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"content": "3. (6 points) A construction company was building a tunnel. When $\\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \\%$ and reduced the working hours to $80 \\%$ of the original. As a result, it took a ... |
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"content": "4. Given the three sides of an obtuse triangle are 3, $4, x$, then the range of values for $x$ is ( ).\n(A) $1<x<7$.\n(B) $5 \\ll x<7$.\n(C) $1<x<\\sqrt{7}$.\n(D) $5<x<7$ or $1<x<\\sqrt{7}$.",
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"content": "<think>\nOkay, so I have this problem here where I need to fi... |
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"content": "1. Solve the equation: $\\frac{8 x+13}{3}=\\frac{19-12 x}{2}-\\left(16 x-\\frac{7-4 x}{6}\\right)$.",
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"content": "<think>\nOkay, so I need to solve this equation: (8x + 13)/3 = (19 - 12x)/2 - (16x - (7 - 4x)/6). Hmm, that looks a bit complicated with all the fractio... |
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"content": "A right-angled triangle has side lengths that are integers. What could be the last digit of the area's measure, if the length of the hypotenuse is not divisible by 5?",
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"content": "<think>\nOkay, so I need to figure out the possible last digits of the area of a righ... |
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"content": "Let $a_n\\ (n\\geq 1)$ be the value for which $\\int_x^{2x} e^{-t^n}dt\\ (x\\geq 0)$ is maximal. Find $\\lim_{n\\to\\infty} \\ln a_n.$",
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"content": "<think>\nOkay, so I need to find the limit as n approaches infinity of the natural logarithm of a_n, where a_n is the... |
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"content": "11. Given that the internal angles $A, B, C$ of $\\triangle ABC$ have opposite sides $a, b, c$ respectively, and $\\sqrt{3} b \\cos \\frac{A+B}{2}=c \\sin B$.\n(1) Find the size of $\\angle C$;\n(2) If $a+b=\\sqrt{3} c$, find $\\sin A$.",
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"content": "<think>\nOkay, ... |
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"content": "Example 6 The rules of a \"level-up game\" stipulate: On the $n$-th level, a die must be rolled $n$ times. If the sum of the points obtained from these $n$ rolls is greater than $2^{n}$, the level is considered passed. Questions:\n(1) What is the maximum number of levels a person can pass in this ... |
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"content": "2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation.",
"role": "user"
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"content": "<think>\nOkay, let's tackle this problem. So,... |
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"content": "Problem 6. (8 points) In the plane, there is a non-closed, non-self-intersecting broken line consisting of 31 segments (adjacent segments do not lie on the same straight line). For each segment, the line defined by it is constructed. It is possible for some of the 31 constructed lines to coincide.... |
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"content": "7.1. Solve the equation $\\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \\cdot 2 \\cdot \\ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a solution is a trip... |
TsukiOwO/Translation-Sample-Dataset
Data Source
This dataset is a portion of open-r1/OpenR1-Math-220k.
Purpose
Serves as sample data for translation text.
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