{n+1} = sum_{k=1}^n x_k. Then prove: x_{n+1} * sum_{k=1}^n (x_{n+1} - x_k) >= [ sum_{k=1}^n sqrt{ x_k (x_{n+1} - x_k) } ]^2. This is reminiscent of Cauchy-Schwarz or rearrangement. Let’s denote S = sum_{k=1}^n x_k = x_{n+1}. Then inequ...	analysisWe need to prove inequality: For positive real numbers x_1,...,x_n, define x_{n+1} = sum_{i=1}^n x_i. Show: x_{n+1} * sum_{k=1}^n (x_{n+1} - x_k) >= [ sum_{k=1}^n sqrt{x_k (x_{n+1} - x_k)} ]^2. Simplify: Let S = x_{n+1} = sum_{k=1}^n x_k. Then the inequality is: S * sum_{k=1}^n (S - x_k) >= ( sum_{k=1}^n sqr...	analysisWe need to prove inequality: Let x1,...,xn >0, define x_{n+1} = sum_{i=1}^n x_i. Prove: x_{n+1} * sum_{k=1}^n (x_{n+1} - x_k) >= [sum_{k=1}^n sqrt{x_k (x_{n+1} - x_k)}]^2. Essentially, inequality: x_{n+1} * sum_{k=1}^n (x_{n+1} - x_k) >= (sum sqrt{x_k (x_{n+1} - x_k)})^2. We can attempt to use Cauchy-Schwar...	analysisWe need to prove inequality: For positive reals x1,...,xn, let x_{n+1} = sum_{i=1}^n x_i. Prove: x_{n+1} * sum_{k=1}^{n} (x_{n+1} - x_k) >= [ sum_{k=1}^n sqrt( x_k (x_{n+1} - x_k) ) ]^2. We can rewrite: Let S = sum_{k=1}^n x_k = x_{n+1}. Then inequality becomes: S * sum_{k=1}^n (S - x_k) >= [ sum_{k=1}^n sqr...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "inequalities" }
3B. Solve the equation $\log _{x} 2-\log _{4} x+\frac{7}{6}=0$.	8	0	analysisWe need to solve equation: \log_x 2 - \log_4 x + 7/6 = 0. We must find x satisfying domain: base of log_x 2: base x > 0, x ≠ 1. Also argument 2 > 0 always. For log_4 x: base 4 > 0 ≠ 1, argument x > 0. So domain: x>0, x≠1. Also x>0. Equation: log_x 2 - log_4 x + 7/6 = 0. We can express logs in terms of natura...	analysisWe need solve equation: log_x 2 - log_4 x + 7/6 = 0. Let’s denote base x >0, x≠1. Also log_x 2 defined for x>0, x≠1. log_4 x defined for x>0. We can express logs in terms of natural logs: log_x 2 = ln 2 / ln x. log_4 x = ln x / ln 4. Equation: (ln 2)/(ln x) - (ln x)/(ln 4) + 7/6 = 0. Let t = ln x. Then t≠0....	analysisWe need solve equation: log_x 2 - log_4 x + 7/6 = 0. Let’s parse: log base x of 2 minus log base 4 of x plus 7/6 = 0. Domain: x>0, x≠1, because log base x requires x>0 and x≠1. Also log_4 x requires x>0. So domain: x>0, x≠1. We need solve equation: log_x 2 - log_4 x + 7/6 = 0. Let’s denote A = log_x 2. Then...	analysisWe need solve equation: log_x 2 - log_4 x + 7/6 = 0. Solve for x. We must consider domain: log_x 2 defined for x>0, x≠1, and base x positive not 1. Also log_4 x defined for x>0. So domain: x>0, x≠1. We need to solve equation: log_x 2 - log_4 x + 7/6 = 0. We can express logs with same base. Let's use natural ...	analysisWe need to solve equation: log_x 2 - log_4 x + 7/6 = 0. Domain: base x > 0, x ≠ 1. Also argument of log_x 2: 2 > 0, fine. Argument of log_4 x: x > 0. So domain: x > 0, x ≠ 1. Also base of log_x 2 must be >0, ≠1. So x > 0, x ≠ 1. Also base of log_4 x is 4 > 0, ≠1, fine. Equation: log_x 2 - log_4 x + 7/6 = 0. W...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
What is the sum of all possible values of $\cos\left(2\theta\right)$ if $\cos\left(2\theta\right)=2\cos\left(\theta\right)$ for a real number $\theta$? [i]2019 CCA Math Bonanza Team Round #3[/i]	2	0	analysisWe need to solve equation: cos(2θ) = 2 cos θ. For real θ. Need sum of all possible values of cos(2θ). That is, find all solutions for cos(2θ) that satisfy equation, then sum them. Let x = cos θ. Then cos 2θ = 2x^2 - 1. Equation: 2x^2 - 1 = 2x. Rearranged: 2x^2 - 2x - 1 = 0. Solve: x = [2 ± sqrt(4 + 8)]/(4) = [2...	analysisWe need sum of all possible values of cos(2θ) given equation cos(2θ) = 2 cos θ for real θ. We can use double-angle identity: cos 2θ = 2 cos^2 θ - 1. Set equal to 2 cos θ: 2 cos^2 θ - 1 = 2 cos θ. Rearrange: 2 cos^2 θ - 2 cos θ - 1 = 0. Let x = cos θ. Then 2x^2 - 2x - 1 = 0. Solve: x = [2 ± sqrt(4 + 8)] / (4)...	analysisWe need sum of all possible values of cos(2θ) satisfying cos(2θ) = 2 cos θ. For real θ. We know identity: cos(2θ) = 2 cos^2 θ - 1. So equation becomes 2 cos^2 θ - 1 = 2 cos θ. Let x = cos θ. Then 2x^2 - 1 = 2x. Solve: 2x^2 - 2x - 1 = 0. Solve: x = [2 ± sqrt(4 + 8)]/(4) = [2 ± sqrt12]/4 = [2 ± 2√3]/4 = (1 ± √3)...	analysisWe need to find all real θ satisfying cos(2θ)=2 cos θ. Then compute sum of all possible values of cos(2θ). Since cos(2θ) = 2 cos θ is an equation linking cos(2θ) and cos θ. But we can express cos(2θ) in terms of cos θ: cos(2θ) = 2 cos^2 θ - 1. So equation becomes 2 cos^2 θ - 1 = 2 cos θ. Let x = cos θ. Then 2x^...	analysisWe need sum of all possible values of cos(2θ) satisfying cos(2θ)=2 cos θ. Let x=cos θ. Then cos(2θ)=2 cos^2 θ -1 = 2x^2-1. Equation: 2x^2 -1 = 2x => 2x^2 - 2x -1 =0 => x^2 - x - 1/2 =0 => x = [1 ± sqrt(1 +2)]/2 = [1 ± sqrt3]/2. So cos θ = (1 ± sqrt3)/2. Need to find cos(2θ) values. Using formula cos(2θ)=2x^2-1....	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "aops_forum" }
1. In triangle $A B C$, there is a point $D$ on side $B C$ such that $B A=A D=D C$. Suppose $\angle B A D=80^{\circ}$. Determine the size of $\angle A C B$.	25	0	analysisWe need to solve geometry problem: triangle ABC, point D on BC such that BA = AD = DC. So D is on BC, with AD equal to BA, and DC equal to AD, so AD = BA = DC. So D is somewhere along BC such that DC = AD, and AD = BA. So BA = AD = DC. Also given angle BAD = 80°. Need angle ACB. We need to find measure of angl...	analysisWe need to solve geometry problem: triangle ABC, point D on BC such that BA = AD = DC. So BA = AD = DC. So AD = BA, AD = DC. So AD = DC, so D is on BC such that DC = AD, but also BA = AD. So BA = AD = DC. So AD = BA, DC = AD, so DC = BA. So D is on BC such that DC = BA. But also AD = BA. So D is such that AD = ...	analysisWe need to solve geometry problem: Triangle ABC, point D on side BC such that BA = AD = DC. So BA = AD = DC. So D is on BC between B and C. Also given angle BAD = 80°. We need to find angle ACB. We need to analyze geometry. Let’s denote unknown angles. Let’s set up triangle ABC with known constraints. D on BC...	analysisWe need to solve: In triangle ABC, point D on BC such that BA = AD = DC. So BA = AD = DC. D lies on BC between B and C. So we have BA = AD = DC. Also given ∠BAD = 80°. Find ∠ACB. We need to find angle at C in triangle ABC. Let's denote angles: Let ∠ABC = B, ∠ACB = C, ∠BAC = A. We need C. Given D on BC. BA = A...	analysisWe need to solve: In triangle ABC, point D on side BC such that BA = AD = DC. So BA = AD = DC. So D is on BC, between B and C? It says "point D on side BC" so yes between B and C. Conditions: BA = AD = DC. So BA = AD and AD = DC. So BA = AD = DC. So D divides BC such that DC = AD = BA. BA is side from B to A. S...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Frankin 5. In triangle $ABC$, the altitude and median drawn from vertex $A$ form (together with line $BC$) a triangle in which the bisector of angle $A$ is a median, and the altitude and median drawn from vertex $B$ form (together with line $AC$) a triangle in which the bisector of angle $B$ is a bisector. Find the ra...	1:2\sqrt{2}:3	0	analysisWe need to parse the problem: "Frankin 5" maybe from some contest. We have triangle ABC. The altitude and median drawn from vertex A form (together with line BC) a triangle in which the bisector of angle A is a median, and the altitude and median drawn from vertex B form (together with line AC) a triangle in wh...	analysisWe have a problem: In triangle ABC, the altitude and median drawn from vertex A form (together with line BC) a triangle in which the bisector of angle A is a median, and the altitude and median drawn from vertex B form (together with line AC) a triangle in which the bisector of angle B is a bisector. Find the r...	analysisWe need to parse the problem: In triangle ABC, altitude from A and median from A together with BC form a triangle (call it triangle formed by AH (altitude from A to BC) and AM (median from A to BC? Wait median from A to BC is to midpoint of BC). Actually median from A goes to midpoint of BC. So AH and AM are li...	analysisWe need to parse the problem. In triangle ABC, altitude and median drawn from vertex A form (together with line BC) a triangle in which the bisector of angle A is a median, and the altitude and median drawn from vertex B form (together with line AC) a triangle in which the bisector of angle B is a bisector. Fin...	analysisWe need to parse the problem carefully. In triangle ABC, the altitude and median drawn from vertex A form (together with line BC) a triangle in which the bisector of angle A is a median, and the altitude and median drawn from vertex B form (together with line AC) a triangle in which the bisector of angle B is a...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
The points $A, B, C, D, P$ lie on an circle as shown in the figure such that $\angle AP B = \angle BPC = \angle CPD$. Prove that the lengths of the segments are denoted by $a, b, c, d$ by $\frac{a + c}{b + d} =\frac{b}{c}$. [img]https://cdn.artofproblemsolving.com/attachments/a/2/ba8965f5d7d180426db26e8f7dd5c7ad02c...	\frac{a + c}{b + d} = \frac{b}{c}	0	analysisWe need to analyze problem: points A,B,C,D,P lie on a circle; angles APB, BPC, CPD are equal. So arcs? Let's denote arcs between consecutive points? The figure likely shows quadrilateral A,B,C,D on circle with point P inside? Wait P also on circle? It says points A,B,C,D,P lie on a circle. So all five points on...	analysisWe need to analyze the problem: We have a circle with points A, B, C, D, P on it such that ∠APB = ∠BPC = ∠CPD. The picture likely shows a circle with chords AB, BC, CD, and the diagonals? Let's parse: Points A, B, C, D, P lie on a circle. The angles at P formed by chords from A to B, B to C, C to D are equal. S...	analysisWe need to parse the problem carefully. We have points A,B,C,D,P on a circle. The angles APB, BPC, CPD are equal. So P is a point on the circle? Wait: The points A, B, C, D, P lie on a circle as shown in the figure such that ∠APB = ∠BPC = ∠CPD. Prove that the lengths of the segments are denoted by a, b, c, d b...	analysisWe need to parse the problem: There is a circle with points A, B, C, D, P on it such that angles APB, BPC, CPD are equal. So P is a point on the circle? Wait: The points A, B, C, D, P lie on a circle as shown in the figure such that ∠APB = ∠BPC = ∠CPD. So all 5 points on the circle. The angles at P between chor...	analysisWe need to analyze the problem: There are points A, B, C, D, P on a circle such that angle APB = angle BPC = angle CPD. So P is some point on circle? Wait: All points lie on a circle. So we have a cyclic quadrilateral with point P on circle? Actually the figure: maybe A, B, C, D are four points on circle in ord...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "aops_forum" }
