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[
{
"content": "## Task 5 - 100935\n\nA triangular pyramid with vertices $A, B, C, D$ and apex $D$ has edge lengths $A B=4$ $\\mathrm{cm}, A C=3 \\mathrm{~cm}, B C=5 \\mathrm{~cm}, B D=12 \\mathrm{~cm}, C D=13 \\mathrm{~cm}$, and $\\angle A B D$ is a right angle.\n\nCalculate the volume $V$ of this pyramid.\n\nHi... | 5C |
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"content": "The range of values for $k$ in the function $y=(k+1)x^{2}-2x+1$ to have intersections with the $x$-axis is ______.\n\nHint: the answer is (-\\infty, 0]. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo determine the... | 3C |
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"content": "Let $a$ and $b$ be positive integers that satisfy $ab-7a-11b+13=0$. What is the minimum possible value of $a+b$?\n\nHint: the answer is 34. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand... | 3C |
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"content": "2. $50 N$ is an integer, its base $b$ representation is 777, find the smallest positive integer $b$, such that $N$ is an integer to the fourth power.\n\nHint: the answer is 18. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<s... | 2C |
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"content": "M. Murraikin\n\nOn an island, there live 100 knights and 100 liars, each of whom has at least one friend. Knights always tell the truth, while liars always lie. One morning, each resident either said the phrase \"All my friends are knights\" or the phrase \"All my friends are liars,\" and exactly ... | 3C |
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"content": "3. The members of the Monster Family have their health points (HP) double every morning at 8 o'clock, but to sustain their lives, they consume 5 HP every evening at 8 o'clock. Baby Monster Pinky was born with 6 HP at 6 o'clock in the morning on the first day, and her HP became 7 at 6 o'clock in th... | 5C |
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"content": "Let $a=\\log_{0.6}0.9$, $b=\\ln0.9$, $c=20.9$, then the order of $a$, $b$, $c$ from smallest to largest is .\n\nHint: the answer is b < a < c. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo determine the order of ... | 4C |
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"content": "2. You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1 , the value of each blue card is equal to twice the number ... | 2C |
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"content": "Given that $x,y \\in \\mathbb{R}_+, x + \\frac{2}{x} + 2y + \\frac{1}{y} = 6$, find the maximum value of $x + 2y$.\n\nHint: the answer is 4. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo maximize \\( x + 2y \\) g... | 1C |
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"content": "4. Given that the line $l$ is the tangent line to the graph of the function $f(x)=2 \\ln x+x^{2}$, when the slope of $l$ is the smallest, the equation of $l$ is\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, the... | 1C |
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"content": "Given $a^{2}+b^{2}-4a-6b+13=0$, find the value of $\\left(a-b\\right)^{2023}$.\n\nHint: the answer is -1. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out... | 5C |
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"content": "Given the function $f(x) = a^x$, where $a > 0$ and $a \\neq 1$, its maximum value on the interval $[-2, 1]$ is 4, and its minimum value is $b$. Additionally, the function $g(x) = (2 - 7b)x$ is a decreasing function. Find the value of $a + b$.\n\nHint: the answer is 1. Pretend you did not see this ... | 5C |
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"content": "The function $f(x)=mx^{3}+nx$ has an extreme value at $x= \\frac {1}{m}$, then $mn=$ _____ .\n\nHint: the answer is -3. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better ... | 3C |
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"content": "3. On the table, there are 140 different cards with numbers $4, 8, 12, \\ldots, 556, 560$ (each card has exactly one number, and each number appears exactly once). In how many ways can 3 cards be chosen so that the sum of the numbers on the selected cards is divisible by 3?\n\nHint: the answer is ... | 5C |
