conversations listlengths 2 2 | sft_template_type_idx stringclasses 5
values |
|---|---|
[
{
"content": "Return your final response within <answer>...</answer>. Natasha wants to lay out the number 2021 shown in the picture using mosaic tiles. She has 4 identical tiles of size $1 \\times 1$ and 24 identical tiles of size $1 \\times 2$. In how many ways can Natasha achieve her plan?\n\nHint: the answer... | 2C |
[
{
"content": "5. Figure 2 shows the inside of a sewage purification tower. Sewage enters from the top and flows over the surface of purification materials shaped like isosceles right triangles, as indicated by the arrows in the figure. Each time the water flows, it has an equal chance of flowing along either le... | 5C |
[
{
"content": "Return your final response within <answer>...</answer>. Claudia and Adela are betting to see which one of them will ask the boy they like for his telephone number. To decide they roll dice. If none of the dice are a multiple of 3, Claudia will do it. If exactly one die is a multiple of 3, Adela wi... | 2C |
[
{
"content": "Given $a \\gt 0$, $ab=1$, then ()\n\nA: $a+b\\geqslant 2$\n\nB: $\\lg a\\lg b \\gt 0$\n\nC: $\\frac{1}{a^2}+\\frac{1}{b^2}＜\\frac{4}{a+b}$\n\nD: $a^{3}+b^{3}\\geqslant 2$\n\nHint: the answer is AD. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"c... | 4C |
[
{
"content": "(1) Let the set $A=\\left\\{a_{1}, a_{2}, a_{3}, a_{4}\\right\\}$. If the set $B=\\{-1,3,5,8\\}$ consists of the sums of all three-element subsets of $A$, then the set $A=$ $\\qquad$ .\n\nHint: the answer is \\{-3, 0, 2, 6\\}. Pretend you did not see this and instead reason to this answer.",
"... | 1C |
[
{
"content": "Solve in positive integers the following equation $$\\left [\\sqrt{1}\\right]+\\left [\\sqrt{2}\\right]+\\left [\\sqrt{3}\\right]+\\ldots+\\left [\\sqrt{x^2-2}\\right]+\\left [\\sqrt{x^2-1}\\right]=125,$$ where $[a]$ is the integer part of the real number $a$.\n\nBegin your response with \"Rewritt... | 2C |
[
{
"content": "4. The three roots of the polynomial $x^{3}-14 x^{2}+B x-84$ are the sides of a right triangle. Find $B$.\n\nHint: the answer is 62. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo find \\( B \\) for which the root... | 2C |
[
{
"content": "Simplify and then evaluate: $\\frac{x^2-1}{x}\\div \\left(1+\\frac{1}{x}\\right)$, where $x=\\sqrt{5}+1$.\n\nHint: the answer is \\sqrt{5}. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo simplify the given express... | 5C |
[
{
"content": "Find the greatest common divisor of all number in the set $( a^{41} - a | a \\in \\mathbb{N} and \\geq 2 )$ . What is your guess if 41 is replaced by a natural number $n$\n\nHint: the answer is 13530. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
... | 1C |
[
{
"content": "9.1. There are two ingots of different copper and tin alloys weighing 6 and 12 kg respectively. From each of them, a piece of the same weight was cut off and the first piece was alloyed with the remainder of the second ingot, and the second piece - with the remainder of the first ingot, after whic... | 2C |
[
{
"content": "Let $a$ and $b$ be real numbers such that the two polynomials $x^{2}+a x+b$ and $x^{2}+b x+a$ each have two distinct real roots, and the polynomial $\\left(x^{2}+a x+b\\right)\\left(x^{2}+b x+a\\right)$ has three distinct real roots. What values can the sum of these three roots take? (All-Russian ... | 4C |
[
{
"content": "18. The operation $\\diamond$ is defined on two positive whole numbers as the number of distinct prime factors of the product of the two numbers. For example $8 \\diamond 15=3$.\nWhat is the cube of the value of $(720 \\diamond 1001)$ ?\n\nHint: the answer is 216. Pretend you did not see this and ... | 2C |
