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"value": "Let \\( a, b, c \\) be positive real numbers. Prove that\n\n$$\n\\frac{1}{a(1+b)}+\\frac{1}{b(1+c)}+\\frac{1}{c(1+a)} \\geq \\frac{3}{1+abc},\n$$\n\nand that equality occurs if and only if \\( a = b = c = 1 \\)."
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"value": "<think>\n\nOkay, so... |
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"value": "Given real numbers \\( a, b, c \\) and a positive number \\( \\lambda \\) such that the polynomial \\( f(x) = x^3 + a x^2 + b x + c \\) has three real roots \\( x_1, x_2, x_3 \\), and the conditions \\( x_2 - x_1 = \\lambda \\) and \\( x_3 > \\frac{1}{2}(x_1 + x_2) \\) are satisf... |
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"value": "In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triangles $ABC$ and $EFC$ share the same incenter. Prove that $DE \\cdot DF = DA^2$."
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"value": "<think>\n\nOkay, so I have this geomet... |
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"value": "Let \\( p = 2^{3009}, q = 3^{2006}, \\) and \\( r = 5^{1003} \\). Which of the following statements is true?\n(A) \\( p < q < r \\)\n(B) \\( p < r < q \\)\n(C) \\( q < p < r \\)\n(D) \\( r < p < q \\)\n(E) \\( q < r < p \\)"
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"value": "<think>... |
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"value": "When \\( a < -1 \\), the nature of the roots for the equation\n$$\n\\left(a^{3}+1\\right) x^{2}+\\left(a^{2}+1\\right) x-(a+1)=0\n$$\nis:\n(A) Two negative roots.\n(B) One positive root and one negative root, with the absolute value of the negative root being larger.\n(C) One pos... |
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"value": "Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, T, U, V$ be the respective midpoints of $AB, AC, AD, BC, BD, CD$. If $QR = 2001$, $SU = 2002$, and $TV = 2003$, find the distance between the midpoints of $QU$ and $RV$."
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... |
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"value": "Given that \\(1 \\leq x, y, z \\leq 6\\), how many cases are there in which the product of natural numbers \\(x, y, z\\) is divisible by 10?"
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"value": "<think>\n\nOkay, let's see. The problem is asking how many ordered triples (x, y, z) there... |
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"value": "Find all prime numbers \\( p \\) such that for any prime number \\( q < p \\), if \\( p = kq + r \\) with \\( 0 \\leq r < q \\), then there does not exist an integer \\( a > 1 \\) such that \\( a^2 \\) divides \\( r \\)."
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"value": "<think>\n\... |
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"value": "Find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfy for all \\( x, y \\in \\mathbb{R} \\):\n\n\\[ \nf(y-f(x)) = f(x) - 2x + f(f(y)) \n\\]"
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"value": "<think>\n\nOkay, so I need to find all functions \\( f: \\mathbb{... |
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"value": "Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of triangle $ABC$. It turns out that the perimeter of $\\triangle AMC$ is equal to the perimeter of $\\triangle CNA$, and the perimeter of $\\triangle ANB$ is equal to the perimeter of $\\triangle CMB$. Prove th... |
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"value": "Let \\( a, b, c \\) be given complex numbers, with \\( |a+b| = m \\) and \\( |a-b| = n \\). Given that \\( m \\cdot n \\neq 0 \\), prove that:\n$$\n\\max \\{|ac+b|, |a+bc|\\} \\geqslant \\frac{mn}{\\sqrt{m^2 + n^2}}\n$$"
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"value": "<think>\n\n... |
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"value": "Write the first $n$ natural numbers in decimal form on a (very long) strip of paper, then cut the strip so that each piece contains exactly one digit. Place these pieces in a box, mix them up, and draw one at random. Let $p_{n}$ denote the probability that the drawn piece of pape... |
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"value": "How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight."
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"value": "<think>\n\nOkay, let's see. I n... |
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"value": "Determine the value of\n\n$$\nz=a \\sqrt{a} \\sqrt[4]{a} \\sqrt[8]{a} \\ldots \\sqrt[2^{n}]{a} \\ldots\n$$\n\nif \\( n \\) is infinitely large."
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"value": "<think>\n\nOkay, so I need to find the value of z, which is this infinite product of a ... |
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"value": "Give the value of \\(0 - 1 + 2 - 3 + 4 - 5 + \\ldots - 49 + 50\\). Only a numerical answer is expected."