25. In the diagram below, $D$ and $E$ lie on the side $A B$, and $F$ lies on the side $A C$ such that $D A=D F=D E, B E=E F$ and $B F=B C$. It is given that $\angle A B C=2 \angle A C B$. Find $x$, where $\angle B F D=x^{\circ}$.	108	80	analysisWe need to parse the geometry problem. We have triangle ABC with angle ABC = 2 * angle ACB. Points D and E lie on AB, F lies on AC such that: - DA = DF = DE - BE = EF - BF = BC We need angle BFD = x degrees. We need to find x. Likely a nice integer. We need to interpret the configuration. Let's set up coord...	analysisWe need to solve geometry problem: triangle ABC with points D, E on AB, F on AC. Conditions: D A = D F = D E, B E = E F, B F = B C. Angle ABC = 2 * angle ACB. Find angle BFD. We need to find angle BFD. Let's denote triangle ABC with AB side containing D and E. AB is base? Let's set coordinates or use geometry....	analysisWe need to parse the problem carefully. There is a diagram: triangle ABC with points D,E on AB and F on AC. Conditions: D A = D F = D E, B E = E F, and B F = B C. Also given angle ABC = 2 * angle ACB. We need angle BFD = x degrees. So we have triangle ABC with AB side, AC side, BC side. D and E on AB such that ...	analysisWe need to analyze geometry problem. Diagram: Triangle ABC with AB side, AC side. Points D and E lie on AB. F lies on AC. Conditions: D A = D F = D E. So D is equidistant from A, F, and E. So D is center of circle through A, F, E. Also B E = E F and B F = B C. So E and F are points such that BE = EF and BF = BC...	analysisWe need to analyze geometry problem: Points D, E on AB; F on AC such that DA=DF=DE, BE=EF, BF=BC. Also ∠ABC = 2 ∠ACB. Find angle BFD = x degrees. We need to find angle at F between BF and FD. We need to identify key structure: There are equalities of lengths leading to isosceles triangles. Let's denote points:...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
21. In a right-angled triangle, the angle bisector of an acute angle divides the opposite side into segments of length 1 and 2 . What is the length of the bisector? A $\sqrt{2}$ B $\sqrt{3}$ C $\sqrt{4}$ D $\sqrt{5}$ E $\sqrt{6}$	2	100	analysisWe need to find length of angle bisector of an acute angle in a right triangle. The bisector divides opposite side into segments of length 1 and 2. We must find length of bisector. We have options: sqrt2, sqrt3, sqrt4=2, sqrt5, sqrt6. We need to determine which matches. Let’s denote triangle ABC right-angled a...	analysisWe need to solve: In a right-angled triangle, an acute angle's bisector divides the opposite side into segments of lengths 1 and 2. Find the length of the bisector. Options: sqrt2, sqrt3, sqrt4=2, sqrt5, sqrt6. We need to find the length of the angle bisector of an acute angle in a right triangle, given that i...	analysisWe need to solve: In a right-angled triangle, the angle bisector of an acute angle divides the opposite side into segments of lengths 1 and 2. Find the length of the bisector. Options: sqrt(2), sqrt(3), sqrt(4), sqrt(5), sqrt(6). We need to find the length of the bisector. Let’s denote the right triangle ABC w...	analysisWe need to solve: In a right-angled triangle, the angle bisector of an acute angle divides the opposite side into segments of length 1 and 2. Find the length of the bisector. We have a right triangle, acute angle at one of the acute vertices, its angle bisector intersects the opposite side (the hypotenuse? Wai...	analysisWe need to solve: In a right-angled triangle, the angle bisector of an acute angle divides the opposite side into segments of length 1 and 2. Find the length of the bisector. Let’s define triangle ABC right-angled at C? Usually right triangle with acute angles at A and B. Let's choose right angle at C. Suppose...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
5) In a right triangle $A B C$, the legs $B C$ and $C A$ measure $7 \mathrm{~cm}$ and $24 \mathrm{~cm}$ respectively. Let $H$ be the projection of $C$ onto the hypotenuse $A B$. What is the perimeter of triangle $H B C$? (A) $\frac{262}{25} \mathrm{~cm}$, (B) $\frac{501}{49} \mathrm{~cm}$ (C) $\frac{392}{25} \mathrm{~c...	\frac{392}{25}\mathrm{~}	0	analysisWe have right triangle ABC with legs BC=7, CA=24, so AB hypotenuse length sqrt(7^2+24^2)=sqrt(49+576)=sqrt(625)=25. So AB=25. H is foot of altitude from C onto hypotenuse AB. We need perimeter of triangle HBC. That is BH + HC + CB. We know CB=7. We need BH and HC. We know property: In right triangle, altitude f...	analysisWe need to find perimeter of triangle HBC where H is foot of altitude from C onto hypotenuse AB in a right triangle with legs BC=7 cm, CA=24 cm. Right angle at C. So AB is hypotenuse length sqrt(7^2+24^2)= sqrt(49+576)= sqrt(625)=25 cm. Good. We have right triangle ABC with right angle at C. H is foot of altit...	analysisWe need to solve: Right triangle ABC with legs BC=7, CA=24. Right angle at C. Hypotenuse AB length? Use Pythagorean: AB = sqrt(7^2+24^2)= sqrt(49+576)= sqrt(625)=25. So AB=25. Projection of C onto hypotenuse AB is point H. So H is foot of altitude from C to AB. We need perimeter of triangle HBC. So we need leng...	analysisWe need to compute perimeter of triangle HBC. Right triangle ABC with legs BC=7, CA=24. So AB = sqrt(7^2+24^2)= sqrt(49+576)= sqrt(625)=25. So ABC is 7-24-25 right triangle. H is projection of C onto hypotenuse AB. So H is foot of altitude from right angle C onto hypotenuse AB. We know that CH^2 = AH * HB. Also...	analysisWe need to compute perimeter of triangle HBC, where ABC is right triangle at C, legs BC=7, CA=24, hypotenuse AB= sqrt(7^2+24^2)= sqrt(49+576)= sqrt(625)=25. So AB=25. We have altitude from C onto hypotenuse AB at point H. So CH is altitude. We need perimeter of triangle HBC: sum of lengths HB + BC + CH. We kn...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
3. The perimeter of a square inscribed in a circle is $p$. What is the area of the square that circumscribes the circle?	\frac{p^{2}}{8}	100	analysisWe need to solve: Given a square inscribed in a circle, perimeter p. Find area of the square that circumscribes the circle (i.e., the square that contains the circle, touching at midpoints of sides). The circle is inscribed in the outer square. So we have circle radius r. The inscribed square is inside the circ...	analysisWe need to analyze the problem: We have a circle, inside it a square is inscribed. The perimeter of that inscribed square is p. We need the area of the square that circumscribes the circle (i.e., the smallest square that contains the circle). We need to express area in terms of p. We need to use geometry: For ...	analysisWe need to solve: The perimeter of a square inscribed in a circle is p. Find the area of the square that circumscribes the circle. So we have a circle, inside it a square inscribed (i.e., the square's vertices lie on the circle). The perimeter of that inscribed square is p. We need area of the circumscribed squ...	analysisWe need to parse problem: "3. The perimeter of a square inscribed in a circle is $p$. What is the area of the square that circumscribes the circle?" So we have a circle. Inside it, a square inscribed: meaning all vertices of square lie on the circle. The perimeter of that inscribed square is p. We need area of ...	analysisWe need to interpret problem: "The perimeter of a square inscribed in a circle is p. What is the area of the square that circumscribes the circle?" So we have a circle, a square inscribed inside it (square inside circle). That square has perimeter p. We need area of square that circumscribes the circle (square ...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
VIII OM - III - Task 2 Prove that between the sides $ a $, $ b $, $ c $ and the opposite angles $ A $, $ B $, $ C $ of a triangle, the following relationship holds	proof	0	analysisWe need to read the problem statement: "VIII OM - III - Task 2: Prove that between the sides a, b, c and the opposite angles A, B, C of a triangle, the following relationship holds". But the relationship is not included in the prompt. We need to deduce from context or typical tasks. The problem is likely from a...	analysisWe need to parse the problem: "VIII OM - III - Task 2: Prove that between the sides a, b, c and the opposite angles A, B, C of a triangle, the following relationship holds". But the statement of the relationship is missing in the prompt. The retrieved example shows relationships between sides and angles: law of...	analysisWe need to parse the problem: "VIII OM - III - Task 2" maybe from a geometry book. "Prove that between the sides a, b, c and the opposite angles A, B, C of a triangle, the following relationship holds". It doesn't state the relationship; maybe it's the law of sines? The retrieved example deals with equivalences...	analysisWe need to parse the problem: "VIII OM - III - Task 2" maybe from some geometry book. The statement: "Prove that between the sides $ a $, $ b $, $ c $ and the opposite angles $ A $, $ B $, $ C $ of a triangle, the following relationship holds". But the relationship is not given in the prompt; maybe omitted? We ...	analysisWe need to interpret the problem: "VIII OM - III - Task 2" maybe refers to a problem about triangle sides and angles. The statement: "Prove that between the sides a, b, c and the opposite angles A, B, C of a triangle, the following relationship holds". But what relationship? The problem statement incomplete. Ne...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