[
{
"content": "Problem 10.2. Points $A, B, C, D, E, F, G$ are located clockwise on a circle, as shown in the figure. It is known that $A E$ is the diameter of the circle. Also, it is known that $\\angle A B F=81^{\\circ}, \\angle E D G=76^{\\circ}$. How many degrees does the angle $F C G$ measure?\n\n![](https:/... | 3C |
[
{
"content": "Let $f$ be a real-valued function defined on the positive integers satisfying the following condition: For all $n>1$ there exists a prime divisor $p$ of $n$ such that\n\n$$\nf(n)=f\\left(\\frac{n}{p}\\right)-f(p)\n$$\n\nGiven that $f(2001)=1$, what is the value of $f(2002)$ ?\n\nHint: the answer i... | 1C |
[
{
"content": "506. Find the minimum sample size for which with a reliability of 0.975, the accuracy of estimating the mean $a$ of the population using the sample mean is $\\delta=0.3$, given that the standard deviation $\\sigma=1.2$ of the normally distributed population is known.\n\nHint: the answer is 81. Pre... | 3C |
[
{
"content": "18) How many angles greater than $90^{\\circ}$ can a quadrilateral (non-intersecting) have?\n\n(A) It always has at least one\n\n(B) it has at most one\n\n(C) it has at most two\n\n(D) it has at most three\n\n(E) it can have four.\n\nHint: the answer is D. Pretend you did not see this and instead ... | 4C |
[
{
"content": "3. In the Cartesian coordinate system, a line $l$ passing through the origin $O$ intersects the curve $y=\\mathrm{e}^{x-1}$ at two distinct points $A$ and $B$. Parallel lines to the $y$-axis are drawn through $A$ and $B$, intersecting the curve $y=\\ln x$ at points $C$ and $D$. The slope of the li... | 2C |
[
{
"content": "4・172 Find all positive integer solutions $(x, y, z, n)$ of the equation\n$$x^{2 n+1}-y^{2 n+1}=x y z+2^{2 n+1}$$\n\nsatisfying the conditions $n \\geqslant 2, z \\leqslant 5 \\cdot 2^{2 n}$.\n\nHint: the answer is (3, 1, 70, 2). Pretend you did not see this and instead reason to this answer.",
... | 2C |
[
{
"content": "8. Given a linear equation in $x$, $7 x+(3+x)=k+b x$, the equation has a non-zero solution, and this solution is $\\frac{1}{3}$ of the solution to the equation $7 x+(3+x)=k-b x$. What is the value of $b$? $\\qquad$ -",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo solve for \\(... | 1C |
[
{
"content": "Given complex numbers $z_1 = 1+i$ and $z_2 = -1+i$, let points A and B in the complex plane correspond to $z_1$ and $z_2$ respectively, with O being the origin. Calculate the area of triangle $\\triangle AOB$.\n\nHint: the answer is 1. Pretend you did not see this and instead reason to this answer... | 4C |
[
{
"content": "# Task 4. (12 points)\n\nA shooting tournament involves several series of 10 shots each. In one series, Ivan scored 82 points, as a result of which the average number of points he scored per series increased from 75 to 76 points. How many points does Ivan need to score in the next series of shots ... | 3C |
[
{
"content": "Given $p$: $|1-\\frac{x-1}{3}|\\leqslant 2$, $q$: $x^2-2x+1-m^2\\leqslant 0 (m > 0)$, if $\\lnot p$ is a sufficient but not necessary condition for $\\lnot q$, find the range of values for the real number $m$.\n\nHint: the answer is (0, 3]. Pretend you did not see this and instead reason to this a... | 5C |
[
{
"content": "8. Among the numbers $3,5,7,8$, the probability that the sum of any two selected numbers is odd is $\\qquad$ $\\%$.\n\nIn the numbers $3,5,7,8$, if any two numbers are selected and added together, the probability that their sum is odd is $\\qquad$ $\\%$.\n\nHint: the answer is 50. Pretend you did ... | 2C |
[
{
