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[
{
"content": "A crew of rowers travels $3.5 \\mathrm{~km}$ downstream and the same distance upstream in a total of 1 hour and 40 minutes in a river. The speed of the river current is $2 \\mathrm{~km}$ per hour. How far would the rowing team travel in 1 hour in still water?",
"role": "user"
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{
"con... | 1,957 | open-r1/OpenR1-Math-220k |
[
{
"content": "19.19 The interior angles of a convex $n$-sided polygon are denoted as $A, B, C, \\cdots$, and\n$$\n\\lg \\sin A+\\lg \\sin B+\\lg \\sin C+\\cdots=0 .\n$$\n\nThen the polygon is\n(A) an isosceles trapezoid.\n(B) a rectangle.\n(C) a right triangle.\n(D) a regular hexagon. (Anhui Province, Anqing an... | 3,074 | open-r1/OpenR1-Math-220k |
[
{
"content": "$10 \\cdot 33$ For $p=1,2, \\cdots, 10$, let $S_{p}$ be the sum of the first 40 terms of an arithmetic sequence with the first term $p$ and common difference $2 p-1$. Then $S_{1}+S_{2}+\\cdots+S_{10}$ is\n(A) 80000.\n(B) 80200.\n(C) 80400.\n(D) 80600.\n(E) 80800.\n(25th American High School Mathem... | 3,050 | open-r1/OpenR1-Math-220k |
[
{
"content": "The expression $\\frac{2021}{2020} - \\frac{2020}{2021}$ is equal to the fraction $\\frac{p}{q}$ in which $p$ and $q$ are positive integers whose greatest common divisor is ${ }1$. What is $p?$\n$(\\textbf{A})\\: 1\\qquad(\\textbf{B}) \\: 9\\qquad(\\textbf{C}) \\: 2020\\qquad(\\textbf{D}) \\: 2021... | 3,621 | open-r1/OpenR1-Math-220k |
[
{
"content": "$\\left.\\quad \\begin{array}{ll}\\text { Coordinate method on the plane }\\end{array}\\right]$\n\nGiven points $A(5 ; 5), B(8 ;-3)$ and $C(-4 ; 1)$. Find the coordinates of the intersection point of the medians of triangle $A B C$.",
"role": "user"
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{
"content": "<think>\nOkay, so I... | 2,187 | open-r1/OpenR1-Math-220k |
[
{
"content": "1. [5] The formula to convert Celsius to Fahrenheit is\n$$\nF^{\\circ}=1.8 \\cdot C^{\\circ}+32 \\text {. }\n$$\n\nIn Celsius, it is $10^{\\circ}$ warmer in New York right now than in Boston. In Fahrenheit, how much warmer is it in New York than in Boston?",
"role": "user"
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{
"conten... | 991 | open-r1/OpenR1-Math-220k |
[
{
"content": "G5.3 In Figure 6, a square-based pyramid is cut into two shapes by a cut running parallel to the base and made $\\frac{2}{3}$ of the way up. Let $1: c$ be the ratio of the volume of the small pyramid to that of the truncated base, find the value of $c$.",
"role": "user"
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{
"content":... | 1,613 | open-r1/OpenR1-Math-220k |
[
{
"content": "The Partridge family pays each of their five children a weekly allowance. The average allowance for each of the three younger children is $\\$ 8$. The two older children each receive an average allowance of $\\$ 13$. The total amount of allowance money paid per week is\n(A) $\\$ 50$\n(B) $\\$ 52.5... | 1,356 | open-r1/OpenR1-Math-220k |
[
{
"content": "If $\\frac{x}{2}-5=9$, what is the value of $\\sqrt{7 x}$ ?",
"role": "user"
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"content": "<think>\nOkay, so I need to solve for x in the equation (x/2) - 5 = 9, and then find the square root of 7 times that x. Let me start with the equation. Hmm, maybe I should first get rid of the... | 999 | open-r1/OpenR1-Math-220k |
[
{
"content": "7.2. If the width of a rectangle is increased by 3 cm and the height is decreased by 3 cm, its area will not change. How will the area change if, instead, the width of the original rectangle is decreased by 4 cm and the height is increased by 4 cm?",
"role": "user"
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"content": "<th... | 1,781 | open-r1/OpenR1-Math-220k |
[
{
"content": "2. (6 points) The 50th term of the arithmetic sequence $3, 5, 7, 9 \\cdots$ is\n\n保留源文本的换行和格式，翻译结果如下：\n\n2. (6 points) The 50th term of the arithmetic sequence $3, 5, 7, 9 \\cdots$ is",
