conversations listlengths 2 2 | sft_template_type_idx stringclasses 5 values |
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"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 |
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"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 |
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"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 |
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"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.",
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"c... | 4C |
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"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 |
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"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 |
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"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.",
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"content": "<think>\n<sample>\nTo find \\( B \\) for which the root... | 2C |
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"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.",
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"content": "<think>\n<sample>\nTo simplify the given express... | 5C |
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"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.",
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... | 1C |
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"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 |
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"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 |
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"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 |
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"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 |
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"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 |
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"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 |
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"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 |
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"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 |
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"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 |
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"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 $b$? $\\qquad$ -",
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"content": "<think>\n<sample>\nTo solve for \\(... | 1C |
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"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 |
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"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 |
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"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 |
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"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 |
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