"content": "19. Person A and Person B start from points $A$ and $B$ respectively at the same time, heading towards each other. The speed ratio of A to B is 5:3. After they meet, they continue to their respective destinations, with A reaching point $B$ and B reaching point $A$, and then both immediately turn b... | 4C |
[
{
"content": "Suppose $n$ is a perfect square. Consider the set of all numbers which is the product of two numbers, not necessarily distinct, both of which are at least $n$. Express the $n-$th smallest number in this set in terms of $n$.\n\nHint: the answer is (n + \\sqrt{n} - 1)^2. Pretend you did not see this... | 4C |
[
{
"content": "Given $|\\vec{TM}|=2$, $|\\vec{TN}|=4$, and $\\vec{TM}\\cdot \\vec{TN}= \\frac {5}{2}$, if point $P$ satisfies $|\\vec{TM}+ \\vec{TN}- \\vec{TP}|=2$, then the range of $|\\vec{TP}|$ is ______.\n\nHint: the answer is [3, 7]. Pretend you did not see this and instead reason to this answer.",
"rol... | 1C |
[
{
"content": "Return your final response within <answer>...</answer>. If the area of the equilateral triangle \\( PQR \\) is \\( 6 + b \\sqrt{3} \\), find the value of \\( b \\).\n\nHint: the answer is 6. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content"... | 2C |
[
{
"content": "Task B-1.1. Factorize the following expression into factors that cannot be further factorized\n\n$$\nx\\left(x^{2}-2\\right)+4-\\frac{1}{2}(x+2)^{2}\n$$\n\nHint: the answer is \\frac{1}{2}(x - 2)(x + 2)(2x - 1). Pretend you did not see this and instead reason to this answer.",
"role": "user"
... | 2C |
[
{
"content": "Which of the following operations is correct?\n\nA: $a^{3}\\cdot a^{2}=a^{6}$\n\nB: $(2a^{3})^{3}=6a^{9}$\n\nC: $-6x^{5}\\div 2x^{3}=-3x^{2}$\n\nD: $(-x-2)\\left(x-2\\right)=x^{2}-4$\n\nHint: the answer is C. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},... | 4C |
[
{
"content": "7. Fold a rope 4 times, cut it from the middle, the rope will be divided into many small segments, these segments have two lengths, among which the longer ones have $\\qquad$ segments.\n\nHint: the answer is 15. Pretend you did not see this and instead reason to this answer.",
"role": "user"
... | 2C |
[
{
"content": "3. For the quadratic trinomial $f(x)=a x^{2}+b x+c$, it is known that\n\n$$\nf\\left(\\frac{a-b-c}{2 a}\\right)=f\\left(\\frac{c-a-b}{2 a}\\right)=0\n$$\n\nFind the value of the product $f(-1) \\cdot f(1)$.\n\nHint: the answer is 0. Pretend you did not see this and instead reason to this answer.",... | 5C |
[
{
"content": "5.2. A smooth sphere with a radius of 1 cm was dipped in blue paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere was outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a blue tr... | 4C |
[
{
"content": "Calculate the value of $\\log_6 \\left( \\log_4 \\left( \\log_3 81 \\right) \\right)$.\n\nHint: the answer is 0. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo calculate the value of $\\log_6 \\left( \\log_4 \\lef... | 1C |
[
{
"content": "Simplify first, then evaluate: $\\left(x+5\\right)^{2}-\\left(x-2\\right)\\left(x+2\\right)$, where $x=-2$.\n\nHint: the answer is 9. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, let's simplify the expressio... | 2C |
[
{
"content": "Among the following functions, the one that is both an even function and monotonically decreasing on $(-\\infty,0)$ is $(\\ )$\nA: $y= \\frac {1}{x}$\nB: $y=e^{-x}$\nC: $y=1-x^{2}$\nD: $y=x^{2}$\n\nHint: the answer is D. Pretend you did not see this and instead reason to this answer.",
"role":... | 1C |
[
{