"role": "user"
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{
"content": "<think>\n嗯，好的，我现在要解决这个问题：算术数列3,5,7,9,...的第50项是什么。让我仔细想想该怎么解决这个问题。首先，... | 1,196 | open-r1/OpenR1-Math-220k |
[
{
"content": "3. 1 Simplify:\n$$\n\\log _{a}\\left[\\left(\\frac{m^{4} n^{-4}}{m^{-1} n}\\right)^{-3} \\div\\left(\\frac{m^{-2} n^{2}}{m n^{-1}}\\right)^{5}\\right]\n$$\n\nHere $m$, $n$, and $a$ are all positive numbers, $a \\neq 1$.",
"role": "user"
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"content": "<think>\nAlright, so I need to s... | 3,712 | open-r1/OpenR1-Math-220k |
[
{
"content": "6. (3 points) A deck of playing cards has 54 cards, with the jokers considered as 0 points, $A$ as 1 point, $J$ as 11 points, $Q$ as 12 points, and $K$ as 13 points. If you draw several cards at random, disregarding the suits, and if it is required that each draw must include 2 cards whose points ... | 3,233 | open-r1/OpenR1-Math-220k |
[
{
"content": "## Task A-1.3. (8 points)\n\nDetermine the largest integer $n$ for which the inequality $3\\left(n-\\frac{5}{3}\\right)-2(4 n+1)>6 n+5$ holds.",
"role": "user"
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"content": "<think>\nOkay, so I need to find the largest integer n that satisfies this inequality: 3(n - 5/3) - 2(4n + 1)... | 1,517 | open-r1/OpenR1-Math-220k |
[
{
"content": "14. $f(x)$ is a function defined on $(0,+\\infty)$ that is increasing, and $f\\left(\\frac{x}{y}\\right)=f(x)-f(y)$.\n(1) Find the value of $f(1)$; (2) If $f(6)=1$, solve the inequality $f(x+3)-f\\left(\\frac{1}{x}\\right)<2$.",
"role": "user"
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"content": "<think>\nOkay, so I have ... | 3,171 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 9 Calculate\n$$\n(2+1)\\left(2^{2}+1\\right)\\left(2^{4}+1\\right) \\cdots \\cdot\\left(2^{2^{n}}+1\\right) .\n$$",
"role": "user"
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"content": "<think>\nOkay, so I have this problem to calculate the product (2+1)(2²+1)(2⁴+1)…(2^{2ⁿ}+1). Hmm, let me see. First, let me write ... | 1,804 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 9. Find $\\lim _{x \\rightarrow 0} \\frac{\\sin x}{\\sqrt{x+9}-3}$.",
"role": "user"
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"content": "<think>\nOkay, so I need to find the limit as x approaches 0 of sin x divided by (sqrt(x + 9) minus 3). Hmm, let me think. First, I remember that when dealing with limits invol... | 1,789 | open-r1/OpenR1-Math-220k |
[
{
"content": "Points $A$ and $B$ are 10 units apart. Points $B$ and $C$ are 4 units apart. Points $C$ and $D$ are 3 units apart. If $A$ and $D$ are as close as possible, then the number of units between them is\n$\\text{(A)}\\ 0 \\qquad \\text{(B)}\\ 3 \\qquad \\text{(C)}\\ 9 \\qquad \\text{(D)}\\ 11 \\qquad... | 3,837 | open-r1/OpenR1-Math-220k |
[
{
"content": "17. Starting from July 1st last year, during the summer vacation, Xiao Hui stayed at her grandmother's house for three consecutive days. The sum of these three dates is 62. Therefore, the dates Xiao Hui stayed at her grandmother's house are from ( ) month ( ) day to ( ) month ( ) day.",
"role"... | 2,964 | open-r1/OpenR1-Math-220k |
[
{
"content": "7.199. $\\log _{2} \\log _{3}\\left(x^{2}-16\\right)-\\log _{1 / 2} \\log _{1 / 3} \\frac{1}{x^{2}-16}=2$.",
"role": "user"
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"content": "<think>\nOkay, so I have this logarithmic equation to solve: log base 2 of log base 3 of (x² - 16) minus log base 1/2 of log base 1/3 of 1/(x² - ... | 2,815 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 6 If $x^{2}+x+2 m$ is a perfect square, then $m=$ $\\qquad$\n$(1989$, Wu Yang Cup Junior High School Mathematics Competition)",
"role": "user"
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"content": "<think>\nOkay, so I need to figure out the value of m that makes the quadratic expression x² + x + 2m a perfect square... | 1,748 | open-r1/OpenR1-Math-220k |