## Subject III In the acute-angled triangle $\mathrm{ABC}$, the altitude $\mathrm{AD}$ is equal to the side $\mathrm{BC}$. The angle bisectors of $\mathrm{ADB}$ and $\mathrm{ADC}$ intersect $\mathrm{AB}$ and $\mathrm{AC}$ at $\mathrm{M}$ and $\mathrm{N}$, respectively. Show that: $$ \frac{M B}{M A}+\frac{N C}{N A}=1 ...	proof	0	analysisWe have triangle ABC acute. AD is altitude from A to BC, so D is foot on BC. AD = BC. Let angle bisectors of triangle ADB and ADC intersect AB at M and AC at N, respectively. We need to show MB/MA + NC/NA = 1. We need to derive relation. Let's denote coordinates or use geometry. Let’s set up triangle ABC with...	analysisWe need to analyze the problem: In acute triangle ABC, altitude AD from A to BC is equal to side BC: AD = BC. Let D be foot of altitude from A onto BC. Consider triangles ADB and ADC. In triangle ADB, its angle bisector from vertex D? Wait: "The angle bisectors of ADB and ADC intersect AB and AC at M and N, res...	analysisWe need to analyze geometry problem: triangle ABC acute, AD altitude from A to BC. AD = BC. Let angle bisector of triangle ADB intersect AB at M. Similarly angle bisector of triangle ADC intersects AC at N. Need to show MB/MA + NC/NA = 1. We must use properties of angle bisector, maybe ratio AB:BD = AM:MD? Wai...	analysisWe need to prove that in acute triangle ABC, altitude AD equals BC. M is intersection of angle bisector of ∠ADB with AB. N is intersection of angle bisector of ∠ADC with AC. Show MB/MA + NC/NA = 1. We have triangle ABC with altitude from A to BC at D. So AD ⟂ BC. Given AD = BC. Also triangle is acute, so D lie...	analysisWe need to solve: In acute-angled triangle ABC, altitude AD from A to BC, with AD = BC. The angle bisectors of ∠ADB and ∠ADC intersect AB and AC at M and N respectively. Show that MB/MA + NC/NA = 1. We have triangle ABC acute, D foot of altitude from A onto BC. So AD ⟂ BC. Given AD = BC. We consider triangle A...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Ex. 77. The median $A A_{1}$, the bisector $B B_{1}$, and the altitude $C C_{1}$ of triangle $A B C$ intersect at one point, which divides the altitude $C C_{1}$ in the ratio 3 : 1, counting from vertex C. Prove that the median $A A_{1}$ and the bisector $B B_{1}$ of the triangle are perpendicular.	proof	0	analysisWe need to prove that median from A to BC and angle bisector from B to AC are perpendicular given concurrency of median from A, bisector from B, altitude from C, with intersection point dividing altitude in ratio 3:1 from C. So intersection point P lies on CC1 such that CP : PC1 = 3:1? Wait altitude from C to A...	analysisWe need to analyze the problem: In triangle ABC, median AA1, bisector BB1, altitude CC1 are concurrent at a point P. P divides altitude CC1 in ratio 3:1 from vertex C. So CP:PC1 = 3:1? Wait: altitude CC1 from vertex C to side AB. So C1 is foot on AB. P lies on CC1 such that CP:PC1 = 3:1. So P is closer to C1? A...	analysisWe need to analyze the problem: In triangle ABC, median from A to side BC, bisector from B to side AC, altitude from C to side AB. They are concurrent at a point, say P, and P divides altitude CC1 in ratio 3:1 counting from vertex C. That is CP:PC1 = 3:1? Wait altitude CC1: C is vertex, C1 foot on AB. So ratio ...	analysisWe need to analyze problem: In triangle ABC, median AA1, bisector BB1, and altitude CC1 are concurrent at a point P. P divides altitude CC1 in ratio 3:1 counting from vertex C. So CP:PC1 = 3:1? Wait: altitude CC1 from C to AB. So point P lies on CC1, dividing it 3:1 from vertex C. That is CP:PC1 = 3:1. So CP = ...	analysisWe need to prove that in triangle ABC, median from A to side BC, bisector from B to side AC, and altitude from C to AB all concurrent at point P. The point divides altitude CC1 in ratio 3:1 from vertex C. So CP:PC1 = 3:1. Need to show that median AA1 and bisector BB1 are perpendicular. We can use coordinates o...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
59. The sum of two angles of a triangle is half of a right angle, and one of these two angles is $5^{\circ}$ less than the other. The degree measure of the smallest angle in the triangle is $\qquad$ $\circ$. A triangle has two interior angles whose sum is half of a right angle, and one of these angles is $5^{\circ}$ l...	20	100	analysisWe need to find smallest angle of triangle given that two angles sum to half of a right angle (i.e., 90°/2 = 45°). So two angles sum to 45°, and one is 5° less than the other. Let larger angle = x, smaller = x-5. Sum: x + (x-5) = 45 => 2x - 5 = 45 => 2x = 50 => x = 25°. So larger angle is 25°, smaller is 20°. B...	analysisWe need to parse the problem: "The sum of two angles of a triangle is half of a right angle, and one of these two angles is 5° less than the other. The degree measure of the smallest angle in the triangle is ___." We have a triangle with interior angles A, B, C. Two of them sum to half of a right angle: half o...	analysisWe need to solve: "The sum of two angles of a triangle is half of a right angle, and one of these two angles is $5^\circ$ less than the other. The degree measure of the smallest angle in the triangle is ___." We have a triangle with three interior angles that sum to 180°. Two of them sum to half of a right ang...	analysisWe need to solve: In a triangle, two interior angles have sum equal to half of a right angle. A right angle is 90°, half is 45°. So two angles sum to 45°. Among them, one is 5° less than the other. Let the larger be x, the smaller be x-5. Then x + (x-5) = 45 => 2x -5 = 45 => 2x = 50 => x = 25. So the two angles...	analysisWe need to solve: "The sum of two angles of a triangle is half of a right angle, and one of these two angles is $5^{\circ}$ less than the other. The degree measure of the smallest angle in the triangle is ___." The problem states: The sum of two angles of a triangle is half of a right angle, and one of these tw...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
In $\triangle ABC$, $AC = BC$, and point $D$ is on $\overline{BC}$ so that $CD = 3\cdot BD$. Let $E$ be the midpoint of $\overline{AD}$. Given that $CE = \sqrt{7}$ and $BE = 3$, the area of $\triangle ABC$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the ...	10	100	analysisWe need to find area of isosceles triangle ABC with AC=BC. D on BC such that CD = 3 BD. So BD:CD = 1:3. Let BD = x, CD = 3x. Then BC = 4x. Let coordinates: Put B at (0,0), C at (4x,0) along x-axis. Since AC=BC, point A somewhere such that distances to C and B equal: AC=BC=4x? Wait BC length is 4x. AC=BC => AC=4...	analysisWe need to solve geometry problem. Triangle ABC with AC = BC, so isosceles with apex at C? Wait AC=BC, so C is vertex between equal sides AC and BC, base AB. D on BC such that CD = 3*BD. So D divides BC in ratio BD:DC = 1:3. E is midpoint of AD. Given CE = sqrt(7) and BE = 3. We need area of triangle ABC. We c...	analysisWe need to solve geometry problem: Triangle ABC is isosceles with AC = BC. D on BC such that CD = 3 BD. E is midpoint of AD. Given CE = sqrt(7), BE = 3. Find area of triangle ABC expressed as m sqrt(n). Compute m+n. We need to find side lengths or area. Let's denote coordinates or use vector geometry. Since AC...	analysisWe need to find area of triangle ABC given AC=BC (isosceles with apex A? Wait: AC = BC, so C is apex? Actually AC=BC means sides from C to A and B equal, so triangle is isosceles with vertex C at apex. So base AB. D is on BC such that CD = 3*BD. So D is closer to B, dividing BC in ratio BD:DC = 1:3. E is midpoi...	analysisWe need to find area of triangle ABC given AC=BC, D on BC with CD=3*BD. Let’s set up coordinates or use geometry. Let’s denote triangle ABC is isosceles with AC=BC. Let base AB. D lies on BC such that CD=3 BD. So D divides BC in ratio BD:DC = 1:3, so D is closer to B. Let’s set coordinates: Place B at (0,0), C...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "amc_aime" }
\left.\begin{array}{l}{[\quad \text { Inscribed quadrilaterals (other). }} \\ \text { [Central angle. Arc length and circumference }]\end{array}\right] Three consecutive angles of an inscribed quadrilateral are in the ratio 1:2:3. Find all the angles of the quadrilateral.	45,90,135,90	0	analysisWe need to solve: "Three consecutive angles of an inscribed quadrilateral are in the ratio 1:2:3. Find all the angles of the quadrilateral." We have a cyclic quadrilateral. Let the four angles be A, B, C, D in order around the circle. We are told that three consecutive angles are in ratio 1:2:3. That means per...	analysisWe need to find all angles of an inscribed quadrilateral where three consecutive angles are in ratio 1:2:3. Let's denote angles A, B, C, D in order around quadrilateral. Three consecutive angles: perhaps A, B, C are consecutive. Or maybe B, C, D. But "three consecutive angles" could be any triple; but we can ch...	analysisWe have a problem: "Three consecutive angles of an inscribed quadrilateral are in the ratio 1:2:3. Find all the angles of the quadrilateral." We need to find the four angles of a cyclic quadrilateral (inscribed in a circle) where three consecutive angles are in ratio 1:2:3. We need to find all possible sets of...	analysisWe need to solve: A quadrilateral inscribed in a circle (cyclic quadrilateral). Three consecutive angles are in ratio 1:2:3. We need to find all angles of the quadrilateral. Let's denote the angles in order A, B, C, D. Suppose A, B, C consecutive are in ratio 1:2:3. So A : B : C = 1 : 2 : 3. Let A = x, B = 2x, ...	analysisWe need to parse problem: "Inscribed quadrilaterals (other). [Central angle. Arc length and circumference]" Probably just a problem: "Three consecutive angles of an inscribed quadrilateral are in the ratio 1:2:3. Find all the angles of the quadrilateral." So we have a cyclic quadrilateral (inscribed in a circle...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