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"content": "1. Variant 1.\n\nPetya has four cards with digits $1,2,3,4$. Each digit appears exactly once. How many natural numbers greater than 2222 can Petya make from these cards?\n\nHint: the answer is 16. Pretend you did not see this and instead reason to this answer.",
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"co... | 4C |
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"content": "1. Answer: $19890+8716=28606$.\n\nHint: the answer is 28606. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nLet's break down the problem into steps.\n\nStep 1: Identify the numbers given for addition.\n- The first nu... | 3C |
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"content": "## Task A-2.2.\n\nInside the triangle $A B C$ are points $S$ and $T$. The distances from point $S$ to the lines $A B, B C$, and $C A$ are 10, 7, and 4, respectively. The distances from point $T$ to these lines are 4, 10, and 16, respectively.\n\nDetermine the radius of the inscribed circle of tria... | 3C |
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"content": "Among the following surveys, the one suitable for using a census method is (__).\n\nA: To understand the lethal radius of the latest batch of shells produced in 2016 \nB: To determine the viewership rating of the “XX” column on Yangquan TV station \nC: To identify the species of fish in the Yell... | 5C |
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"content": "Given that $\\frac{\\bar{z}}{1-i} = 2+i$, find the imaginary part of the complex number $z$.\n\nHint: the answer is 1. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better b... | 2C |
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"content": "Find the gcd of $n^{17}-n$ for $n$ an integer.\n\nHint: the answer is 510. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo find the greatest common divisor (gcd) of \\( n^{17} - n \\) for any integer \\( n \\), we ... | 1C |
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"content": "Determine the constant term in the expansion of \\\\((2 \\sqrt {x}- \\frac {1}{x})^{9}\\\\).\n\nHint: the answer is -5376. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo determine the constant term in the expansio... | 1C |
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"content": "4. Solve the following equation for positive $x$.\n\n$$\nx^{2014} + 2014^{2013} = x^{2013} + 2014^{2014}\n$$\n\nHint: the answer is 2014. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo solve the equation \\( x^{20... | 3C |
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"content": "## 192. Math Puzzle $5 / 81$\n\nThe students of class 7 b collected a total of $336 \\mathrm{~kg}$ of waste paper. From $1 \\mathrm{~kg}$ of waste paper, a paper factory produces $700 \\mathrm{~g}$ of pure white paper, and from every $30 \\mathrm{~g}$ of this, a writing pad is made.\n\nWhat is the... | 3C |
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"content": "1. Let $x^{2}+y^{2} \\leqslant 2$. Then the maximum value of $\\left|x^{2}-2 x y-y^{2}\\right|$ is $\\qquad$ .\n\nHint: the answer is 2\\sqrt{2}. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to unde... | 3C |
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"content": "42 Let $k$ be a natural number. Try to determine the smallest natural number $n$ such that: in any $n$ integers, there must be two numbers whose sum or difference is divisible by $2 k+1$.\n\n\n\nHint: the answer is k + 2. Pretend you did not see this and instead reason to this answer.",
"role"... | 2C |
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"content": "Convert 89 to binary.\n\nHint: the answer is 1011001. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo convert the decimal number 89 to binary, we can use the method of successive division by 2 and recording the rem... | 3C |
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"content": "4. Given the curve $C_{1}: y=\\sqrt{-x^{2}+10 x-9}$ and point $A(1,0)$. If there exist two distinct points $B$ and $C$ on curve $C_{1}$, whose distances to the line $l: 3 x+1=0$ are $|A B|$ and $|A C|$, respectively, then $|A B|+|A C|=$\n\nHint: the answer is 8. Pretend you did not see this and in... | 1C |