[
{
"content": "166. The device consists of three independently operating elements. The probability of failure of each element in one trial is 0.1. Formulate the distribution law of the number of failed elements in one trial.",
"role": "user"
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"content": "<think>\nAlright, so I need to figure out ... | 2,212 | open-r1/OpenR1-Math-220k |
[
{
"content": "4. If $a$ and $b$ are integers and $\\sqrt{7-4 \\sqrt{3}}$ is one of the roots of the equation $x^{2}+a x+b=0$, find the value of $a+b$.\n(A) -3\n(B) -2\n(C) 0\n(D) 2\n(E) 3",
"role": "user"
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{
"content": "<think>\nOkay, let's see. The problem says that √(7 - 4√3) is one of the roots... | 2,186 | open-r1/OpenR1-Math-220k |
[
{
"content": "6.032. $\\sqrt{x+\\sqrt{x+11}}+\\sqrt{x-\\sqrt{x+11}}=4$.\n\n6.032. $\\sqrt{x+\\sqrt{x+11}}+\\sqrt{x-\\sqrt{x+11}}=4$.",
"role": "user"
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{
"content": "<think>\nOkay, so I need to solve the equation √(x + √(x + 11)) + √(x - √(x + 11)) = 4. Hmm, that looks a bit complicated with nested... | 2,079 | open-r1/OpenR1-Math-220k |
[
{
"content": "1. If $n=3^{0}+3^{1}+3^{2}+\\cdots+3^{2015}+3^{2016}$, then the remainder when $n$ is divided by 4 is\nA. 3\nB. 2\nC. 1\nD. 0",
"role": "user"
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{
"content": "<think>\nOkay, so I need to find the remainder when n is divided by 4, where n is the sum of a geometric series: 3^0 + 3^1 + 3... | 3,293 | open-r1/OpenR1-Math-220k |
[
{
"content": "List 3. Regardless of the value of the real number $k$, the line $y=2 k x$ $+k^{2}$ is tangent to the parabola $y=a x^{2}+b x+c$. Find the equation of the parabola.",
"role": "user"
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{
"content": "<think>\nOkay, so I have this problem here where I need to find the equation of a parab... | 3,329 | open-r1/OpenR1-Math-220k |
[
{
"content": "Four consecutive odd numbers, when their sum is added to their product, as well as all possible two-factor and three-factor products formed from these numbers without repetition, result in 26,879. Which four numbers are these?",
"role": "user"
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{
"content": "<think>\nOkay, let me try... | 3,628 | open-r1/OpenR1-Math-220k |
[
{
"content": "In Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below.\n\\[\\begin{array}{c@{\\hspace{8em}} c@{\\hspace{6pt}}c@{\\hspace{6pt}}c@{\\hspace{6pt}}c@{\\hspace{4pt}}c@{\\hspace{2pt}} c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\hspace{2pt}... | 3,731 | open-r1/OpenR1-Math-220k |
[
{
"content": "2. If the polynomial $P=a^{2}+4 a+2014$, then the minimum value of $P$ is ( ).\n(A) 2010\n(B) 2011\n(C) 2012\n(D) 2013",
"role": "user"
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"content": "<think>\nOkay, so I need to find the minimum value of the polynomial \\( P = a^2 + 4a + 2014 \\). The options given are 2010, 2011, 2... | 1,811 | open-r1/OpenR1-Math-220k |
[
{
"content": "3. If $a+b=4, a^{3}+b^{3}=28$, then the value of $a^{2}+b^{2}$ is ( ).\n(A) 8\n(B) 10\n(C) 12\n(D) 14",
"role": "user"
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{
"content": "<think>\nOkay, so I have this problem here: if a + b = 4 and a³ + b³ = 28, then what is the value of a² + b²? The options are A) 8, B) 10, C) 12, D) 1... | 1,441 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 4 Let the set $A=\\{1,2,3,4,5,6\\}$, and the mapping $f: A \\rightarrow A$, such that its third composite mapping $f \\cdot f \\cdot f$ is the identity mapping. How many such $f$ are there?\n(1996. Japan Mathematical Olympiad Preliminary)",
"role": "user"
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{