5. In an acute triangle $A B C$, let $D$ be the foot of the altitude from vertex $C$ and $P, Q$ be the feet of the perpendiculars from point $D$ to sides $A C$ and $B C$. The areas of triangles $A D P, D C P, D B Q, C D Q$ are denoted by $S_{1}, S_{2}, S_{3}, S_{4}$, respectively. Calculate $S_{1}: S_{3}$, given that $...	88:45	20	analysisWe need to parse the problem. Triangle ABC acute. D is foot of altitude from C onto AB? Wait: "let D be the foot of the altitude from vertex C" meaning altitude from C to side AB, so D lies on AB. Good. Then P, Q are feet of perpendiculars from point D to sides AC and BC, respectively. So from D drop perpendic...	analysisWe need to parse the problem carefully. We have acute triangle ABC. D is foot of altitude from vertex C to side AB? Wait: "foot of the altitude from vertex C" meaning altitude from C to side AB? Usually foot of altitude from vertex C onto AB. So D is foot of perpendicular from C to AB. So CD ⟂ AB. Then P and ...	analysisWe need to analyze geometry. We have acute triangle ABC. D is foot of altitude from vertex C onto AB? Wait: "foot of the altitude from vertex C" meaning drop perpendicular from C to side AB. So D lies on AB. Then P, Q are feet of perpendiculars from point D to sides AC and BC respectively. So from D drop perpen...	analysisWe need to parse the problem. We have acute triangle ABC. D is foot of altitude from vertex C onto side AB? Wait: "foot of the altitude from vertex C" typically means drop perpendicular from C to side AB. So D lies on AB such that CD ⟂ AB. Then P, Q are feet of perpendiculars from point D to sides AC and BC. ...	analysisWe need to parse the geometry. Triangle ABC acute. D is foot of altitude from C onto AB? Wait: "foot of the altitude from vertex C" means D is foot of altitude from C onto side AB. So D lies on AB. Then P, Q are feet of perpendiculars from point D to sides AC and BC respectively. So from D drop perpendicular t...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Line $ DE $ cuts through triangle $ ABC $, with $ DF $ parallel to $ BE $. Given that $ BD = DF = 10 $ and $ AD = BE = 25 $, find $ BC $.	14	40	analysisWe need to parse the problem: "Line DE cuts through triangle ABC, with DF parallel to BE. Given that BD = DF = 10 and AD = BE = 25, find BC." We have triangle ABC. Points D and E presumably lie on sides? Let's interpret: "Line DE cuts through triangle ABC" meaning line segment DE is inside triangle? Usually D ...	analysisWe need to parse the problem: "Line DE cuts through triangle ABC, with DF parallel to BE. Given that BD = DF = 10 and AD = BE = 25, find BC." We have triangle ABC. There's a line DE that cuts through the triangle. So D and E are points on sides? Not specified. We have DF parallel to BE. So D and F? Wait DF is ...	analysisWe need to parse problem: "Line DE cuts through triangle ABC, with DF parallel to BE. Given that BD = DF = 10 and AD = BE = 25, find BC." We need to interpret geometry. We have triangle ABC. Points D and E presumably lie on sides? We need to figure configuration: "Line DE cuts through triangle ABC" meaning D an...	analysisWe need to parse the problem: "Line DE cuts through triangle ABC, with DF parallel to BE. Given that BD = DF = 10 and AD = BE = 25, find BC." Let's interpret geometry. We have triangle ABC. There is a line DE that cuts through the triangle. We have points D and E presumably on sides? Let's analyze: "Line DE cu...	analysisWe need to parse the problem: "Line DE cuts through triangle ABC, with DF parallel to BE. Given that BD = DF = 10 and AD = BE = 25, find BC." We have a triangle ABC. Points: D on side? Not specified. But we have BD, DF, AD, BE. D is on AB? Let's interpret: Usually D is on AB, E on AC? But we have DF parallel t...	{ "competition": "jhmt", "dataset": "Ours", "posts": null, "source": "Geometry2005.md" }
Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\angle A = \frac 52 \angle B$. What is the degree measure of $\angle D$? $\text{(A) } 52.5 \qquad \text{(B) } 55 \qquad \text{(C) } 57.7 \qquad \text{(D) } 60 \qquad \text{(E) } 62.5$	52.5	20	analysisWe need to analyze geometry problem. We have segment BD and AE intersect at C. AB = BC = CD = CE. So points A,B,D,E are presumably on lines? Let's picture: We have two segments BD and AE crossing at C. So B and D are endpoints of one segment, A and E endpoints of the other. They intersect at C. AB=BC=CD=CE. Wai...	analysisWe need to parse problem: "Segment BD and AE intersect at C, as shown, AB=BC=CD=CE, and angle A = (5/2) angle B. What is degree measure of angle D?" There's a figure presumably: We have quadrilateral? Points A, B, C, D, E? BD and AE intersect at C. So B-D is a segment; A-E is another segment; they cross at C. A...	analysisWe need to parse problem: There is a diagram with segments BD and AE intersecting at C. AB=BC=CD=CE. Angle A = (5/2) angle B. Need measure of angle D. Likely quadrilateral? Let's imagine geometry: Points A, B, C, D, E such that BD and AE intersect at C. AB, BC, CD, CE equal segments. So we have a chain A-B-C-D-...	analysisWe have a problem: "Segment BD and AE intersect at C, as shown, AB=BC=CD=CE, and ∠A = (5/2)∠B. What is the degree measure of ∠D? Options: 52.5, 55, 57.7, 60, 62.5." We need to find angle D. We need geometry. Let's parse the diagram: There are points A, B, C, D, E. BD and AE intersect at C. AB=BC=CD=CE. So AB = ...	analysisWe need to parse problem: There is a quadrilateral with intersecting diagonals? Let's understand geometry: Segment BD and AE intersect at C. AB = BC = CD = CE. So AB, BC, CD, CE all equal. So we have a shape where points A, B, C, D, E such that AB is a segment between A and B, BC between B and C, CD between C a...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "amc_aime" }
In $\triangle P Q R$, the sum of $\angle P$ and $\angle Q$ is $60^{\circ}$. The measure of $\angle R$ is (A) $60^{\circ}$ (B) $300^{\circ}$ (C) $120^{\circ}$ (D) $30^{\circ}$ (E) $40^{\circ}$	120	60	analysisWe need to solve the problem: In triangle PQR, the sum of angles P and Q is 60 degrees. Find measure of angle R. Options: 60°, 300°, 120°, 30°, 40°. We need to think about triangle interior angles sum to 180°. So P + Q + R = 180°. Given P + Q = 60°, then R = 180° - 60° = 120°. So answer is 120°, option (C). But...	analysisWe have a triangle PQR. Sum of angles P + Q = 60°. In any triangle, sum of interior angles is 180°. So angle R = 180° - (P+Q) = 180° - 60° = 120°. So answer: 120°. But we must check: The options include 120°, yes option (C). So answer: 120°. But we must produce rigorous reasoning. But the problem might have so...	analysisWe need to analyze the problem: In triangle PQR, sum of angles P and Q is 60 degrees. What is measure of angle R? Options: 60°, 300°, 120°, 30°, 40°. We know in any triangle, sum of interior angles = 180°. So if P+Q = 60°, then R = 180-60 = 120°. So answer should be 120°, option (C). But let's double-check: Are...	analysisWe need to analyze the problem: In triangle PQR, the sum of angles P and Q is 60°. The measure of angle R? The options: 60°, 300°, 120°, 30°, 40°. But triangle angles sum to 180°. So if P+Q=60°, then R = 180° - 60° = 120°. Option (C). But need to consider that angles are interior angles of a triangle, so each a...	analysisWe need to parse the problem: "In triangle P Q R, the sum of angle P and angle Q is 60°. The measure of angle R is (options)." We know that in a triangle, sum of interior angles equals 180°. So if angle P + angle Q = 60°, then angle R = 180° - 60° = 120°. Option (C) 120°. But maybe there's a twist: the sum of ...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Right triangle $XYZ$, with hypotenuse $YZ$, has an incircle of radius $\frac{3}{8}$ and one leg of length 3. Find the area of the triangle. If the answer is of the form of an irreducible fraction $\frac{a}{b}$, compute the value of $a + b$.	37	100	analysisWe need to solve: Right triangle XYZ with hypotenuse YZ. Incircle radius r=3/8. One leg length 3. Find area. Let legs: maybe XY = 3? or XZ? unspecified. Let's denote legs a and b, with a=3 (one leg). Hypotenuse c. Right triangle. Inradius r = (a + b - c)/2. That's formula for right triangle: r = (a + b - c)/2. ...	analysisWe have a right triangle XYZ with right angle at X? Not specified but typical: right triangle XYZ, hypotenuse YZ. So legs are XY and XZ. One leg length 3. Incircle radius r = 3/8. We need area of triangle. Let legs a and b: a = XY, b = XZ. Without loss, a = 3, b unknown. Hypotenuse c = sqrt(a^2 + b^2). For righ...	analysisWe need to solve: Right triangle XYZ with legs X? Let's denote right triangle with legs X? Usually right angle at X? But not specified. Let's assume right triangle XYZ with hypotenuse YZ. So right angle at X. So legs are XY and XZ. One leg length is 3. Incircle radius r = 3/8. Find area of triangle. Then comput...	analysisWe need to solve: Right triangle XYZ, hypotenuse YZ. Incircle radius r = 3/8. One leg length = 3. Find area. Then compute a+b if area expressed as irreducible fraction a/b. We must identify the key mathematical structure: right triangle with incircle radius r, leg length known. For a right triangle with legs a...	analysisWe need to solve: right triangle XYZ, right angle at X? Usually right triangle XYZ, with hypotenuse YZ. So right angle at X. So legs: XY and XZ. One leg length 3. Incircle radius r = 3/8. Find area. Then answer as irreducible fraction a/b, compute a+b. We need to determine legs lengths a, b (say XY = a, XZ = b...	{ "competition": "hmmt", "dataset": "Ours", "posts": null, "source": "gen2_nov_2008.md" }