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"content": "9. find all injective functions $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ such that for all natural numbers $n$ holds\n\n$$\nf(f(n)) \\leq \\frac{f(n)+n}{2}\n$$\n\n## 1st solution\n\nHint: the answer is f(n) = n. Pretend you did not see this and instead reason to this answer.",
"role": "user"
... | 2C |
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"content": "Problem 5. Three regular octagons have a common center, their sides are respectively parallel. The sides of two octagons are 7 cm and 42 cm. The third octagon divides the area of the figure enclosed between the first two in the ratio $1: 6$, counting from the smaller octagon. Find the side of the ... | 5C |
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"content": "10.4 Let $n$ - be a natural number greater than 10. What digit can stand immediately after the decimal point in the decimal representation of the number $\\sqrt{n^{2}+n}$? Provide all possible answers and prove that there are no others.\n\nHint: the answer is 4. Pretend you did not see this and in... | 1C |
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"content": "15. On the number line, points with coordinates\n$1,2, \\cdots, 2006$ are called marked points. A frog starts from point 1, and after 2006 jumps, it visits all the marked points and returns to the starting point. What is the maximum length of the total path the frog has jumped? Explain your reason... | 1C |
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"content": "2. An 8-row $n$-column array, if formed into several 15-row 15-column squares, has 3 people left over, one person holding a flag, and 2 people guarding the flag. Then the smallest $n$ equals $\\qquad$\n\nHint: the answer is 141. Pretend you did not see this and instead reason to this answer.",
... | 3C |
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"content": "There are $281 n(n>3)$ football teams that have played $k$ matches, with each pair of teams playing at most one match, and there are no ties. It turns out that exactly one team remains undefeated. Additionally, it is found that for any two teams $A$ and $B$, there exists another team $C$ such that... | 4C |
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"content": "If $y=\\left(a+1\\right)x^{|a+3|}-x+3$ is a quadratic function with respect to $x$, then the value of $a$ is ______.\n\nHint: the answer is -5. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo determine if the given... | 1C |
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"content": "Given the function $f(x)= \\frac{4^{x}-1}{2^{x+1}}$, the correct statement about the function $f(x)$ is ($\\quad\\quad$).\nA: It is an odd function and an increasing function on $R$.\nB: It is an even function and an increasing function on $R$.\nC: It is an odd function and a decreasing function o... | 5C |
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"content": "In the arithmetic sequence $\\{a_n\\}$, $a_1 > 0$, $a_{10} \\cdot a_{11} < 0$, if the sum of the first $10$ terms of this sequence is $S_{10} = 36$, and the sum of the first $18$ terms is $S_{18} = 12$, then the sum of the first $18$ terms of the sequence $\\{|a_n|\\}$, denoted as $T_{18}$, is ___... | 4C |
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"content": "3. As shown in the figure, a toy ant starts crawling from point $O$. On the $n$-th time, it first crawls $n$ units to the right, then $n$ units upwards, reaching point $A_{n}$. Then, it continues crawling from point $A_{n}$. If point $O$ is denoted as $(0,0)$, point $A_{1}$ as $(1,1)$, point $A_{2... | 3C |
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"content": "In the sequence $\\{a_n\\}$, the sum of the first $n$ terms, denoted by $S_n$, is given by $S_n = 4^n + a$. If the sequence $\\{a_n\\}$ is a geometric sequence, determine the value of the constant $a$.\n\nHint: the answer is -1. Pretend you did not see this and instead reason to this answer.",
... | 5C |