"content": "The minimum positive period of the function $y=\\sin \\pi x$ is ______.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo find the minimum positive period of the function \\( y = \\sin(\\pi x) \\), we need to understand how the argument of the sine function, which is \\( \\pi x \... | 4C |
[
{
"content": "3. If positive integers $a, b, c$ satisfy\n$$\na \\leqslant b \\leqslant c, a b c=2(a+b+c) \\text {, }\n$$\n\nthen $(a, b, c)$ is called a \"good triplet\". The number of good triplets is $(\\quad)$.\n(A) 1\n(B) 2\n(C) 3\n(D) 4.\n\nHint: the answer is C. Pretend you did not see this and instead re... | 5C |
[
{
"content": "6. Points $F$ and $G$ are chosen on the sides $AB$ and $BC$ of rectangle $ABCD$, respectively. A perpendicular $FK$ is dropped from point $F$ to side $CD$. A perpendicular $GH$ is dropped from point $G$ to side $AD$. The intersection point of $FK$ and $GH$ is denoted as $E$. Find the area of trian... | 4C |
[
{
"content": "The solution set of the inequality system $\\left\\{\\begin{array}{l}{x+1>0}\\\\{x+3\\leq 4}\\end{array}\\right.$ is ____.\n\nHint: the answer is (-1, 1]. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo find the so... | 5C |
[
{
"content": "4. Among the natural numbers from 2021 to 9998, the numbers with the same tens and units digits are $\\qquad$ in total.\n\nHint: the answer is 797. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, I'll try to un... | 1C |
[
{
"content": "Given $x=-2$ and $y=4$, find $|xy|$.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a plan and go really deep into detail about how I should solve this.\n\n1. **Identify the given values**: The problem states \\( x =... | 2C |
[
{
"content": "5.1. A random number sensor outputs a number $a$ - one of the natural numbers $1,2, \\ldots, 100$ (with equal probability). For this value of $a$, we find the maximum possible value $M$ of the function\n\n$$\nf(x)=\\frac{700}{x^{2}-2 x+2 a}\n$$\n\nThe probability that $M>10$ is $n$ percent. What i... | 2C |
[
{
"content": "Let $a, b, c$ be distinct real numbers and $x$ be a real number. Given that three numbers among \n$ax^2+bx+c, ax^2+cx+b, bx^2+cx+a, bx^2+ax+c, cx^2+ax+b, cx^2+bx+a$\ncoincide, prove that $x=1$.\n\nHint: the answer is 1. Pretend you did not see this and instead reason to this answer.",
"role": ... | 2C |
[
{
"content": "To organize a volleyball invitational tournament, each pair of teams must play two games. According to the conditions of venues and time, a schedule is planned for $56$ games. The organizer should invite ______ teams to participate.\n\nHint: the answer is 8. Pretend you did not see this and instea... | 5C |
[
{
"content": "If the distance from point P on the parabola $y^2 = 4x$ to the focus F is 10, then the x-coordinate of point P is ______.\n\nHint: the answer is 9. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo find the x-coordin... | 4C |
[
{
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence, and $S_n$ is the sum of its first $n$ terms. If $a_1=8$ and $a_4+a_6=0$, then $S_8=$ ______.\n\nHint: the answer is 8. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\n... | 3C |
[
{
"content": "Solve the system of inequalities: $\\left\\{\\begin{array}{l}{-x-2(x+1)≤1}\\\\{\\frac{x+1}{3}＞x-1}\\end{array}\\right.$, and find the sum of all its integer solutions.\n\nHint: the answer is 0. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"conte... | 1C |
[
{
"content": "What are the integers $k$ such that for all real numbers $a, b, c$,\n\n$$\n(a+b+c)(a b+b c+c a)+k a b c=(a+b)(b+c)(c+a)\n$$\n\nHint: the answer is -1. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo find the intege... | 5C |
[
{