A line parallel to side $AB$ of triangle $ABC$ intersects side $BC$ at point $M$ and side $AC$ at point $N$. The area of triangle $MCN$ is twice the area of trapezoid $ABMN$. Find $CM: MB$.	2+\sqrt{6}	20	analysisWe need to analyze the problem. We have triangle ABC. A line parallel to side AB intersects BC at M and AC at N. So line MN is parallel to AB. So we have a triangle ABC with base AB? Actually AB is one side. We have line MN parallel to AB, crossing BC at M and AC at N. So M is on BC, N is on AC. The line is par...	analysisWe need to analyze the problem: Triangle ABC, line parallel to AB intersects BC at M and AC at N. So MN ∥ AB. We are given area of triangle MCN is twice area of trapezoid ABMN. Need ratio CM:MB. Let’s denote triangle ABC with vertices A,B,C. AB is a side. MN ∥ AB. So MN is a segment from M on BC to N on AC. W...	analysisWe need to solve problem: In triangle ABC, line parallel to side AB intersects BC at M and AC at N. So line MN is parallel to AB. So MN is a segment connecting BC at M and AC at N. Wait: line parallel to AB through some point? The statement: "A line parallel to side AB of triangle ABC intersects side BC at poin...	analysisWe need to analyze the problem: In triangle ABC, a line parallel to AB intersects BC at M and AC at N. So MN is parallel to AB. So we have triangle ABC, with base AB presumably. M lies on BC, N lies on AC. MN ∥ AB. So we have a smaller triangle MCN at the vertex C with base MN, and trapezoid ABMN between AB and...	analysisWe need to solve: Triangle ABC, line parallel to AB intersects BC at M and AC at N. So MN is parallel to AB. So M on BC, N on AC. We know area of triangle MCN is twice area of trapezoid ABMN. Find CM:MB. Let’s denote coordinates or use similar triangles. Let’s set triangle ABC with AB horizontal base? Not nece...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
10.050. Find the radius of the circle circumscribed around a right-angled triangle if the radius of the circle inscribed in this triangle is $3 \mathrm{~cm}$, and one of the legs is $10 \mathrm{~cm}$.	7.25	20	analysisWe need to find circumradius R of right triangle given inradius r=3 cm, one leg length a=10 cm. We need the other leg b and hypotenuse c. For right triangle, formulas: inradius r = (a + b - c)/2. Circumradius R = c/2. Also area K = (1/2)ab. Also r = K / s, where s = (a+b+c)/2. So r = (ab/2)/s = ab/(a+b+c). Sinc...	analysisWe need to parse the problem: "10.050. Find the radius of the circle circumscribed around a right-angled triangle if the radius of the circle inscribed in this triangle is 3 cm, and one of the legs is 10 cm." So we have a right triangle with legs a and b, hypotenuse c. We know inradius r = 3 cm, one leg a = 10 ...	analysisWe need to solve: For a right triangle, inradius r=3 cm, one leg length a=10 cm. Find circumradius R. We know for any triangle, R = abc/(4Δ). For right triangle, circumradius R = hypotenuse/2. Also, inradius r = (a + b - c)/2 where c is hypotenuse. Or r = (Δ)/s where s = semiperimeter. Let legs: a=10 cm, b un...	analysisWe need to find circumradius R of right triangle given inradius r=3 cm, one leg a=10 cm. Let other leg b unknown, hypotenuse c. In right triangle, circumradius R = c/2. Also inradius r = (a + b - c)/2? Wait formula: For any triangle, r = (2Δ)/(a+b+c). For right triangle, area Δ = (a*b)/2. Also c = sqrt(a^2 + b^...	analysisWe need to find circumradius of a right triangle given inradius r=3 cm and one leg length a=10 cm. Let triangle right-angled with legs a and b, hypotenuse c. Inradius formula: r = (a + b - c)/2 (since area = rs = (1/2)ab; semiperimeter s = (a+b+c)/2; r = A/s). But for right triangle, circumradius R = c/2. We kn...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
[ Area of a triangle (using two sides and the angle between them).] Law of Cosines Two sides of a triangle are equal to $2 \sqrt{2}$ and 3, and the area of the triangle is 3. Find the third side.	\sqrt{29},\sqrt{5}	80	analysisWe need to solve: triangle sides a=2√2, b=3, angle between them? Wait: area = 3. We have two sides lengths and area. We can find the included angle using area formula: area = (1/2) ab sin(C) where C is angle between sides a and b. So sin(C) = 2Area/(ab) = 23/( (2√2)*3) = 6/(6√2) = 1/√2 = √2/2. So sin(C)=√2/2 ...	analysisWe need to solve triangle with sides a=2√2, b=3, included angle between them? Wait: area formula uses two sides and included angle: area = (1/2) ab sin(C). We are given area=3. So (1/2)(2√2)(3) sin(C)=3 => (1/2)(6√2) sin(C)=3 => 3√2 sin(C)=3 => sin(C)=3/(3√2)=1/√2 = √2/2. So angle C = 45° or 135°. But since sid...	analysisWe need to find third side length of triangle given two sides a=2√2 and b=3, area = 3. Find c. We know area formula: A = 1/2 * a * b * sin(C), where C is angle between a and b. So sin(C) = 2A/(ab). Compute: ab = 2√2 * 3 = 6√2. 2A = 6. So sin(C) = 6 / (6√2) = 1/√2 = √2/2. So C = 45° or 135°? Since sin(C)=√2/2. F...	analysisWe need to solve triangle with sides a=2√2, b=3, included angle between them? Wait area given: area = 3. We know area formula: area = 1/2 * a * b * sin C, where C is angle between sides a and b. So sin C = 2Area/(ab). Compute: 23/(2√2 * 3) = 6/(6√2)=1/√2 = √2/2. So sin C = √2/2. That implies angle C = 45° or ...	analysisWe need to solve triangle with sides a=2√2, b=3, c unknown. Area is 3. We need to find c. We have two sides and area. The area formula: For sides a, b, with included angle C between them, area = (1/2)ab sin(C). So sin(C) = 2Area/(ab) = 23/( (2√2)*3 ) = 6/(6√2) = 1/√2 = √2/2. So angle C = 45° or 135°? Since s...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Two sides of an isosceles triangle $\triangle ABC$ have lengths 9 and 4. What is the area of $\triangle ABC$?	2 \sqrt{77}	100	analysisWe need to solve: Two sides of an isosceles triangle ABC have lengths 9 and 4. Which sides? We know two sides: 9 and 4. In an isosceles triangle, two sides are equal. So either the equal sides are 9 and 9, or 4 and 4. But we are told two sides have lengths 9 and 4; that suggests that the sides of the triangle a...	analysisWe need area of isosceles triangle with side lengths 9 and 4. But which sides are equal? In isosceles triangle, two sides equal. We have two sides of lengths 9 and 4. That means either 9=4? No. So the equal sides must be of one of these lengths: either the two equal sides are both 9, or both 4. But we only have...	analysisWe need area of isosceles triangle with two sides lengths 9 and 4. In an isosceles triangle, two sides equal. Which two? Could be the two sides of lengths 9 and 9, or 4 and 4, or 9 and 4 equal? But only two sides given: 9 and 4. So the triangle has sides 9, 9, 4 (isosceles with equal sides 9) or 4, 4, 9 (equal ...	analysisWe need to parse the problem: Two sides of an isosceles triangle ABC have lengths 9 and 4. What is the area of triangle ABC? We are not given which sides are equal. In an isosceles triangle, two sides are equal. So we have two equal sides of unknown length, and the third side is either 9 or 4. So possibilities:...	analysisWe need to find area of isosceles triangle with two sides lengths 9 and 4. But isosceles triangle means at least two equal sides. Which sides are equal? It says two sides have lengths 9 and 4. So the other side could be either 9 or 4? Wait: "Two sides of an isosceles triangle have lengths 9 and 4." Means that a...	{ "competition": "bmt", "dataset": "Ours", "posts": null, "source": "IF2015S.md" }
Problem 4. Let triangle $ABC$ and points $D, E, F \in (BC)$ such that $[AD$ is the bisector of angle $BAC$, $[AE$ is the bisector of angle $BAD$, and $[AF$ is the bisector of angle $CAD$. Show that: $$ AE \cdot \left(\frac{1}{AB} + \frac{1}{AD}\right) = AF \cdot \left(\frac{1}{AC} + \frac{1}{AD}\right) $$ Ion Neaţă, ...	proof	0	analysisWe need to prove identity: AE(1/AB + 1/AD) = AF(1/AC + 1/AD). Equivalent: AE(AB+AD)/(ABAD) = AF(AC+AD)/(ACAD). Multiply both sides by ABACAD: AE(AB+AD)AC = AF(AC+AD)AB. So need to show: AEAC(AB+AD) = AFAB(AC+AD). Or rearrange: AEACAB + AEACAD = AFABAC + AFABAD. Cancel common term AB*AC? ...	analysisWe need to prove identity involving lengths AE, AF, AB, AD, AC. In triangle ABC, D on BC is angle bisector of BAC. E on BC? Wait: points D, E, F in (BC). So all are on BC? D, E, F are on side BC. But E is intersection of AE bisector of angle BAD. So AE is a ray from A to BC at point E, such that AE bisects angl...	analysisWe need to prove identity involving lengths along bisectors. Let triangle ABC, D, E, F on BC? Wait: D, E, F in (BC) meaning points on BC? But AD bisects angle BAC, AE bisects angle BAD, AF bisects angle CAD. So E and F are on BC? Actually AE and AF are angle bisectors of sub-angles at A? Wait: AE is bisector of...	analysisWe need to prove equality involving lengths AE, AF, AB, AC, AD. The configuration: Triangle ABC, D, E, F on BC such that AD is bisector of angle BAC. AE is bisector of angle BAD. AF is bisector of angle CAD. So E and F are points on BC? Wait: D, E, F ∈ (BC). That means points on segment BC. AD bisects angle BAC...	analysisWe need to prove equality involving lengths of segments along the angle bisectors. Triangle ABC, points D, E, F on side BC? Wait: Problem statement: "Let triangle ABC and points D, E, F ∈ (BC) such that [AD is the bisector of angle BAC, [AE is the bisector of angle BAD, and [AF is the bisector of angle CAD." So...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Source: 1976 Euclid Part B Problem 1 ----- Triangle $ABC$ has $\angle{B}=30^{\circ}$, $AB=150$, and $AC=50\sqrt{3}$. Determine the length of $BC$.	50\sqrt{3}	0	analysisWe need to solve triangle ABC with given side AB=150, AC=50√3, angle B=30°. Determine BC. We know side AB opposite angle C? Wait labeling: Triangle ABC, angle B = 30°, side AB is between A and B, side AC between A and C. So AB is side adjacent to angle B? Actually angle B is at vertex B; side AB is adjacent to...	analysisWe need to find BC in triangle ABC with angle B=30°, AB=150, AC=50√3. Use Law of Cosines or Sines. Need to identify which side is opposite which angle: AB opposite angle C, AC opposite angle B? Wait: AB is side between A and B. AC is side between A and C. So side BC is between B and C. Angle B is at vertex B be...	analysisWe need to solve triangle ABC with given side AB=150, AC=50√3, angle at B=30°, find BC. We can use law of cosines or law of sines. We know side AB opposite angle C? Let's label: triangle ABC with vertices A, B, C. AB is side between A and B. AC side between A and C. BC side between B and C. Angle B is at vertex...	analysisWe need to find BC given triangle ABC with angle B=30°, AB=150, AC=50√3. We need BC. We can use law of cosines or law of sines. But we have two sides and included angle? Wait we have AB opposite angle C? Let's define vertices: Triangle ABC with AB=150, AC=50√3, angle B=30°. We need BC. We know side AB between v...	analysisWe need to solve triangle ABC with given side AB=150, AC=50√3, angle B=30°. We need BC. This is a triangle with known side AB opposite angle C? Wait labeling: Triangle ABC, sides AB, BC, CA. Angle B at vertex B. Side opposite B is AC? Actually side opposite angle B is AC. Indeed side AC is opposite angle B. So ...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "aops_forum" }