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"content": "2. (Mathematics Teaching Problem 538) Given $a>1, b>1, c>1$, and $a^{2}+b^{2}+c^{2}=12$, prove: $\\frac{1}{a-1}+\\frac{1}{b-1}+\\frac{1}{c-1} \\geqslant$\n3.\n\nHint: the answer is 3. Pretend you did not see this and instead reason to this answer.",
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"content": "<thi... | 3C |
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"content": "In [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) $ABC$ with [right angle](https://artofproblemsolving.com/wiki/index.php/Right_angle) $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. [Points](https://artofproblemsolving.com/w... | 3C |
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"content": "Return your final response within <answer>...</answer>. Given that 2 is both the square root of \\( x-2 \\) and the cube root of \\( 2x - y + 1 \\), find the arithmetic square root of \\( x^2 - 4y \\).\n\nHint: the answer is 4. Pretend you did not see this and instead reason to this answer.",
... | 5C |
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"content": "Return your final response within <answer>...</answer>. What is the product of the solutions of the equation $-35=-x^2-2x?$\n\nHint: the answer is -35. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo find the produ... | 1C |
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"content": "Return your final response within <answer>...</answer>. $r(x)$ has domain $\\{-1,0,1,2\\}$ and range $\\{0,2,4,6\\}$. $s(x)$ has domain $\\{1,2,3,4\\}$ and is defined by $s(x)=x+1$. What is the sum of all possible values of $s(r(x))$?\n\nHint: the answer is 8. Pretend you did not see this and inst... | 5C |
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"content": "154. $(0,25)^{2-x}=\\frac{256}{2^{x+3}}$.\n\n154. $(0.25)^{2-x}=\\frac{256}{2^{x+3}}$.",
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"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.\n1. Recognize... | 1C |
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"content": "7. Define the operation symbol “ * \" to mean $a * b=\\frac{a+b}{a b}$ (where $a$ and $b$ are not 0).\nGiven the following two conclusions:\n(1) The operation “ * \" satisfies the commutative law;\n(2) The operation “ * \" satisfies the associative law.\n\nWhich of the following is correct? ( ).\n... | 5C |
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"content": "Given $a=\\log _{1.5}3$, $b=e^{0.4}$, $c=\\log _{0.8}9$, the relationship between $a$, $b$, and $c$ is ( ).\n\nA: $c \\lt a \\lt b$\n\nB: $b \\lt c \\lt a$\n\nC: $b \\lt a \\lt c$\n\nD: $c \\lt b \\lt a$\n\nHint: the answer is D. Pretend you did not see this and instead reason to this answer.",
... | 3C |
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"content": "4. Given non-zero real numbers $a_{1}, a_{2}, \\cdots, a_{n}$ satisfy $\\frac{1}{a_{1}}+\\frac{1}{a_{2}}+\\cdots+\\frac{1}{a_{n}} \\neq 0$,\n$$\nS=a_{1}+a_{2}+\\cdots+a_{n}=-2020,\n$$\n\nthen the solution to $\\frac{x+S-a_{1}}{a_{1}}+\\frac{x+S-a_{2}}{a_{2}}+\\cdots+\\frac{x+S-a_{n}}{a_{n}}=-n$ is... | 4C |
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"content": "10. In Rt $\\triangle A B C$, $\\angle C=90^{\\circ}, D, E$ are points on sides $B C, A C$ respectively. Given $A D=17$, $B E=13, D E=\\sqrt{97}$, then $A B=$ $\\qquad$ .\n\nHint: the answer is 19. Pretend you did not see this and instead reason to this answer.",
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"c... | 5C |
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"content": "Given \\\\(p\\\\): \\\\(x^{2}-8x-20\\leqslant 0\\\\); \\\\(q\\\\): \\\\(x^{2}-2x+1-m^{2}\\leqslant 0(m > 0)\\\\); If \\\\(\\neg p\\\\) is a sufficient but not necessary condition for \\\\(\\neg q\\\\), find the range of values for \\\\(m\\\\).\n\nHint: the answer is (0, 3]. Pretend you did not see... | 5C |
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"content": "7、A river flows at a uniform speed, with docks A and B located upstream and downstream, respectively, 200 kilometers apart. Boats A and B depart from docks A and B simultaneously and head towards each other. After meeting, they continue to their respective destinations, immediately turn around, an... | 5C |