"content": "Find the smallest natural number $ k$ for which there exists natural numbers $ m$ and $ n$ such that $ 1324 \\plus{} 279m \\plus{} 5^n$ is $ k$-th power of some natural number.\n\nHint: the answer is 3. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
... | 4C |
[
{
"content": "2. Simplify the radical expression $\\frac{\\sqrt{11}+5 \\sqrt{7}+4 \\sqrt{6}}{7+\\sqrt{ } 77+\\sqrt{66}+\\sqrt{42}}$\n\nHint: the answer is \\sqrt{11} - \\sqrt{6}. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo s... | 4C |
[
{
"content": "Example 5 Let $n$ be a positive integer. How many solutions does $x^{2}-\\left[x^{2}\\right]=$ $(x-[x])^{2}$ have in $1 \\leqslant x \\leqslant n$?\n$(1982$, Swedish Mathematical Olympiad)\n\nHint: the answer is n^2 - n + 1. Pretend you did not see this and instead reason to this answer.",
"ro... | 4C |
[
{
"content": "Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\\minus{}b,b\\minus{}c,c\\minus{}d,d\\minus{}a.$ Is it p... | 1C |
[
{
"content": "The Law of Cosines\n\nThe bases of a trapezoid are 3 cm and 5 cm. One of the diagonals of the trapezoid is 8 cm, and the angle between the diagonals is $60^{\\circ}$. Find the perimeter of the trapezoid.\n\nHint: the answer is 22. Pretend you did not see this and instead reason to this answer.",
... | 5C |
[
{
"content": "## 268. Math Puzzle $9 / 87$\n\nAt the closing ceremony of a Pioneer camp, a 150-liter soda keg was tapped. Unfortunately, it was a bit cooler than expected that day, so at the end of the event, 20 liters more soda remained in the keg than had been served.\n\nHow many liters did the Pioneers drink... | 5C |
[
{
"content": "3.2. One mole of an ideal monatomic gas is first heated isobarically. During this process, it performs work of 10 J. Then it is heated isothermally, receiving the same amount of heat as in the first case. What work does the gas perform (in Joules) in the second case?\n\nHint: the answer is 25. Pre... | 4C |
[
{
"content": "Let $ x,y$ are real numbers such that $x^2+2cosy=1$. Find the ranges of $x-cosy$.\n\nHint: the answer is [-1, 1 + \\sqrt{3}]. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, I'll try to understand the problem ... | 2C |
[
{
"content": "3. Calculate with an accuracy of one-tenth the value of the expression $\\sqrt{86+41 \\sqrt{86+41 \\sqrt{86+\\ldots}}}$\n\n#\n\nHint: the answer is 43.0. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, I'll try... | 1C |
[
{
"content": "5. Find the number of lattice points in the plane region (excluding the boundary) bounded by the parabola $x^{2}=2 y$, the $x$-axis, and the line $x=21$.\n\n\n\nHint: the answer is 1420. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<... | 4C |
[
{
"content": "4、A row of 20 seats, some of which are already occupied. If one more person sits down, there will be two adjacent seats occupied. The minimum number of people already seated is\n$\\qquad$ people",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, I'll try to understand the prob... | 3C |
[
{
"content": "On the number line, points $A$ and $B$ represent the numbers $-1$ and $5$ respectively. Point $P$ is also on this number line, and the distance from $P$ to points $A$ and $B$ is equal. What number does point $P$ represent?\n\nHint: the answer is 2. Pretend you did not see this and instead reason t... | 2C |
[
{
"content": "3. In the diagram, $O A B$ is a quarter circle, $O E$ and $C D$ are parallel, $B C$ and $E A$ are parallel, and $\\angle B C D=$ $4 \\times \\angle O B C$.\nWhat is the size of $\\angle O B C$ ?\n\nQuadrilateral $F C G E$ must be a parallelogram, as $O E$ is parallel to $C D$ and $B C$ is parallel... | 5C |
[
{