Given that $ BC = CD $, find the measure of $\angle CBD$ and $\angle CDB$ if $\angle BCD = 160^\circ$.	10^\circ	0	analysisWe need to analyze the geometry: There is triangle BCD, with BC = CD. So it's isosceles with base BD. Given angle BCD = 160°, which is at vertex C. We need to find angle CBD and angle CDB. In an isosceles triangle, base angles are equal: ∠CBD = ∠CDB. Sum of angles in triangle = 180°. So let each base angle be x...	analysisWe need to parse the problem: We have a triangle or quadrilateral? Points B, C, D. Given BC = CD, so C is a vertex with two equal sides BC and CD. So triangle BCD is isosceles with BC = CD. Given angle BCD = 160°. Find measures of angle CBD and angle CDB. We need to interpret: Triangle BCD is isosceles with BC...	analysisWe need to analyze the geometry problem: Given that BC = CD, find measure of angle CBD and angle CDB if angle BCD = 160°. So we have a triangle BCD where BC = CD, so it's isosceles with vertex at C? Wait: BC and CD are equal, so the equal sides are BC and CD. The vertex between them is C? Actually BC and CD sha...	analysisWe need to analyze the problem: "Given that $BC = CD$, find the measure of $\angle CBD$ and $\angle CDB$ if $\angle BCD = 160^\circ$." We have a triangle BCD with BC = CD, so it's isosceles with vertex at C? Wait: BC = CD, so sides BC and CD are equal, so triangle BCD is isosceles with base BD. The ver...	analysisWe need to analyze the problem. We have triangle BCD? Actually points B, C, D. BC = CD. So triangle BCD is isosceles with BC = CD. Angle BCD = 160°. That is angle at C between BC and CD. In triangle BCD, the vertex at C has angle 160°. Since BC = CD, the base is BD. So the base angles at B and D are equal. Let ...	{ "competition": "hmmt", "dataset": "Ours", "posts": null, "source": "guts_nov_2012_0.md" }
Task B-3.1. If $a, b, c$ are the lengths of the sides of an arbitrary triangle and $\alpha$ is the angle between the sides of lengths $b$ and $c$, prove that: $$ a^{2}=(b+c)^{2} \cdot \sin ^{2} \frac{\alpha}{2}+(b-c)^{2} \cdot \cos ^{2} \frac{\alpha}{2} $$	proof	0	analysisWe need to prove identity for sides of triangle: a^2 = (b+c)^2 sin^2(α/2) + (b-c)^2 cos^2(α/2). Here α is angle opposite side a? Wait: "angle between the sides of lengths b and c" means angle at vertex between sides b and c, which is the angle opposite side a. Yes α is angle opposite side a. So law of cos: a^2 ...	analysisWe need to prove identity: For triangle with side lengths a,b,c, and angle α between sides b and c. So a is opposite α? Wait: "angle between sides of lengths b and c" means α is the angle between sides b and c, so α is opposite side a. So standard notation: side a opposite angle α. So law of cosines: a^2 = b^2 ...	analysisWe need to prove identity: a^2 = (b+c)^2 sin^2(α/2) + (b-c)^2 cos^2(α/2). Here a is opposite α? Wait: a is side length opposite angle α? Problem says a, b, c lengths of sides of an arbitrary triangle and α is angle between sides of lengths b and c. So α is angle between sides b and c, i.e., α is the angle at ve...	analysisWe need to prove identity: For triangle with sides a opposite angle alpha? Wait: The problem states: "a, b, c are lengths of sides of an arbitrary triangle and α is the angle between the sides of lengths b and c." So α is angle between sides b and c, meaning α is the angle opposite side a. So side a is opposite...	analysisWe need to prove identity: For triangle with sides a,b,c, angle α opposite side a? Wait, angle between sides of lengths b and c is α, so α is angle at vertex between sides b and c, opposite side a. So a is opposite α. We need to prove: a^2 = (b+c)^2 sin^2(α/2) + (b-c)^2 cos^2(α/2). We can use law of cosines: ...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
7. Adam has a triangular field $A B C$ with $A B=5, B C=8$ and $C A=11$. He intends to separate the field into two parts by building a straight fence from $A$ to a point $D$ on side $B C$ such that $A D$ bisects $\angle B A C$. Find the area of the part of the field $A B D$. (A) $\frac{4 \sqrt{21}}{11}$; (B) $\frac{4 \...	\frac{5\sqrt{21}}{4}	100	analysisWe need area of triangle ABD where AD is angle bisector of angle BAC in triangle ABC with sides AB=5, BC=8, CA=11. We can compute area of triangle ABC first, then use angle bisector theorem to find BD length, then compute area of ABD via base BD and height from A or via ratio of areas using angle bisector prop...	analysisWe need area of triangle ABD, where D is on BC such that AD bisects angle BAC. Triangle ABC sides: AB=5, BC=8, CA=11. We need area of ABD. We know angle bisector theorem: BD/DC = AB/AC = 5/11. Also BD + DC = BC = 8. So BD = (5/ (5+11)) * 8 = (5/16)*8 = 40/16 = 5/2 = 2.5. DC = 8 - 2.5 = 5.5 = 11/2. Good. We ne...	analysisWe need area of triangle ABD given triangle ABC side lengths AB=5, BC=8, CA=11. D is on BC such that AD bisects angle BAC. Find area of triangle ABD. We can use Angle Bisector Theorem: BD/DC = AB/AC = 5/11. Since BC=8, let BD = (5/(5+11))8 = (5/16)8 = 5/2 = 2.5. DC = 8 - 2.5 = 5.5 = 11/2. Now we need area o...	analysisWe need area of triangle ABD where AD bisects angle BAC. We have triangle ABC with side lengths AB=5, BC=8, CA=11. We need to find area of triangle ABD. Let’s denote sides: AB=c=5, BC=a=8, CA=b=11. Vertex A opposite side a? Wait standard notation: side a=BC opposite A, side b=CA opposite B, side c=AB opposite ...	analysisWe need area of triangle ABD where AD is angle bisector at A of triangle ABC with sides AB=5, BC=8, CA=11. Need to find length BD or ratio BD:DC from angle bisector theorem. Then find area ABD. Given triangle sides: AB=5, AC=11, BC=8. Check triangle inequality: 5+8>11? 13>11 yes; 5+11>8? 16>8 yes; 8+11>5? 19>5...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
7. Given in $\triangle A B C$, $a=2 b, \cos B=\frac{2 \sqrt{2}}{3}$, then $\sin \frac{A-B}{2}+\sin \frac{C}{2}=$ $\qquad$	\frac{\sqrt{10}}{3}	40	analysisWe need to solve: In triangle ABC, side lengths satisfy a = 2b, where a = BC? Wait: Usually in triangle ABC, side a opposite A, b opposite B, c opposite C. So a=BC opposite A. So a=2b. Also cos B = 2√2 / 3. Need value of sin((A-B)/2) + sin(C/2). We need to compute this expression in terms of known values. Use ...	analysisWe need to solve: In triangle ABC, side lengths satisfy a = 2b (where a opposite A, b opposite B, c opposite C). Also cos B = 2√2 / 3. We need to find value of sin((A - B)/2) + sin(C/2). We need to use triangle relationships. We know sides ratio a: b: c. We have a = 2b. Use Law of Cosines to relate sides to an...	analysisWe need to solve: In triangle ABC, side lengths a=2b, cos B = 2√2 / 3. Find sin((A-B)/2) + sin(C/2). We need to express angles A,B,C. Known side ratio a:2b. Use law of cosines? We have side lengths opposite angles: a opposite A, b opposite B, c opposite C. We know a=2b. Cos B given. We can find sides ratio or ...	analysisWe need to find expression value: sin((A-B)/2)+ sin(C/2). Given triangle ABC with side lengths a,b,c opposite angles A,B,C. Given a=2b. Cos B = 2√2/3. We need to compute sin((A-B)/2)+ sin(C/2). Use known relations. We know side ratio a=2b. By law of sines: a/sin A = b/sin B = c/sin C = 2R. So a/b = sin A / si...	analysisWe need to compute expression given triangle ABC with side lengths relation a = 2b, and cos B = 2√2/3. We must find sin((A-B)/2)+sin(C/2). We need to use law of sines, cos rule, etc. Let’s denote standard notation: sides a opposite A, b opposite B, c opposite C. Given a = 2b. Also cos B = 2√2/3. Goal: compute...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
The sides of a triangle are equal to a, b, c. It is known that $\mathrm{a}^{3}=\mathrm{b}^{3}+\mathrm{c}^{3}$. Prove that this triangle is acute.	proof	0	analysisWe need to show that if a^3 = b^3 + c^3, then triangle with sides a,b,c is acute. Equivalent to a^3 > b^3 + c^3? Wait equality given. So a^3 = b^3 + c^3. Since sides are positive, a is largest? Let's check: If a^3 = b^3 + c^3, then a > b, a > c. Because b^3 + c^3 > b^3, so a^3 > b^3, so a > b. Similarly a > c. ...	analysisWe need to analyze the problem: For a triangle with side lengths a, b, c, we are given that a^3 = b^3 + c^3. Prove that the triangle is acute-angled. So we need to show that all angles are less than 90°, i.e., the triangle is acute. Equivalent: show that a^2 < b^2 + c^2, b^2 < a^2 + c^2, c^2 < a^2 + b^2. Since ...	analysisWe need to prove: For a triangle with side lengths a,b,c satisfying a^3 = b^3 + c^3, the triangle is acute. Equivalent: show all angles < 90°. We can use cosine law: cos A = (b^2 + c^2 - a^2)/(2bc). Since a is largest? Actually from a^3 = b^3 + c^3, we can deduce a > b and a > c? Let's check. Suppose b,c positi...	analysisWe need to prove that if a^3 = b^3 + c^3 for sides of a triangle, then the triangle is acute. That is, all angles less than 90 degrees. Equivalent to show that for each pair of sides, the square of the longest side is less than sum of squares of other two sides. Since a^3 = b^3 + c^3, we can deduce something ab...	analysisWe need to prove that if a^3 = b^3 + c^3 for sides of a triangle, then triangle is acute-angled. We can use inequality to show all angles are less than 90 degrees. We have triangle with sides a, b, c, satisfying a^3 = b^3 + c^3. Since triangle inequality requires a < b + c, etc. But we need to deduce something...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