"content": "<thin... | 3,579 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 1 Find the largest constant $c$, such that for real numbers $x, y$ satisfying $x>0, y>0, x^{2}+y^{2}=1$, it always holds that $x^{6}+y^{6} \\geqslant c x y$.\n\n",
"role": "user"
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{
"content": "<think>\nOkay, so I need to find the largest constant c such that for all positive... | 3,245 | open-r1/OpenR1-Math-220k |
[
{
"content": "13. The sum of two natural numbers is 1000 less than their product, and one of these natural numbers is a perfect square. Then, the larger of these two natural numbers is $\\qquad$ .",
"role": "user"
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{
"content": "<think>\nOkay, let's see. I need to find two natural numbers where th... | 2,015 | open-r1/OpenR1-Math-220k |
[
{
"content": "A2. What is the sum of all real numbers that solve the equation $|x-2011|+|x-2012|=3$?\n(A) 2011\n(B) 2012\n(C) 2013\n(D) 4021\n(E) 4023",
"role": "user"
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{
"content": "<think>\nOkay, so I need to solve the equation |x - 2011| + |x - 2012| = 3 and find the sum of all real numbers tha... | 2,391 | open-r1/OpenR1-Math-220k |
[
{
"content": "I1.4 Let $f(x)$ be a function such that $f(n)=(n-1) f(n-1)$ and $f(n) \\neq 0$ hold for all integers $n \\geq 6$. If $U=\\frac{f(T)}{(T-1) f(T-3)}$, find the value of $U$.",
"role": "user"
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{
"content": "<think>\nOkay, let's try to figure out the value of U here. So the problem says ... | 2,253 | open-r1/OpenR1-Math-220k |
[
{
"content": "1. In a pile of 200 coins, $2 \\%$ are gold coins and the rest are silver. Simple Simon removes one silver coin every day until the pile contains $20 \\%$ gold coins. How many silver coins does Simon remove?",
"role": "user"
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{
"content": "<think>\nOkay, let's see. There's this probl... | 1,810 | open-r1/OpenR1-Math-220k |
[
{
"content": "An international firm has 250 employees, each of whom speaks several languages. For each pair of employees, $(A,B)$, there is a language spoken by $A$ and not $B$, and there is another language spoken by $B$ but not $A$. At least how many languages must be spoken at the firm?",
"role": "user"
... | 3,086 | open-r1/OpenR1-Math-220k |
[
{
"content": "12. As shown in the figure, points $A(m, 8), B(n, 2)$ lie on the graph of the inverse proportion function $y=\\frac{k}{x}$, $A D \\perp x$-axis at point $D$, $B C \\perp x$-axis at point $C$, and $D C=6$. Then $k=$ $\\qquad$",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to... | 1,573 | open-r1/OpenR1-Math-220k |
[
{
"content": "3. A two-digit number has the following property: if it is added to the number with the same digits but in reverse order, the result is a perfect square.\n\nFind all such two-digit numbers.",
"role": "user"
},
{
"content": "<think>\nAlright, so I need to find all two-digit numbers that... | 2,481 | open-r1/OpenR1-Math-220k |
[
{
"content": "Determine all pairs $(a,b)$ of real numbers with $a\\leqslant b$ that maximise the integral $$\\int_a^b e^{\\cos (x)}(380-x-x^2) \\mathrm{d} x.$$",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all pairs (a, b) of real numbers where a ≤ b that maximize the integral f... | 2,387 | open-r1/OpenR1-Math-220k |
[
{
"content": "7.1.1. (12 points) Find the greatest negative root of the equation\n\n$$\n\\frac{\\sin \\pi x - \\cos 2 \\pi x}{(\\sin \\pi x - 1)^2 + \\cos^2 \\pi x - 1} = 0\n$$",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the greatest negative root of the equation given. Let me... | 3,667 | open-r1/OpenR1-Math-220k |
[
{
"content": "3、Given $2014=\\left(a^{2}+b^{2}\\right) \\times\\left(c^{3}-d^{3}\\right)$, where $\\mathbf{a} 、 \\mathbf{b} 、 \\mathbf{c} 、 \\mathbf{d}$ are four positive integers, please write a multiplication equation that satisfies the condition: $\\qquad$",
"role": "user"
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{