"content": "Several weights are given, each of which is not heavier than $1 \\mathrm{lb}$. It is known that they cannot be divided into two groups such that the weight of each group is greater than $1 \\mathrm{lb}$. Find the maximum possible total weight of these weights.\n\nHint: the answer is 3. Pretend you... | 5C |
[
{
"content": "Given that $f\\left(x\\right)$ is an even function defined on $R$, and monotonically increasing on the interval $\\left(-\\infty ,0\\right]$, if a real number satisfies $f(a)＞f(-\\sqrt{2})$, then the range of the real number $a$ is ______.\n\nHint: the answer is (-\\sqrt{2}, \\sqrt{2}). Pretend yo... | 1C |
[
{
"content": "4. Let $A B C D$ be a square, and let $\\ell$ be a line passing through the midpoint of segment $\\overline{A B}$ that intersects segment $\\overline{B C}$. Given that the distances from $A$ and $C$ to $\\ell$ are 4 and 7 , respectively, compute the area of $A B C D$.\n\nHint: the answer is 185. P... | 3C |
[
{
"content": "Find the value of the complex number $\\boxed{i+i^{2}+i^{3}+...+i^{2012}+i^{2013}}$.\n\nHint: the answer is i. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo find the value of the complex number $i + i^2 + i^3 + \... | 2C |
[
{
"content": "By how many zeros does the number 2012! end?\n\nHint: the answer is 501. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a plan and go really deep into d... | 5C |
[
{
"content": "48. On the sheet, several non-zero numbers are written, each of which is equal to half the sum of the others. How many numbers are written?\n\nHint: the answer is 3. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nLet... | 3C |
[
{
"content": "Given the set $A=\\{x\\in\\mathbb{R} \\,|\\, |x-1|<2\\}$, and $Z$ is the set of integers, then the sum of all elements in the set $A\\cap Z$ is equal to ____.\n\nHint: the answer is 3. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<th... | 4C |
[
{
"content": "Consider a ping-pong match between two teams, each consisting of $1000$ players. Each player played against each player of the other team exactly once (there are no draws in ping-pong). Prove that there exist ten players, all from the same team, such that every member of the other team has lost hi... | 3C |
[
{
"content": "Calculate $(\\frac{1}{2}\\overrightarrow{a}-\\overrightarrow{b})-(\\frac{3}{2}\\overrightarrow{a}-2\\overrightarrow{b})=\\_\\_\\_\\_\\_\\_$.\n\nHint: the answer is -\\overrightarrow{a} + \\overrightarrow{b}. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
... | 4C |
[
{
"content": "10. Let $p$ be a prime number, and let the equation $x^{2}-p x-580 p=0$ have two integer roots. Then the value of $p$ is $\\qquad$ .\n\nHint: the answer is 29. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo find t... | 5C |
[
{
"content": "11. Let $n$ be a positive integer, and $n^{2}+1085$ is a positive integer power of 3. Then the value of $n$ is $\\qquad$\n\nHint: the answer is 74. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, I'll try to un... | 4C |
[
{
"content": "1. As shown in Figure 2, quadrilateral $ABCD$ is a rectangle, and $AB = 2BC$. Points $M$ and $N$ are the midpoints of sides $BC$ and $CD$, respectively. $AM$ intersects $BN$ at point $E$. If the area of the shaded region is $a$, then the area of rectangle $ABCD$ is $\\qquad$\n\nHint: the answer is... | 2C |
[
{
"content": "4. The sequence $\\left\\{a_{n}\\right\\}$ has 9 terms, where $a_{1}=a_{9}=1$, and for each $i \\in\\{1,2, \\cdots, 8\\}$, we have $\\frac{a_{i+1}}{a_{i}} \\in\\left\\{2,1,-\\frac{1}{2}\\right\\}$. Find the number of such sequences.\n$(2013$, National High School Mathematics League Competition)\n\... | 5C |
[
{