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"content": "Return your final response within <answer>...</answer>. The curves \\( y = x^{2}(x-3)^{2} \\) and \\( y = (x^{2}-1)(x-2) \\) intersect at a number of points in the real plane. Determine the sum of the \\( x \\)-coordinates of these points of intersection.\n\nHint: the answer is 7. Pretend you did ... | 2C |
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"content": "515. Calculate approximately $1.998^{5}$.\n\nHint: the answer is 31.84. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a plan and go really deep into de... | 4C |
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"content": "17. (MON 1) Given seven points in the plane, some of them are connected by segments so that:\n(i) among any three of the given points, two are connected by a segment;\n(i) the number of segments is minimal.\nHow many segments does a figure satisfying (i) and (ii) contain? Give an example of such a... | 1C |
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"content": "A population is divided into two strata, A and B. A stratified sampling method is used to draw a sample of size 10 from the population. It is known that the probability of each individual in stratum B being selected is $\\frac{1}{12}$. The total number of individuals in the population is \\_\\_\\_... | 3C |
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"content": "13. In the triangle $A B C, \\angle A=20^{\\circ}, \\angle C=90^{\\circ}, O$ is a point on $A B$ and $D$ is the midpoint of $O B$. Suppose the circle centered at $O$ with radius $O D$ touches the side $A C$ at $T$. Determine the size of $\\angle B C D$ in degrees.\n\nHint: the answer is 35. Preten... | 5C |
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"content": "Example 2 Suppose point $O$ is inside $\\triangle A B C$, and $\\overrightarrow{O A}+2 \\overrightarrow{O B}+3 \\overrightarrow{O C}=0$.\nThen $S_{\\triangle A B C}: S_{\\triangle A O C}=(\\quad$.\n(A) 2\n(B) $\\frac{3}{2}$\n(C) 3\n(D) $\\frac{5}{3}$\n(2004, National High School Mathematics Compet... | 4C |
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"content": "Given that $f(x)$ is a monotonically increasing function defined on $[-1, +\\infty)$, find the solution set of the inequality $f(e^{x-2}) \\geq f(2 - \\frac{x}{2})$.\n\nHint: the answer is [2, 6]. Pretend you did not see this and instead reason to this answer.",
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"co... | 1C |
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"content": "【Question 3】There are weights of 1 gram, 2 grams, 3 grams, and 5 grams, each one piece. These can be used to weigh from 1 to 11 grams. Some weights can be measured in more than one way, for example, 3 grams can be measured using the 3-gram weight, or using the 1-gram and 2-gram weights. Therefore,... | 1C |
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"content": "1. Function\n$$\nf(x)=27^{x}-3^{x+3}+1\n$$\n\nThe minimum value of the function on the interval $[0,3]$ is $\\qquad$",
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"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 sh... | 5C |
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"content": "In the sequence $\\{a_n\\}$, $a_{1}=- \\frac {4}{3}$, $a_{n+2}= \\frac {1}{a_{n}+1}$, then $a_{7}=$ \\_\\_\\_\\_\\_\\_.\n\nHint: the answer is 2. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to unde... | 5C |
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"content": "Return your final response within <answer>...</answer>. East and West Towns are 45 kilometers apart. Person A and Person B start from the two towns simultaneously and travel towards each other. Person A travels 1 kilometer per hour faster than Person B. They meet after 5 hours. How many kilometers... | 1C |
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"content": "5-0. A certain quadratic trinomial $x^{2}-p x+q$ has integer roots $x_{1}$ and $x_{2}$. It turns out that the numbers $x_{1}$, $x_{2}$, and $q$ form a decreasing arithmetic progression. Find the sum of all possible values of $x_{2}$.\n\nHint: the answer is -5. Pretend you did not see this and inst... | 4C |