A right triangle has perimeter $2008$, and the area of a circle inscribed in the triangle is $100\pi^3$. Let $A$ be the area of the triangle. Compute $\lfloor A\rfloor$.	31541	100	analysisWe need to solve: right triangle with perimeter 2008, area of incircle is 100π^3. Wait area of circle inscribed is 100π^3. Usually area of incircle is π r^2. So π r^2 = 100π^3 => r^2 = 100π^2 => r = 10π. But radius is 10π. That seems huge relative to sides. But perimeter is 2008. Let's analyze. Let sides of ri...	analysisWe need to solve: right triangle with perimeter P=2008. Inscribed circle area = 100π^3. Wait area of circle inscribed: π r^2 = 100 π^3? That seems odd: area of circle is π r^2. If area equals 100 π^3, then r^2 = 100 π^2 => r = 10 π. But radius of incircle of right triangle is r = (a + b - c)/2? Wait formula: Fo...	analysisWe need to parse problem: Right triangle perimeter 2008. Area of incircle = 100π^3. Find floor of area of triangle A. Let sides a, b, c with c hypotenuse. Perimeter: a + b + c = 2008. Right triangle: a^2 + b^2 = c^2. Inradius r: formula r = (a + b - c)/2 (for right triangle). Also r = A / s where s = semiperim...	analysisWe need to solve: Right triangle with perimeter 2008, inscribed circle area = 100π^3. Wait area of circle inscribed: area = π r^2. So π r^2 = 100 π^3. That implies r^2 = 100 π^2, so r = 10 π. Since radius positive. Given right triangle with sides a,b,c (c hypotenuse). Perimeter a+b+c = 2008. For a right triang...	analysisWe need to analyze the problem: Right triangle with perimeter 2008. Area of incircle is 100 π^3. Need area of triangle A. Compute floor of A. We know formula: radius r of incircle of right triangle with legs a, b, hypotenuse c: r = (a + b - c)/2. Wait is that correct? For any triangle, r = (2Δ)/(a+b+c). For ri...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "aops_forum" }
Segments $AA'$, $BB'$, and $CC'$ are the bisectrices of triangle $ABC$. It is known that these lines are also the bisectrices of triangle $A'B'C'$. Is it true that triangle $ABC$ is regular?	\triangle ABC	0	analysisWe need to analyze the problem: In triangle ABC, AA', BB', CC' are bisectors (internal angle bisectors) of triangle ABC. It's known that these lines are also the bisectors of triangle A'B'C'. So the internal angle bisectors of ABC also bisect angles of triangle A'B'C'. Does that imply triangle ABC is regular (i...	analysisWe need to analyze: We have triangle ABC, with internal angle bisectors AA', BB', CC'. These meet at incenter I of ABC. They also are bisectors of triangle A'B'C' (the triangle formed by the feet of the bisectors? Wait: "Segments AA', BB', CC' are the bisectrices of triangle ABC. It is known that these lines ar...	analysisWe need to analyze the problem: In triangle ABC, we have lines AA', BB', CC' that are the bisectors of triangle ABC. So A' is intersection of BB' and CC'? Wait: Usually, in triangle ABC, the internal angle bisectors meet at incenter. But here they say "segments AA', BB', CC' are the bisectrices of triangle ABC"...	analysisWe need to analyze the problem: In triangle ABC, we have segments AA', BB', CC' as the bisectors of triangle ABC. So A', B', C' are points where the internal angle bisectors of triangle ABC meet the opposite sides? Wait: "Segments AA', BB', CC' are the bisectrices of triangle ABC" likely means that AA', BB', CC...	analysisWe need to analyze problem: Given triangle ABC, with bisectors of its sides? Wait "bisectrices" maybe means angle bisectors? Usually "bisectors" of triangle are internal angle bisectors. So AA', BB', CC' are the internal angle bisectors of triangle ABC. And it is known that these lines are also the bisectors of...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "aops_forum" }
13. In the diagram shown, sides $P Q$ and $P R$ are equal. Also $\angle Q P R=40^{\circ}$ and $\angle T Q P=\angle S R Q$. What is the size of $\angle T U R$ ? A $55^{\circ}$ B $60^{\circ}$ C $65^{\circ}$ D $70^{\circ}$ E $75^{\circ}$	70	40	analysisWe need to parse the problem. "In the diagram shown, sides PQ and PR are equal. Also ∠QPR = 40° and ∠TQP = ∠SRQ. What is the size of ∠TUR?" We need to imagine a diagram. There are points P, Q, R, T, S, U. We know PQ = PR, so triangle PQR is isosceles with base QR, apex at P. Angle at P is 40°. So base angles at...	analysisWe need to analyze problem 13: "In the diagram shown, sides PQ and PR are equal. Also angle QPR=40° and angle TQP = angle SRQ. What is the size of angle TUR? A 55°, B 60°, C 65°, D 70°, E 75°." We don't have diagram. Need to deduce geometry. Likely points: P, Q, R maybe forming triangle PQR is isosceles with P...	analysisWe need to parse problem: "In the diagram shown, sides PQ and PR are equal. Also ∠QPR = 40° and ∠TQP = ∠SRQ. What is the size of ∠TUR?" So we have a diagram with points P, Q, R, T, S, U? We need to deduce geometry. The diagram likely has some lines: maybe P is vertex of an isosceles triangle with Q and R. Point...	analysisWe need to parse the problem. There's a diagram with points P, Q, R, S, T, U? We need to understand geometry. The problem states: sides PQ and PR are equal. Also angle QPR = 40°. And angle TQP = angle SRQ. We are asked for size of angle TUR. We have choices 55°,60°,65°,70°,75°. We need to deduce the geometry. ...	analysisWe need to parse the problem: We have a diagram, presumably with points P, Q, R, S, T, U. Sides PQ and PR equal, angle QPR = 40°. Also angle TQP = angle SRQ. We need angle TUR. Options: 55°,60°,65°,70°,75°. We need to figure geometry. We don't have diagram. We must reconstruct configuration. Likely triangle PQ...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
8,9 In an isosceles triangle $ABC (AB = BC)$, the bisectors $BD$ and $AF$ intersect at point $O$. The ratio of the area of triangle $DOA$ to the area of triangle $BOF$ is $\frac{3}{8}$. Find the ratio $\frac{AC}{AB}$.	\frac{1}{2}	40	analysisWe have isosceles triangle ABC with AB = BC. So vertex B is apex? Actually AB=BC so B is vertex at apex of isosceles with base AC. So triangle ABC with AB=BC. We have bisectors BD and AF intersect at point O. Which bisectors? BD is bisector of angle B? Since BD is a bisector, likely BD bisects angle B. AF bise...	analysisWe need to parse problem: In an isosceles triangle ABC with AB = BC. So B is apex? Wait AB=BC, so B is vertex between equal sides AB and BC. So base is AC. So triangle is isosceles with vertex at B. Altitude? Not altitude. We have bisectors BD and AF: BD is bisector of angle B? Wait BD is a bisector of angle B?...	analysisWe need to parse the problem. Triangle ABC is isosceles with AB = BC. So B is apex? Usually AB=BC means B is vertex with equal sides AB and BC. So base is AC. So B is apex at top. The bisectors BD and AF intersect at O. Which bisectors? BD is the bisector of angle B? Actually BD is the bisector of angle B? Wait...	analysisWe need to parse the problem: Isosceles triangle ABC with AB = BC. So B is the vertex at base? Wait: AB=BC indicates B is the apex? Actually AB and BC are adjacent sides; so B is the vertex connecting equal sides AB and BC. So triangle ABC is isosceles with apex at B, base AC. So AB=BC. Then BD and AF are angle...	analysisWe need to solve: In isosceles triangle ABC with AB=BC, BD is bisector of angle B? Wait BD is bisector: "the bisectors BD and AF intersect at point O". Since triangle ABC is isosceles with AB=BC, vertex B is at apex? Actually AB=BC so B is vertex at equal sides AB and BC. So base AC. BD is angle bisector of ang...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
Let $ABC$ be a right triangle at $A$ such that $AB=156$ and $AC=65$. We denote $H$ as the foot of the altitude from $A$. Determine the value of $AH$. ## 4 Junior Eliminatory: Solutions	60	80	analysisWe need to find altitude from right angle to hypotenuse? Wait: Right triangle at A means angle A is 90 degrees. AB=156, AC=65. So legs are AB and AC? But usually right triangle at A, so AB and AC are legs. Hypotenuse is BC. So altitude from A to BC? Actually H is foot of altitude from A onto BC? Wait: "foot of ...	analysisWe need to parse problem: Right triangle at A: so angle A = 90°. AB = 156, AC = 65. So AB and AC are legs? Wait, A is right angle; AB and AC are legs adjacent to right angle. So AB = 156, AC = 65. Hypotenuse BC = sqrt(156^2 + 65^2). We need altitude from A to hypotenuse BC. So AH is altitude from right angle to...	analysisWe have a right triangle at A. So angle A = 90 degrees. AB and AC are legs? Wait, right triangle at A, so sides AB and AC are legs (perpendicular). So AB=156, AC=65. We need the altitude from A to hypotenuse BC? Wait, H is foot of altitude from A onto BC? Wait, the problem: "Let ABC be a right triangle at A suc...	analysisWe have a right triangle at A. So angle A is 90°, AB and AC are legs? Wait: Right triangle at A means A is the right angle, so AB and AC are legs. AB=156, AC=65. So the hypotenuse is BC. We need AH, where H is foot of altitude from A onto BC? Wait: "foot of the altitude from A" means the altitude from A to the ...	analysisWe need to solve: Right triangle at A, AB=156, AC=65. So AB and AC are legs? Wait, right triangle at A means angle A = 90°. So AB and AC are the legs adjacent to A. So AB = 156, AC = 65. Hypotenuse BC? Compute BC = sqrt(AB^2 + AC^2) = sqrt(156^2 + 65^2). Let's compute: 156^2 = 24336. 65^2 = 4225. Sum = 28561. s...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