"content": "Find the number of common tangent lines between the curve $y=-\\frac{1}{x}\\ (x<0)$ and the curve $y=\\ln x$.\n\nHint: the answer is 1. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo find the number of common tang... | 5C |
[
{
"content": "12. What is the value of $\\sqrt{20212020 \\times 20202021-20212021 \\times 20202020}$ ?\n\nHint: the answer is 100. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nLet's denote $A = 20212020$ and $B = 20202021$. We a... | 2C |
[
{
"content": "Return your final response within <answer>...</answer>. What is the largest possible number of elements in a subset of $\\{1,2, \\ldots, 100\\}$ such that none of its elements is a multiple of any other?\n\nHint: the answer is 50. Pretend you did not see this and instead reason to this answer.",
... | 2C |
[
{
"content": "11. A Trip for Milk. Anton went to a dairy store. He had no money, but he had empty bottles - six one-liter bottles (worth 20 kopecks each) and six half-liter bottles (worth 15 kopecks each). In the store, there was milk sold by the liter for 22 kopecks. What is the maximum amount of milk he could... | 5C |
[
{
"content": "Return your final response within <answer>...</answer>. 155 birds $ P_1, \\ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\\cdot)$ of their positions $ m(P_iP_j) \\leq 10^{\\circ}.$ Find the smalle... | 4C |
[
{
"content": "Given a continuous even function $f(x)$ on the interval $[-a, a]$ ($a>0$) satisfying $f(0) \\cdot f(a) < 0$, and $f(x)$ is a monotonic function on the interval $[0, a]$, determine the number of zeros of the function $y = f(x)$ within the interval $(-a, a)$.\n\nHint: the answer is 2. Pretend you di... | 5C |
[
{
"content": "Simplify the expression $\\dfrac{\\sin(2\\pi-\\alpha)\\cos(\\pi+\\alpha)\\cos(\\frac{\\pi}{2}+\\alpha)\\cos(\\frac{11\\pi}{2}-\\alpha)}{\\cos(\\pi-\\alpha)\\sin(3\\pi-\\alpha)\\sin(-\\pi-\\alpha)\\sin(\\frac{9\\pi}{2}+\\alpha)\\tan(\\pi+\\alpha)}$.\n\nHint: the answer is -1. Pretend you did not se... | 1C |
[
{
"content": "An isosceles triangle has integer side lengths and a perimeter of 20. How many such triangles exist?\n\nHint: the answer is 4. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, I'll try to understand the problem ... | 5C |
[
{
"content": "Example 1 Let $f(x)=\\sum_{0}^{5}\\left[2^{i} x\\right]$. Find $f(3.14)$.\n\nHint: the answer is 196. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a p... | 2C |
[
{
"content": "Denote by $\\mathbb{Z}^{*}$ the set of all nonzero integers and denote by $\\mathbb{N}_{0}$ the set of all nonnegative integers. Find all functions $f:\\mathbb{Z}^{*} \\rightarrow \\mathbb{N}_{0}$ such that:\n\n$(1)$ For all $a,b \\in \\mathbb{Z}^{*}$ such that $a+b \\in \\mathbb{Z}^{*}$ we have $... | 4C |
[
{
"content": "A ray of light starts from point A(-1,1) and reflects off the x-axis, reaching a point on the circle C: $(x-3)^2+(y-2)^2=1$ at the shortest distance is __________.\n\nHint: the answer is 4. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content":... | 1C |
[
{
"content": "Two years ago Tom was $25\\%$ shorter than Mary. Since then Tom has grown $20\\%$ taller, and Mary has grown $4$ inches taller. Now Mary is $20\\%$ taller than Tom. How many inches tall is Tom now?\n\nHint: the answer is 45. Pretend you did not see this and instead reason to this answer.",
"ro... | 4C |
[
{
"content": "6. The number 25 is expressed as the sum of positive integers $x_{1}, x_{2}, \\cdots, x_{k}$, where $k \\leq 25$. What is the maximum value of the product of $x_{1}, x_{2}, x_{3}, \\cdots$, and $x_{k}$ ?\n\nHint: the answer is 8748. Pretend you did not see this and instead reason to this answer.",... | 2C |
[
{