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"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. It is known that $\\sin ^{2}B+\\cos ^{2}A-\\cos ^{2}C= \\sqrt {3}\\sin B\\sin C$, and the area of the circumcircle of triangle $ABC$ is $4\\pi$. Find the value of $a$.\n\nHint: the answer is 2. Pretend yo... | 2C |
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"content": "6. In the Thrice-Tenth Kingdom, there are 17 islands, each inhabited by 119 people. The inhabitants of the kingdom are divided into two castes: knights, who always tell the truth, and liars, who always lie. During the census, each person was first asked: \"Excluding you, are there an equal number ... | 5C |
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"content": "4. If the acute angle $\\alpha$ satisfies\n$$\n\\frac{1}{\\sqrt{\\tan \\frac{\\alpha}{2}}}=\\sqrt{2 \\sqrt{3}} \\cdot \\sqrt{\\tan 10^{\\circ}}+\\sqrt{\\tan \\frac{\\alpha}{2}},\n$$\n\nthen $\\alpha=$ $\\qquad$\n\nHint: the answer is 50^\\circ. Pretend you did not see this and instead reason to th... | 3C |
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"content": "Determine the range of values for $a$ such that the function $f(x) = x^2 - 2ax + a$ has a minimum value in the interval $(-\\infty, 1)$.\n\nHint: the answer is (-\\infty, 1). Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sam... | 3C |
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"content": "Return your final response within <answer>...</answer>. Consider the quadratic functions \\( y = x^2 + px + q \\) for which \\( p + q = 2002 \\).\n\nShow that the parabolas, which are the graphs of these functions, intersect at one point.\n\nHint: the answer is (1, 2003). Pretend you did not see t... | 5C |
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"content": "Is there a 6-digit number that becomes 6 times larger when its last three digits are moved to the beginning of the number, keeping the order of the digits?\n\nHint: the answer is 142857. Pretend you did not see this and instead reason to this answer.",
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"content": "<... | 5C |
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"content": "Solve the equation: $4^{x}-6\\cdot2^{x}-16=0$\n\nHint: the answer is 3. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nTo solve the equation \\(4^x - 6 \\cdot 2^x - 16 = 0\\), we can start by expressing all terms wit... | 3C |
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"content": "A square wall has side length $n$ metres. Guillaume paints $n$ non-overlapping circular targets on the wall, each with radius 1 metre. Mathilde is going to throw a dart at the wall. Her aim is good enough to hit the wall at a single point, but poor enough that the dart will hit a random point on t... | 4C |
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"content": "Round 0.3989 to the nearest hundredth.\n\nHint: the answer is 0.40. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a plan and going really deep into det... | 5C |
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"content": "In the polar coordinate system, point $A$ is a moving point on the line $3\\rho \\cos \\theta + 4\\rho \\sin \\theta + 13 = 0$, and point $B$ is a moving point on the curve $\\rho + 2\\cos \\theta = 0$. The minimum value of the distance $|AB|$ is\n\nHint: the answer is 1. Pretend you did not see t... | 1C |
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"content": "What is the geometric place of the vertex points of the parabolas given by the equation $y=x^{2}+t x+1$? ( $t$ is a real parameter.)\n\nHint: the answer is y = -x^2 + 1. Pretend you did not see this and instead reason to this answer.",
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"content": "<think>\n<sample>\... | 1C |
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"content": "The function $f(x)= \\begin{cases} x^{2}+2x, & x\\geqslant 0 \\\\ x^{2}-2x, & x < 0 \\end{cases}$, if $f(-a)+f(a)\\leqslant 2f(3)$, then the range of values for the real number $a$ is \\_\\_\\_\\_\\_\\_.\n\nHint: the answer is [-3, 3]. Pretend you did not see this and instead reason to this answer... | 4C |
