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 |
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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)\nLet $0<x_{1}<x_{2}<\\cdots<x_{n}<1$, prove that:\n$$\n\\left(1-x_{n}\\right)^{2}\\(...TRUNCATED) | proof | 0 | "analysisWe need to prove inequality:\n\nGiven 0 < x1 < x2 < ... < xn < 1.\n\nWe need to show:\n\n(1(...TRUNCATED) | "analysisWe need to prove inequality: for increasing sequence 0< x1 < x2 < ... < xn < 1, prove\n\n(1(...TRUNCATED) | "analysisWe need to prove inequality: for 0 < x1 < x2 < ... < xn < 1, we have\n\n(1 - x_n)^2 * sum_{(...TRUNCATED) | "analysisWe need to prove inequality:\n\nFor 0 < x1 < x2 < ... < xn < 1, prove\n\n(1 - x_n)^2 * [ su(...TRUNCATED) | "analysisWe need to prove inequality:\n\nGiven 0 < x1 < x2 < ... < xn < 1. Show that:\n\n(1 - x_n)^2(...TRUNCATED) | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} |
"$4.30 \\sin 10 \\alpha \\sin 8 \\alpha + \\sin 8 \\alpha \\sin 6 \\alpha - \\sin 4 \\alpha \\sin 2 (...TRUNCATED) | 2\cos2\alpha\sin6\alpha\sin10\alpha | 0 | "analysisWe need to simplify expression: 4.30 sin10α sin8α + sin8α sin6α - sin4α sin2α. The no(...TRUNCATED) | "analysisWe need to evaluate expression: \\(4.30 \\sin 10 \\alpha \\sin 8 \\alpha + \\sin 8 \\alpha (...TRUNCATED) | "analysisWe need to simplify expression: $4.30 \\sin 10\\alpha \\sin 8\\alpha + \\sin 8\\alpha \\sin(...TRUNCATED) | "analysisWe need to simplify expression: 4.30 sin10α sin8α + sin8α sin6α - sin4α sin2α. Wait 4(...TRUNCATED) | "analysisWe need to compute expression: 4.30 sin10α sin8α + sin8α sin6α - sin4α sin2α. Probabl(...TRUNCATED) | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} |
"Example 6.49 (2007 Serbia Mathematical Olympiad) $x, y, z>0, x+y+z=1$, prove that\n$$\\frac{x^{k+2}(...TRUNCATED) | \frac{1}{7} | 0 | "analysisWe need to prove inequality:\n\nFor positive reals x,y,z>0 with sum 1, prove that\n\nS = su(...TRUNCATED) | "analysisWe need to prove inequality for positive x,y,z with sum 1, and exponent k? k presumably int(...TRUNCATED) | "analysisWe need to prove inequality for positive reals x,y,z summing to 1. For general k? The state(...TRUNCATED) | "analysisWe need to prove inequality: For positive x,y,z with sum 1, for integer k? Problem states k(...TRUNCATED) | "analysisWe need to prove inequality: For positive reals x,y,z >0 with sum 1, and integer k? It says(...TRUNCATED) | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "inequalities"
} |
"### 3.485 Find the maximum value of the expression\n\n$$\nA=\\frac{1}{\\sin ^{6} \\alpha+\\cos ^{6}(...TRUNCATED) | 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. Le(...TRUNCATED) | "analysisWe need maximum of A=1/(sin^6 α + cos^6 α) for α in [0, π/2]. Equivalent to minimize de(...TRUNCATED) | "analysisWe need to find maximum of A = 1/(sin^6 α + cos^6 α) for α in [0, π/2]. Equivalent to m(...TRUNCATED) | "analysisWe need to find max of A = 1/(sin^6 α + cos^6 α) for α in [0, π/2]. Equivalent to minim(...TRUNCATED) | "analysisWe need to find maximum of A = 1/(sin^6 α + cos^6 α) for α in [0, π/2]. Equivalent to m(...TRUNCATED) | {
"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(...TRUNCATED) | (4,1),(1,4) | 100 | "analysisWe need to solve system: (x-y)(x^2 - y^2) = 45, and x + y = 5.\n\nSimplify: x^2 - y^2 = (x-(...TRUNCATED) | "analysisWe need to solve system: (x - y)(x^2 - y^2) = 45, and x + y = 5. Let's analyze.\n\nWe can r(...TRUNCATED) | "analysisWe need to solve system:\n\n(1) (x - y)(x^2 - y^2) = 45\n(2) x + y = 5\n\nWe need to find r(...TRUNCATED) | "analysisWe need to solve system: (x-y)(x^2 - y^2) = 45, and x + y = 5. Let's analyze.\n\nLet x + y (...TRUNCATED) | "analysisWe need to solve system:\n\n(x-y)(x^2 - y^2) = 45\nx + y = 5\n\nSimplify: x^2 - y^2 = (x-y)(...TRUNCATED) | {
"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}(...TRUNCATED) | proof | 0 | "analysisWe need to analyze the inequality: compare \\(x^3 + y^3 + z^3 + 3xyz\\) with \\(x^2(y+z)+ y(...TRUNCATED) | "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(...TRUNCATED) | "analysisWe need to compare two expressions: A = x^3 + y^3 + z^3 + 3xyz; B = x^2(y+z) + y^2(z+x) + z(...TRUNCATED) | "analysisWe need to compare two expressions: S1 = x^3 + y^3 + z^3 + 3xyz, and S2 = x^2(y+z) + y^2(z+(...TRUNCATED) | "analysisWe need to compare two expressions: S1 = x^3 + y^3 + z^3 + 3xyz, and S2 = x^2(y+z)+y^2(z+x)(...TRUNCATED) | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} |
"## Problem 1\n\nLet $\\mathrm{a}, \\mathrm{b}, \\mathrm{c}$ be strictly positive real numbers. Show(...TRUNCATED) | proof | 0 | "analysisWe need to prove inequality:\n\nFor positive a,b,c,\n\nS = a/(a^2 + bc) + b/(b^2 + ac) + c/(...TRUNCATED) | "analysisWe need to prove inequality for positive real a,b,c:\n\nSum a/(a^2+bc) ≤ 1/2 (1/a + 1/b +(...TRUNCATED) | "analysisWe need to prove inequality for positive reals a,b,c:\n\nSum a/(a^2+bc) ≤ 1/2 (1/a + 1/b (...TRUNCATED) | "analysisWe need to prove inequality:\n\nFor positive a,b,c,\n\nSum_{cyc} a/(a^2 + bc) <= 1/2 * (1/a(...TRUNCATED) | "analysisWe need to prove inequality: for positive a,b,c:\n\nSum_{cyc} a/(a^2+bc) <= 1/2*(1/a + 1/b (...TRUNCATED) | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} |
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