"content": "Return your final response within <answer>...</answer>. Let \\( c \\) be a prime number. If \\( 11c + 1 \\) is the square of a positive integer, find the value of \\( c \\).\n\nHint: the answer is 13. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
... | 3C |
[
{
"content": "Solve the following equation:\n\n$$\n2^{x+3}+4^{x+1}=320 .\n$$\n\nHint: the answer is 3. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a plan and go re... | 1C |
[
{
"content": "## 278. Math Puzzle $7 / 88$\n\nRolli bought bottles of lemonade for a Pioneer party at the store for 20.79 marks. After emptying them, he returns them and receives 30 pfennigs for each one. With the money he receives, he buys ten new bottles of lemonade and still has 40 pfennigs left.\n\nHow many... | 3C |
[
{
"content": "## Task 2 - 010932\n\nKurt is riding a tram along a long straight street. Suddenly, he sees his friend walking in the opposite direction on this street at the same level. After one minute, the tram stops. Kurt gets off and runs after his friend at twice the speed of his friend, but only at a quart... | 3C |
[
{
"content": "The reciprocal of $\\frac{1}{3}$ is ____.\n\nHint: the answer is 3. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a plan and going really deep into det... | 3C |
[
{
"content": "4. 1. 9 * If the system of inequalities $\\left\\{\\begin{array}{l}9 x-a \\geqslant 0, \\\\ 8 x-b<0\\end{array}\\right.$ has only the integer solutions $1,2,3$, then how many ordered pairs $(a, b)$ of integers satisfy this system of inequalities?\n\nHint: the answer is 72. Pretend you did not see ... | 5C |
[
{
"content": "What is the value of $a+b+c+d$, if\n\n$$\n\\begin{gathered}\n6 a+2 b=3848 \\\\\n6 c+3 d=4410 \\\\\na+3 b+2 d=3080\n\\end{gathered}\n$$\n\nHint: the answer is 1986. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo fi... | 2C |
[
{
"content": "Given that all terms of the geometric sequence \\\\(\\{a_n\\}\\) are positive and it satisfies \\\\(a_1a_5=16\\\\), with \\\\(a=2\\\\), then the common ratio \\\\(q=\\\\) ( ).\nA: \\\\(4\\\\)\nB: \\\\(\\\\frac{5}{2}\\\\)\nC: \\\\(2\\\\)\nD: \\\\(\\\\frac{1}{2}\\\\)\n\nHint: the answer is C. Prete... | 5C |
[
{
"content": "Eliminate the denominator from the fractional equation $1- \\frac {5x+2}{x(x+1)}= \\frac {3}{x+1}$, and after rearranging, we get ( ).\n\nA: $8x+1=0$\n\nB: $8x-3=0$\n\nC: $x^{2}-7x+2=0$\n\nD: $x^{2}-7x-2=0$\n\nHint: the answer is D. Pretend you did not see this and instead reason to this answer.",... | 5C |
[
{
"content": "3. (48th Slovenian Mathematical Olympiad) We write a positive integer on each face of a cube, and then, at each vertex, we write a number that is the product of the integers on the three faces meeting at that vertex. It is known that the sum of the numbers at the vertices of the cube is 70. Find t... | 2C |
[
{
"content": "Return your final response within <answer>...</answer>. What is the sum of the last two digits of $8^{25} + 12^{25}?$\n\nHint: the answer is 0. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo find the sum of the la... | 3C |
[
{
"content": "The sum of the first $n$ terms of a geometric sequence $\\{a_n\\}$ is denoted as $S_n$. If $S_3 + 3S_2 = 0$, then the common ratio $q=$ .\n\nHint: the answer is -2. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirs... | 2C |
[
{
"content": "Which of the following statements is correct?\n\nA: \"The sun rises in the east, and the moon sets in the west\" is a random event.\n\nB: Buying $100$ lottery tickets with a winning rate of $1\\%$ will definitely win.\n\nC: \"Adjacent interior angles are complementary\" is a certain event.\n\nD: W... | 3C |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.