[
{
"content": "Return your final response within <answer>...</answer>. A creative contest at the institute consisted of four tasks. There were 70 applicants in total. 35 applicants successfully completed the first task, 48 completed the second task, 64 completed the third task, and 63 completed the fourth task. ... | 4C |
[
{
"content": "12. Restore the digits. On the board, there was the product of three consecutive even numbers. During the break, Vasya erased some digits. As a result, on the board remained $87 * * * * * 8$. Help Petya find the missing digits in the product.\n\nHint: the answer is 87526608. Pretend you did not se... | 5C |
[
{
"content": "The slope of a tangent line to the curve $y=x^3-3x^2+2x$ is $-1$. Find the coordinates of the point of tangency.\n\nHint: the answer is (1, 0). Pretend you did not see this and instead reason to this answer.",
"role": "user"
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{
"content": "<think>\n<sample>\nTo find the coordinates o... | 1C |
[
{
"content": "Return your final response within <answer>...</answer>. Given $ n$ points arbitrarily in the plane $ P_{1},P_{2},\\ldots,P_{n},$ among them no three points are collinear. Each of $ P_{i}$ ( $1\\le i\\le n$ ) is colored red or blue arbitrarily. Let $ S$ be the set of triangles having $ \\{... | 5C |
[
{
"content": "Let the maximum value of the function $f(x)= \\frac {(x+2)^{2}+\\sin x}{x^{2}+4}$ be $M$, and the minimum value be $m$, then $M+m=$ \\_\\_\\_\\_\\_\\_.\n\nHint: the answer is 2. Pretend you did not see this and instead reason to this answer.",
"role": "user"
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{
"content": "<think>\n<... | 5C |
[
{
"content": "Which of the following equations is correct?\n\nA: $x^{2}+x^{2}=x^{4}$\n\nB: $x^{2}\\cdot x^{3}=x^{6}$\n\nC: $(-x)^{2}+\\left(-x\\right)^{4}=-x^{6}$\n\nD: $(-x)^{3}\\cdot \\left(-x\\right)^{4}=-x^{7}$\n\nHint: the answer is D. Pretend you did not see this and instead reason to this answer.",
"... | 4C |
[
{
"content": "If the distance from point P(2,0) to an asymptote of the hyperbola $\\frac {x^{2}}{a^{2}} - y^{2} = 1$ ($a > 0$) is 1, then find the value of $a$.\n\nHint: the answer is \\sqrt{3}. Pretend you did not see this and instead reason to this answer.",
"role": "user"
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{
"content": "<think>... | 3C |
[
{
"content": "7.44 A finite set of polygons in the plane is said to be properly placed if for any two of them, there is a line through the origin that intersects both.\n\nFind the smallest natural number $m$ such that for any properly placed set of polygons, one can draw $m$ lines through the origin so that eac... | 5C |
[
{
"content": "Return your final response within <answer>...</answer>. Let \\(ABCD\\) be a rectangle with \\(AB=20\\) and \\(AD=23\\). Let \\(M\\) be the midpoint of \\(CD\\), and let \\(X\\) be the reflection of \\(M\\) across point \\(A\\). Compute the area of triangle \\(XBD\\).\n\nHint: the answer is 575. Pr... | 3C |
[
{
"content": "3. If $z$ is a complex number with a non-zero real part, then the minimum value of $\\frac{\\operatorname{Re}\\left(z^{4}\\right)}{(\\operatorname{Re}(z))^{4}}$ is $\\qquad$\n\nHint: the answer is -8. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
... | 2C |
[
{
"content": "Given proposition p: \"There exists some $x_0 \\in [1, +\\infty)$ such that $(\\log_2 3)^{x_0} \\geq 1$\", what would be the negation of proposition p?\n\nA: There exists some $x_0 \\in [1, +\\infty)$ such that $(\\log_2 3)^{x_0} < 1$ \nB: There exists some $x_0 \\in [1, +\\infty)$ such that $(\\... | 3C |
[
{
"content": "8. (5 points) As shown in the figure, a square wall with a side length of 50 meters. Person A and Person B start from points $A$ and $C$ respectively at the same time, moving along the wall in a clockwise direction. It is known that A walks 5 meters per second, and B walks 3 meters per second. The... | 5C |
[
{
"content": "3. On the table, there are 4 stacks of coins. The first stack has 9 coins, the second has 7, the third has 5, and the fourth has 10. In one move, you are allowed to add one coin to three different stacks. What is the minimum number of moves required to make the number of coins in all stacks equal?... | 1C |