"content": "<think... | 3,163 | open-r1/OpenR1-Math-220k |
[
{
"content": "Task 6. (15 points) On the plane xOy, indicate all points through which no curve passes, given by the equation\n\n$$\n\\mathrm{a} x^{2}+(1-6 a) x+2-a-2 y+a y^{2}=0\n$$",
"role": "user"
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{
"content": "<think>\nOkay, let's see. The problem is asking to indicate all points on the plane ... | 3,791 | open-r1/OpenR1-Math-220k |
[
{
"content": "3. Given that $a$, $b$, and $c$ are all integers, and satisfy\n$$\na^{2}+b^{2}+c^{2}+3 < a b + 3 b + 2 c \\text{.}\n$$\n\nThen the quadratic equation with roots $a+b$ and $c-b$ is ( ).\n(A) $x^{2}-3 x+2=0$\n(B) $x^{2}+2 x-8=0$\n(C) $x^{2}-4 x-5=0$\n(D) $x^{2}-2 x-3=0$",
"role": "user"
},
{... | 3,662 | open-r1/OpenR1-Math-220k |
[
{
"content": "1. The original price of a commodity is $a$ yuan. During the Spring Festival promotion, it is reduced by $20 \\%$. If the price is to be restored to the original price after the festival, the current selling price should be increased by ( ).\n(A) $15 \\%$\n(B) $20 \\%$\n(C) $25 \\%$\n(D) $30 \\%$"... | 1,644 | open-r1/OpenR1-Math-220k |
[
{
"content": "13.079. The first tourist, after riding a bicycle for 1.5 hours at a speed of $16 \\mathrm{~km} / \\mathrm{h}$, makes a stop for 1.5 hours, and then continues the journey at the initial speed. After 4 hours from the departure of the first tourist, the second tourist sets off in pursuit on a motorc... | 2,824 | open-r1/OpenR1-Math-220k |
[
{
"content": "[ [Motion Problems ]\n\nAuthor: Shnol D. $\\cdot$.\n\nDaddy, Masha, and Yasha are walking to school together. While Daddy takes 3 steps, Masha takes 5 steps. While Masha takes 3 steps, Yasha takes 5 steps. Masha and Yasha counted that together they made 400 steps. How many steps did Daddy take?",
... | 3,566 | open-r1/OpenR1-Math-220k |
[
{
"content": "5. One side of a certain triangle is twice as large as another, and the perimeter of this triangle is 56. The quadrupled smallest side is 21 units longer than the largest side. Find the sides of this triangle.",
"role": "user"
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{
"content": "<think>\nOkay, let me try to solve this pr... | 2,763 | open-r1/OpenR1-Math-220k |
[
{
"content": "32. For what $m$, the inequality $\\frac{x^{2}-m x-2}{x^{2}-3 x+4}$ $>-1$ holds for all $x$?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out for what values of m the inequality (x² - m x - 2)/(x² - 3x + 4) > -1 holds for all x. Hmm. Let me start by understandin... | 2,216 | open-r1/OpenR1-Math-220k |
[
{
"content": "A circle with a radius of 4 is inscribed in a right triangle, the hypotenuse of which is 26. Find the perimeter of the triangle.\n\n#",
"role": "user"
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{
"content": "<think>\nOkay, so I need to find the perimeter of a right triangle that has a circle inscribed in it. The radius of th... | 1,877 | open-r1/OpenR1-Math-220k |
[
{
"content": "139. On a meadow, grass grows. If 9 cows were let onto the meadow, they would empty it in 4 days. If 8 cows were let onto the meadow, they would eat all the grass in 6 days. How many cows can graze on the meadow all the time while the grass grows?",
"role": "user"
},
{
"content": "<thi... | 3,683 | open-r1/OpenR1-Math-220k |
[
{
"content": "## Task Condition\n\nCalculate the limit of the numerical sequence:\n\n$$\n\\lim _{n \\rightarrow \\infty} \\frac{(3-n)^{2}+(3+n)^{2}}{(3-n)^{2}-(3+n)^{2}}\n$$",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out this limit problem. The sequence is given by the exp... | 3,439 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 1. Find the equation of the tangent line at point $P\\left(x_{1}, y_{1}\\right)$ on the ellipse $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the equation of the tangent line at a point P(x₁, y₁) on the ellipse... | 1,928 | open-r1/OpenR1-Math-220k |
[
{