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"value": "<think>\n\nOkay, let me try to figure out the value of this series: 0 - 1 + 2 - 3 + 4 - 5 + ... - 49 + 50. Hmm, that's an alter... |
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"value": "Verify that 2 is a solution of the equation:\n\n$$\n(x+1)^{3}+(x+2)^{3}+(x+3)^{3}=(x+4)^{3}\n$$\n\nDoes this equation have any other integer solutions?"
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"value": "<think>\n\nOkay, so I need to verify if 2 is a solution to the equation (x+1)^3... |
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"value": "The sequence $\\left\\{a_{n}\\right\\}$ is defined as follows: $a_{1}=\\frac{1}{2}$, and $a_{n+1}=\\frac{a_{n}^{2}}{a_{n}^{2}-a_{n}+1}$ for $n=1,2,\\ldots$. Prove that for every positive integer $n$, the inequality $a_{1}+a_{2}+\\cdots+a_{n}<1$ holds."
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"value": "If the integer \\( n \\) is neither a multiple of 2 nor a multiple of 5, prove that \\( n \\) must be able to divide a number composed entirely of the digit 1."
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"value": "<think>\n\nOkay, so I need to prove that if an integer n isn't a multip... |
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"value": "A game begins with seven coins lined up on a table, all showing heads up. To win the game, you need to flip some coins such that, in the end, two adjacent coins always show different faces. The rule of the game is to flip two adjacent coins in each move. What is the minimum numbe... |
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"value": "Find the function $F(x)$, whose graph passes through the point $M_{0}(3 ;-2)$, given that $F'(x) = 4x^{2} + 9x^{-2}$."
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"value": "<think>\n\nOkay, so I need to find the function F(x) whose graph passes through the point M₀(3, -2), and we know ... |
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"value": "In an isosceles trapezoid with bases \\(a = 21\\), \\(b = 9\\) and height \\(h = 8\\), find the radius of the circumscribed circle."
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"value": "<think>\n\nOkay, so I need to find the radius of the circumscribed circle around an isosceles trape... |
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"value": "Given two linear functions \\( f(x) \\) and \\( g(x) \\) such that the graphs \\( y = f(x) \\) and \\( y = g(x) \\) are parallel lines that are not parallel to the coordinate axes. Find the minimum value of the function \\( 3(g(x))^2 + 2 f(x) \\), given that the minimum value of ... |
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"value": "Buratino calculated the time accurately and left Papa Carlo's house at 13:40 to reach the Field of Miracles and plant 4 coins exactly at sunset. If he had walked $25 \\%$ faster, he would have arrived at the Field of Miracles 1.5 hours earlier and waited. At what time did Buratin... |
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"value": "Given five distinct real numbers, consider all their pairwise sums. The smallest two sums are 20 and 200, while the largest two sums are 2014 and 2000. Determine the range of the smallest number, \\( a \\), among these five real numbers."
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"va... |
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"value": "On a board, nine quadratic trinomials are written: \\(x^{2}+a_{1} x+b_{1}, x^{2}+a_{2} x+b_{2}, \\ldots, x^{2}+a_{9} x+b_{9}\\). It is known that the sequences \\(a_{1}, a_{2}, \\ldots, a_{9}\\) and \\(b_{1}, b_{2}, \\ldots, b_{9}\\) are arithmetic progressions. It turns out that... |
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"value": "Find the variance of the discrete random variable $X$ that is distributed according to the Poisson distribution:\n\n$$\n\\begin{array}{ccccccc}\nX & 0 & 1 & 2 & \\cdots & k & \\cdots \\\\\nP & \\mathrm{e}^{-\\lambda} & \\lambda \\mathrm{e}^{-\\lambda} / 1! & \\lambda^{2} \\mathrm... |
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"value": "Prove that\n\n$$\n\\int_{0}^{\\pi}|\\ln \\sin x| \\, dx \\leq \\frac{\\pi^{3}}{12}\n$$\n\nwhere $\\ln$ denotes the natural logarithm with base $e = 2.71\\ldots$."
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"value": "<think>\n\nOkay, I need to prove that the integral of the absolute va... |
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"value": "Prove that in a right-angled triangle, the distance between the incenter and the circumcenter is at least $(\\sqrt{2}-1)$ times the radius of the circumcircle."
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"value": "<think>\n\nOkay, so I need to prove that in a right-angled triangle, th... |
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"value": "Calculate the volume of the body bounded above by the surface \\(z = xy^2\\) and below by the rectangle \\(0 \\leq x \\leq 1\\), \\(0 \\leq y \\leq 2\\)."