[
{
"content": "A rabbit, a skunk and a turtle are running a race.\n\nThe skunk finishes the race in 6 minutes.\n\nThe rabbit runs 3 times as quickly as the skunk.\n\nThe rabbit runs 5 times as quickly as the turtle.\n\nHow long does the turtle take to finish the race?\n\nHint: the answer is 10. Pretend you did n... | 4C |
[
{
"content": "10.1. Determine all natural numbers $n$ for which the number $\\sqrt{n+\\sqrt{n-2}}$ is rational.\n\nHint: the answer is 3. Pretend you did not see this and instead reason to this answer.",
"role": "user"
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{
"content": "<think>\n<sample>\nTo determine all natural numbers \\( n \\) fo... | 4C |
[
{
"content": "The fractions of the form $\\frac{n}{n+1}$, with $n$ a positive integer are:\n\n$$\n\\underbrace{\\frac{1}{2}}_{n=1} ; \\quad \\underbrace{\\frac{2}{3}}_{n=2} ; \\quad \\underbrace{\\frac{3}{4}}_{n=3} ; \\quad \\underbrace{\\frac{4}{5}}_{n=4} ; \\quad \\underbrace{\\frac{5}{6}}_{n=5} \\cdots\n$$\n... | 5C |
[
{
"content": "In the sequence $\\{a_n\\}$, $a_1=3$, $a_2=6$, and $a_{n+2}$ is the units digit of $a_n \\cdot a_{n+1}$. Find $a_{2013}$.\n\nHint: the answer is 8. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nFirst, I'll try to un... | 2C |
[
{
"content": "Simplify: $3x^{2}-3(2x^{2}+4y)+2(x^{2}-y)$.\n\nHint: the answer is -x^2 - 14y. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo simplify the expression \\(3x^2 - 3(2x^2 + 4y) + 2(x^2 - y)\\), we will follow these st... | 5C |
[
{
"content": "Let $n$ be a positive integer. For a positive integer $m$, we partition the set $\\{1, 2, 3,...,m\\}$ into $n$ subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of $n$, fi\fnd the largest positive integer $m$ for which such a partition ex... | 5C |
[
{
"content": "14. (6 points) $\\overline{\\mathrm{ab}}, \\overline{\\mathrm{cde}}$ represent a two-digit number and a three-digit number respectively. If $\\overline{\\mathrm{ab}}+\\overline{\\mathrm{cde}}=1079$, then $a+b+c+d+e=$\n\n untranslated part:\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。\n\nThe above text has ... | 2C |
[
{
"content": "Return your final response within <answer>...</answer>. When $p(x) = Ax^5 + Bx^3 + Cx + 4$ is divided by $x - 3,$ the remainder is 11. Find the remainder when $p(x)$ is divided by $x + 3.$\n\nHint: the answer is -3. Pretend you did not see this and instead reason to this answer.",
"role": "us... | 1C |
[
{
"content": "Given that the mapping $f: x \\to ax - 1$ is from set $A$ to set $B$, and $f(2) = 3$, find $f(3)$ = \\_\\_\\_\\_\\_\\_.\n\nHint: the answer is 5. Pretend you did not see this and instead reason to this answer.",
"role": "user"
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{
"content": "<think>\n<sample>\nWe are given a function... | 2C |
[
{
"content": "1. On the board, five different positive numbers are written. Determine the maximum number of ways in which pairs can be formed from them, the sum of which equals one of the five numbers written on the board.\n\nHint: the answer is 4. Pretend you did not see this and instead reason to this answer.... | 4C |
[
{
"content": "Among the following groups of objects, which one cannot form a set? ( )\nA: The more difficult problems in the first-year high school math textbook\nB: All parents of students in Class 2 of the second year of high school\nC: All courses offered in the third year of high school\nD: Students in Clas... | 1C |
[
{
"content": "The equation $\\left(m+2\\right)x^{|m|}+3mx+1=0$ is a quadratic equation in $x$. What is the value of $m$?\n\nHint: the answer is 2. Pretend you did not see this and instead reason to this answer.",
"role": "user"
},
{
"content": "<think>\n<sample>\nTo determine the value of \\( m \\) ... | 3C |
[
{
"content": "Given the sequence $\\{a_n\\}$ that satisfies $$a_{1}=2，a_{n}=2- \\frac {1}{a_{n-1}}(n≥2，n∈N^{*})$$, let $\\prod_{n}$ denote the product of the first $n$ terms of the sequence. Find the value of $\\prod_{2019}$.\n\nHint: the answer is 2020. Pretend you did not see this and instead reason to this a... | 4C |
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