"content": "2.274. $\\frac{8-m}{\\sqrt[3]{m}+2}:\\left(2+\\frac{\\sqrt[3]{m^{2}}}{\\sqrt[3]{m}+2}\\right)+\\left(\\sqrt[3]{m}+\\frac{2 \\sqrt[3]{m}}{\\sqrt[3]{m}-2}\\right) \\cdot \\frac{\\sqrt[3]{m^{2}}-4}{\\sqrt[3]{m^{2}}+2 \\sqrt[3]{m}}$.",
"role": "user"
},
{
"content": "<think>\nAlright, let ... | 3,369 | open-r1/OpenR1-Math-220k |
[
{
"content": "A homogeneous sphere made of a material has a weight that is twice as much when it is half submerged in water compared to when it is completely submerged. What is the specific gravity of the material?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the specific gravi... | 2,542 | open-r1/OpenR1-Math-220k |
[
{
"content": "6. On each face of a cube, write a positive integer, and at each vertex, write the product of the positive integers on the three faces meeting at that vertex. If the sum of the numbers written at the eight vertices is 2014, then the sum of the numbers written on the six faces is $\\qquad$ .\n\n",
... | 3,670 | open-r1/OpenR1-Math-220k |
[
{
"content": "1.18. Solve the system of equations using the matrix method\n\n$$\n\\left\\{\\begin{array}{l}\nx_{1}+2 x_{2}+4 x_{3}=5 \\\\\n2 x_{1}+x_{2}+5 x_{3}=7 \\\\\n3 x_{1}+2 x_{2}+6 x_{3}=9\n\\end{array}\\right.\n$$",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to solve this system... | 3,106 | open-r1/OpenR1-Math-220k |
[
{
"content": "4. A large bag contains coins of 1 cent, 10 cents, and 25 cents. The number of 10 cent coins is twice the number of 1 cent coins, and the number of 25 cent coins is three times the number of 10 cent coins. The total amount of money (in cents) in the bag could be\n(A) 306 ;\n(B) 333 ;\n(C) 342 ;\n(... | 1,869 | open-r1/OpenR1-Math-220k |
[
{
"content": "6. If $a>0, b>0$, and $\\arcsin a+\\arcsin b=\\frac{\\pi}{2}, m=\\log _{2} \\frac{1}{a}+\\log _{2} \\frac{1}{b}$, then the range of $m$ is $\\qquad$ .",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out the range of m, where m is defined as the sum of the logarithm... | 2,982 | open-r1/OpenR1-Math-220k |
[
{
"content": "8.047. $\\sin ^{3} z \\cos z-\\sin z \\cos ^{3} z=\\frac{\\sqrt{2}}{8}$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to solve the equation: sin³z cosz - sinz cos³z = √2 / 8. Hmm, let's see. First, maybe I can factor something out from the left side. Both terms have sinz ... | 3,212 | open-r1/OpenR1-Math-220k |
[
{
"content": "251. $16 x^{2}+2000=680 x$.",
"role": "user"
},
{
"content": "<think>\nAlright, so I have this quadratic equation here: 16x² + 2000 = 680x. Hmm, okay, quadratic equations. Let me remember how to solve them. First, I think I need to rearrange the equation so that it's in standard form, ... | 3,666 | open-r1/OpenR1-Math-220k |
[
{
"content": "19. Let $a, b, c, d, e$ be five numbers satisfying the following conditions:\n$$\n\\begin{array}{l}\na+b+c+d+e=0, \\quad \\text { and } \\\\\na b c+a b d+a b e+a c d+a c e+a d e+b c d+b c e+b d e+c d e=2008 .\n\\end{array}\n$$\n\nFind the value of $a^{3}+b^{3}+c^{3}+d^{4}+e^{3}$.",
"role": "us... | 3,186 | open-r1/OpenR1-Math-220k |
[
{
"content": "7.4. On the island, there live knights and liars. Knights always tell the truth, and liars always lie. Some of the inhabitants claimed that there is an even number of knights on the island, while the others claimed that there is an odd number of liars on the island. Is the number of inhabitants on... | 2,085 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 6. Solve the equation\n\n$$\n\\sqrt[3]{2 x-1}+\\sqrt[3]{x-1}=1\n$$",
"role": "user"
},
{
"content": "<think>\nAlright, let me try to solve this equation: the cube root of (2x - 1) plus the cube root of (x - 1) equals 1. Hmm, cube roots can be tricky, but maybe if I let each cub... | 3,454 | open-r1/OpenR1-Math-220k |
[
{
"content": "If $P$ is a negative integer, which of the following is always positive?\n(A) $P^{2}$\n(B) $\\frac{1}{P}$\n(C) $2 P$\n(D) $P-1$\n(E) $P^{3}$",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. The problem says that P is a negative integer, and we need to determine which of the ... | 1,031 | open-r1/OpenR1-Math-220k |