6. On the same side of line $l$, there are three adjacent equilateral triangles $\triangle A D E, \triangle A F G, \triangle A B C$, and $G, A, B$ are all on line $l$. Let the side lengths of these triangles be $b, c, a$ respectively. Connect $G D$ to intersect $A E$ at $N$, and connect $B N$ to intersect $A C$ at $L$....	AL=\frac{}{++}	0	analysisWe need to prove that AL = (a b c)/(ab + bc + ac) given adjacent equilateral triangles on same side of line l. Triangles ADE, AFG, ABC share vertex A and base along line l? Wait: G, A, B are all on line l. So line l contains G, A, B. So A is between G and B? "On the same side of line l, there are three adjacent...	analysisWe need to analyze geometry problem: There are three adjacent equilateral triangles on same side of line l. Triangles ADE, AFG, ABC. G, A, B are all on line l. So line l is a common base? Let's parse: Triangles share vertex A? Wait: Triangles ADE, AFG, ABC: each triangle has vertex A? Actually triangles: ADE: v...	analysisWe need to analyze geometry. We have three adjacent equilateral triangles on same side of line l. Triangles: ADE (side b), AFG (side c), ABC (side a). G, A, B are all on line l. So line l passes through G, A, B. Triangles are adjacent: meaning they share sides? Let's picture: Equilateral triangle ADE has side A...	analysisWe need to analyze the geometry problem: There are three adjacent equilateral triangles on same side of line l: ΔADE, ΔAFG, ΔABC. G, A, B are all on line l. So line l is a common base line for all three triangles? Let's parse: "On the same side of line l, there are three adjacent equilateral triangles ΔADE, ΔAF...	analysisWe need to analyze geometry problem. Let's restate: On same side of line l, there are three adjacent equilateral triangles: ΔADE, ΔAFG, ΔABC. G, A, B are all on line l. So line l contains points G, A, B in that order? "Adjacent" meaning they share vertices? Let's parse: Triangles share side AD? Wait: Triangles ...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
454. From the vertex $C$ of triangle $A B C$, a median and an angle bisector are drawn, forming an angle $\alpha$ between them. Prove that $$ \operatorname{tg}^{2} \frac{C}{2}=\operatorname{tg} \alpha \cdot \operatorname{tg} \beta $$ where $\beta$ is the acute angle formed by the angle bisector and side $A B$.	proof	0	analysisWe need to prove tan^2(C/2) = tan α * tan β. Setup: triangle ABC, median from C to side AB at point D (midpoint). Angle bisector from C to side AB at point E (point on AB). The angle between median and bisector is α. β is acute angle between bisector and side AB. So angle between bisector CE and AB is β. That m...	analysisWe need to prove relation: tan^2(C/2) = tan α * tan β, where α is angle between median from C and angle bisector from C, β is acute angle between bisector and side AB. Let's restate: Triangle ABC, median from C to side AB (point M midpoint of AB). Angle bisector from C to side AB (point T intersection of bisect...	analysisWe need to parse the problem: In triangle ABC, from vertex C, we draw a median and an angle bisector. The angle between them is α. Prove that tan^2(C/2) = tan α * tan β, where β is the acute angle between the angle bisector and side AB. So we have triangle ABC, with vertex C. Let M be the midpoint of AB. Let D ...	analysisWe need to parse the problem. We have triangle ABC. From vertex C, we draw a median and an angle bisector. The angle between them is α. We need to prove that tan^2(C/2) = tan α * tan β, where β is the acute angle between the angle bisector and side AB. So we need to find relation between these angles. Let's de...	analysisWe need to analyze the problem: In triangle ABC, from vertex C, a median and an angle bisector are drawn. They form an angle alpha between them. We need to prove that tan^2(C/2) = tan alpha * tan beta where beta is the acute angle formed by the angle bisector and side AB. Essentially, we have triangle ABC. L...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
10.345. The radius of the circle inscribed in a triangle is 2 cm. A point of tangency of this circle divides one of the sides into segments of length 4 and $6 \mathrm{~cm}$. Determine the type of the triangle and calculate its area.	24^2	0	analysisWe need to analyze: Triangle with incircle radius r=2 cm. One side is divided by point of tangency into segments of 4 and 6 cm. So the side length is 10 cm. The incircle touches that side at a point dividing it into 4 and 6. So the two segments adjacent to the vertices are 4 and 6. For a triangle with incircle,...	analysisWe have a triangle with inradius r=2. The incircle touches one side at a point that divides that side into segments 4 and 6. Which side? Some side of triangle. The incircle touches each side at a point, dividing each side into segments equal to semiperimeter minus adjacent side lengths? Wait: For triangle with ...	analysisWe need to analyze the problem: "The radius of the circle inscribed in a triangle is 2 cm. A point of tangency of this circle divides one of the sides into segments of length 4 and 6 cm. Determine the type of the triangle and calculate its area." We need to find the type of triangle (likely right, acute, obtus...	analysisWe need to analyze the problem: "10.345. The radius of the circle inscribed in a triangle is 2 cm. A point of tangency of this circle divides one of the sides into segments of length 4 and 6 cm. Determine the type of the triangle and calculate its area." So we have a triangle ABC with incircle radius r = 2. Th...	analysisWe need to solve: Triangle has incircle radius r=2 cm. One side is divided by point of tangency into segments lengths 4 and 6 cm. That side is thus of length 10 cm. We need to determine type of triangle (likely right, obtuse, acute?) and calculate its area. We have triangle ABC with incircle radius r = 2. Supp...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
(treated during the session) Let $ABCD$ be a quadrilateral, $P$ the intersection of $(AC)$ and $(BD)$. Suppose that $\widehat{CAD}=50^{\circ}$, $\widehat{BAC}=70^{\circ}$, $\widehat{DCA}=40^{\circ}$, and $\widehat{ACB}=20^{\circ}$. Calculate the angle $\widehat{CPD}$.	70	60	analysisWe need to find angle CPD where P is intersection of diagonals AC and BD of quadrilateral ABCD. Given angles at vertex A and C: ∠CAD = 50°, ∠BAC = 70°, ∠DCA = 40°, ∠ACB = 20°. Let's denote quadrilateral ABCD with vertices in order? Usually quadrilateral ABCD with vertices in order. So diagonals AC and BD inters...	analysisWe need to analyze geometry problem: quadrilateral ABCD with intersection point P of diagonals AC and BD. Given angles: ∠CAD = 50°, ∠BAC = 70°, ∠DCA = 40°, ∠ACB = 20°. So these are angles at vertex A and C. Let's denote: - ∠CAD is angle between CA and AD at vertex A: 50°. - ∠BAC is angle between BA and AC at v...	analysisWe need to analyze the problem: Quadrilateral ABCD with intersection of diagonals AC and BD at P. Given angles at vertex A: ∠CAD = 50°, ∠BAC = 70°. At vertex C: ∠DCA = 40°, ∠ACB = 20°. We need ∠CPD. The quadrilateral is not necessarily cyclic; we have intersection of diagonals. We need to find angle between CP ...	analysisWe need to find angle CPD given quadrilateral ABCD with intersection of diagonals at P. Given angles at vertex A and C? Let's parse: ∠CAD = 50°, ∠BAC = 70°, ∠DCA = 40°, ∠ACB = 20°. These are angles at vertex A and C with respect to triangle ABC? Actually ABCD quadrilateral, but we only have angles at A and C in...	analysisWe need to solve geometry problem: quadrilateral ABCD, with intersection point P of diagonals AC and BD. Given angles at vertex A and C: ∠CAD=50°, ∠BAC=70°, ∠DCA=40°, ∠ACB=20°. We need ∠CPD. Let's parse: ∠CAD is angle between CA and AD at A? Wait vertex A? Actually CAD: vertex A? Wait notation: ∠CAD: vertex A? ...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }
10.368 Calculate the length of the angle bisector of angle $A$ in triangle $A B C$ with side lengths $a=18$ cm, $b=15$ cm, $c=12$ cm.	10	0	analysisWe need length of angle bisector from vertex A of triangle with sides a=BC=18, b=CA=15, c=AB=12. Standard formula: length of bisector from vertex A: $l_a = \frac{2bc \cos(A/2)}{b+c}$. Or using formula: $l_a^2 = bc \left[1 - \frac{a^2}{(b+c)^2}\right]$. That matches the retrieved solution: \(CC_1^2 = bc[1 - ...	analysisWe need length of angle bisector from vertex A in triangle with side lengths a=BC=18, b=CA=15, c=AB=12. Convention: side a opposite A, side b opposite B, side c opposite C. So a=BC=18, b=CA=15, c=AB=12. Angle A is between sides b and c. We need length of bisector from A to side a (BC). Formula: length of intern...	analysisWe need to calculate length of angle bisector from vertex A to side BC. Standard formula: length of angle bisector from vertex A to side a opposite A? Wait, side a opposite A is BC, length a=18? Wait given sides a=18, b=15, c=12. Convention: side a opposite A is BC, side b opposite B is AC, side c opposite C is...	analysisWe need to compute length of angle bisector from vertex A in triangle ABC with side lengths a,b,c. Usually notation: side a opposite A, side b opposite B, side c opposite C. So a=BC=18, b=CA=15, c=AB=12. We want length of bisector from A to side a? Wait angle A is at vertex A, opposite side a. The bisector from...	analysisWe have a triangle ABC with side lengths: a=BC? Wait standard notation: Usually side a opposite A, b opposite B, c opposite C. So side lengths: a=BC=18 cm, b=CA=15 cm, c=AB=12 cm. We need length of angle bisector of angle A. That is the line from vertex A to side BC, dividing angle A into two equal parts, hitti...	{ "competition": "Numina-1.5", "dataset": "NuminaMath-1.5", "posts": null, "source": "olympiads" }

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