[
{
"content": "Consider the sequence $\\left(a_{n}\\right)$ with the first term $a_{1}=1$ and defined by the following relation: For any positive integer $n$,\n\n$$\na_{n}=a_{\\left\\lfloor\\frac{n}{2}\\right\\rfloor}+a_{\\left\\lceil\\frac{n}{2}\\right\\rceil}\n$$\n\nExpress $a_{n}$ as a function of $n$.",
... | 2,451 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 3 The parabola $y=a x^{2}+b x+c$ intersects the $x$-axis at points $A$ and $B$, and the $y$-axis at point $C$. If $\\triangle A B C$ is a right triangle, then $a c=$ $\\qquad$\n(2003, National Junior High School Mathematics League)",
"role": "user"
},
{
"content": "<think>\nOka... | 2,696 | open-r1/OpenR1-Math-220k |
[
{
"content": "30. The sum of the two diagonals of a rhombus is $l$, and the area is $S$. Then its side length is ( ).\n(A) $\\frac{1}{2} \\sqrt{4 S-l^{2}}$\n(B) $\\frac{1}{2} \\sqrt{4 S+l^{2}}$\n(C) $\\frac{1}{2} \\sqrt{l^{2}-4 S}$\n(D) $\\frac{1}{2} \\sqrt{l^{2}+4 S}$",
"role": "user"
},
{
"content... | 2,172 | open-r1/OpenR1-Math-220k |
[
{
"content": "1. A bus leaves the station at exactly 7:43 a.m. and arrives at its destination at exactly 8:22 a.m. on the same day. How long, in minutes, was the bus trip?",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. The problem is about figuring out how long the bus trip took. The bu... | 1,437 | open-r1/OpenR1-Math-220k |
[
{
"content": "15. The volume of a cube is $V \\mathrm{~cm}^{3}$. The surface area of the cube is $2 V \\mathrm{~cm}^{2}$. What is the value of $V$ ?\nA 8\nB 16\nC 27\nD 64\nE 128",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to solve this problem where the volume of a cube is V cm³, and... | 1,294 | open-r1/OpenR1-Math-220k |
[
{
"content": "4. $\\frac{1}{2 \\sqrt{ } 1}+1 \\sqrt{2}+\\frac{1}{3 \\sqrt{2}+2 \\sqrt{3}}$ $+\\cdots+\\frac{1}{100 \\sqrt{99}+99 \\sqrt{100}}$ The value is ( ).\n(A) $\\frac{3}{4}$.\n(B) $\\frac{9}{10}$.\n(C) 1 .\n(D) $\\sqrt{2}$.\n(E) None of the above.",
"role": "user"
},
{
"content": "<think>\nOk... | 2,880 | open-r1/OpenR1-Math-220k |
[
{
"content": "## problem statement\n\nCalculate the limit of the function:\n\n$\\lim _{x \\rightarrow-1} \\frac{\\left(x^{3}-2 x-1\\right)(x+1)}{x^{4}+4 x^{2}-5}$",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the limit as x approaches -1 of [(x³ - 2x -1)(x +1)] divided by (x⁴ +4... | 3,784 | open-r1/OpenR1-Math-220k |
[
{
"content": "94. A system of five linear equations. Solve the following system of equations:\n\n$$\n\\left\\{\\begin{array}{l}\nx+y+z+u=5 \\\\\ny+z+u+v=1 \\\\\nz+u+v+x=2 \\\\\nu+v+x+y=0 \\\\\nv+x+y+z=4\n\\end{array}\\right.\n$$",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this system ... | 1,985 | open-r1/OpenR1-Math-220k |
[
{
"content": "2. Given the equation in terms of $x$\n$$\na x^{2}=k x+b(a b k \\neq 0)\n$$\n\nwith roots $x_{1} 、 x_{2}$, and\n$$\nk x+b=0(k b \\neq 0)\n$$\n\nwith root $x_{3}$. The quantitative relationship satisfied by $x_{1} 、 x_{2} 、 x_{3}$ is $\\qquad$",
"role": "user"
},
{
"content": "<think>\n... | 2,813 | open-r1/OpenR1-Math-220k |
[
{
"content": "4. Vasya and his father are collecting mushrooms in the forest. Father told Vasya: “Go ahead along this straight path. I will quickly check the glade and catch up with you.” The father's step is $20 \\%$ longer than Vasya's. They both walk at a constant speed and do not change the length of their ... | 3,318 | open-r1/OpenR1-Math-220k |
[
{
"content": "If $y=77$, then $\\frac{7 y+77}{77}$ is equal to\n(A) 8\n(B) 12\n(C) 78\n(D) 84\n(E) 540",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. The problem says if y equals 77, then we need to compute (7y + 77)/77 and choose the correct answer from the options given. Alright, let'... | 1,264 | open-r1/OpenR1-Math-220k |
[
{
"content": "5. Country Omega grows and consumes only vegetables and fruits. It is known that in 2014, Country Omega produced 1200 tons of vegetables and 750 tons of fruits. In 2015, 900 tons of vegetables and 900 tons of fruits were produced. Over the year, the price of one ton of vegetables increased from 90... | 2,617 | open-r1/OpenR1-Math-220k |
[
{
"content": "Let $[n] = \\{1, 2, 3, ... ,n\\}$ and for any set $S$, let$ P(S)$ be the set of non-empty subsets of $S$. What is the last digit of $|P(P([2013]))|$?",
"role": "user"
},
{
"content": "<think>\nOkay, let's tackle this problem step by step. The question is asking for the last digit of th... | 3,679 | open-r1/OpenR1-Math-220k |
[
{
"content": "In the future, each country in the world produces its Olympic athletes via cloning and strict training\nprograms. Therefore, in the fi\fnals of the 200 m free, there are two indistinguishable athletes from each\nof the four countries. How many ways are there to arrange them into eight lanes?",
... | 2,409 | open-r1/OpenR1-Math-220k |
[
{
"content": "Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $74$ kilometers after biking for $2$ hours, jogging for $3$ hours, and swimming for $4$ hours, while Sue covers $91$ kilometers after jogging for $2$... | 2,378 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 4. Calculate the area of the figure bounded by the curve $y^{2}=x^{3}$, the line $y=1$, and the $O y$ axis (Fig. 5.8).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the area of the figure bounded by the curve y² = x³, the line y = 1, and the Oy axis, which ... | 2,818 | open-r1/OpenR1-Math-220k |
[
{
"content": "118. A circle is inscribed in a square, and then a new square is inscribed in the circle. Find the ratio of the areas of these squares.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the ratio of the areas of two squares. The problem says there's a circle inscribed ... | 2,012 | open-r1/OpenR1-Math-220k |
[
{
"content": "Find the sum of all positive integers $1\\leq k\\leq 99$ such that there exist positive integers $a$ and $b$ with the property that \\[x^{100}-ax^k+b=(x^2-2x+1)P(x)\\] for some polynomial $P$ with integer coefficients.\n\n[i]Proposed by David Altizio[/i]",
"role": "user"
},
{
"content"... | 2,843 | open-r1/OpenR1-Math-220k |
[
{
"content": "4. For any real numbers $x, y$, define the operation $x * y$ as $x * y=a x+b y+$ $c x y$, where $a, b, c$ are constants, and the operations on the right side of the equation are the usual real number addition and multiplication. It is known that $1 * 2=3,2 * 3=4$, and there is a non-zero real numb... | 1,975 | open-r1/OpenR1-Math-220k |
[
{
"content": "6) What is the value of the square of the square of the square of 8?\n(A) $2^{8}$\n$\\begin{array}{ll}\\text { (B) } 8^{4} & \\text { (C) } 8^{6}\\end{array}$\n(D) $8^{8}$\n(E) $2^{64}$.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. The problem is asking for the value of ... | 2,590 | open-r1/OpenR1-Math-220k |
[
{
"content": "385. $(\\tan \\alpha+\\cot \\alpha)^{2}-(\\tan \\alpha-\\cot \\alpha)^{2}$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this expression to simplify: (tan α + cot α)² - (tan α - cot α)². Hmm, let me think. I remember that when you have something like (a + b)² - (a - b)², ... | 1,597 | open-r1/OpenR1-Math-220k |
[
{
"content": "In a communication system consisting of 2001 subscribers, each subscriber is connected to exactly $n$ others. Determine all possible values of $n$.\n\n#",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. The problem is about a communication system with 2001 subscribers. Each s... | 2,745 | open-r1/OpenR1-Math-220k |
[
{
"content": "10. Given that\n$$\n\\begin{aligned}\n\\alpha+\\beta+\\gamma & =14 \\\\\n\\alpha^{2}+\\beta^{2}+\\gamma^{2} & =84 \\\\\n\\alpha^{3}+\\beta^{3}+\\gamma^{3} & =584\n\\end{aligned}\n$$\nfind $\\max \\{\\alpha, \\beta, \\gamma\\}$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need... | 2,296 | open-r1/OpenR1-Math-220k |
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