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"value": "<think>\n\nOkay, so I need to find the volume bounded above by the surface z ... |
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"value": "\nVasya cut a triangle out of cardboard and numbered its vertices with the digits $1, 2, 3$. It turned out that if Vasya rotates the triangle 12 times clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex, it will return to its original position. If... |
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"value": "On the sides of a convex centrally symmetric hexagon \\( A B C D E F \\), equilateral triangles are constructed externally. Prove that the midpoints of the segments connecting the vertices of adjacent triangles form a regular hexagon."
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"value": "A person writes \\( n \\) letters for \\( n \\) individuals, each receiving one letter, and prepares \\( n \\) envelopes labeled with the recipients' names and addresses. How many different ways are there to place the letters into the envelopes (one letter per envelope) such that... |
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"value": "On February 1, it was 16.2°C outside Jacinta's house at 3:00 p.m. On February 2, it was -3.6°C outside Jacinta's house at 2:00 a.m. If the temperature changed at a constant rate between these times, the rate at which the temperature decreased was\n(A) 1.1°C / h\n(B) 1.8°C / h\n(C... |
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"value": "If \\( n \\) is a positive integer, the notation \\( n! \\) (read \" \\( n \\) factorial\") represents the product of the integers from 1 to \\( n \\) inclusive. For example, \\( 5! = 1 \\times 2 \\times 3 \\times 4 \\times 5 = 120 \\). Which of the following is equal to a perfec... |
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"value": "Find all positive integers \\( n \\) such that the set \\( \\{ n, n+1, n+2, n+3, n+4, n+5 \\} \\) can be partitioned into two subsets so that the product of the numbers in each subset is equal."
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"value": "<think>\n\nOkay, so I need to find al... |
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"value": "The real numbers \\( x_{1}, x_{2}, \\ldots, x_{1996} \\) have the following property: for any polynomial \\( W \\) of degree 2, at least three of the numbers \\( W(x_{1}), W(x_{2}), \\ldots, W(x_{1996}) \\) are equal. Prove that at least three of the numbers \\( x_{1}, x_{2}, \\l... |
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"value": "Given that $O$ is the circumcenter of an acute-angled triangle $ABC$, and $\\angle A = \\theta$. If $\\frac{\\cos B}{\\sin C} AB + \\frac{\\cos C}{\\sin B} AC = 2m \\cdot AO$, then find the value of $m$."
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"value": "<think>\n\nOkay, so I need ... |
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"value": "What is the largest result we can find when we divide a 2-digit number by the sum of its digits?"
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"value": "<think>\n\nOkay, so I need to find the largest result when dividing a two-digit number by the sum of its digits. Let me start by under... |
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"value": "In a class of \\( n \\geq 3 \\) students, the teacher has \\( m = \\frac{1}{2} n(n-1) \\) cards, each with a unique number from \\( 1, 2, 3, \\ldots, m \\). The teacher gives each student one card. For each pair of students, the sum of the numbers on their cards is calculated and... |
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"value": "Show that for every positive integer \\( n \\), there exists a nonzero multiple of \\( n \\) that has at most \\( n \\) digits, all of which are either 0 or 1."
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"value": "<think>\n\nOkay, so I need to show that for every positive integer n, t... |
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"value": "The probability of an event $A$ occurring is $v=0.1$. We conduct $n=998$ trials. Which is more likely; that $A$ occurs 100 times, or that it occurs 99 times?"
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"value": "<think>\n\nOkay, let's try to figure out which is more likely: event A oc... |
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"value": "A person is practicing target shooting, starting at a distance of 100 meters from the target for the first shot. If the first shot misses, they move back 50 meters for the second shot and continue this pattern; each time they miss, they move back 50 meters and take another shot u... |
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"value": "In the regular hexagonal prism \\(ABCDEF - A'B'C'D'E'F'\\), \\(M\\) is the midpoint of \\(DE\\). Determine the shape of the cross-section passing through the points \\(A'\\), \\(C\\), and \\(M\\).\n\nA. Triangle\n\nB. Quadrilateral\n\nC. Pentagon\n\nD. Hexagon"
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"value": "Prove that for any positive integer \\( n \\), the following inequality holds:\n\\[ \\lg n \\geqslant k \\lg 2 \\]\nwhere \\( k \\) is the number of distinct prime factors of \\( n \\)."
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"value": "<think>\n\nOkay, so I need to prove that for ... |
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"value": "Let \\( a_{1}, a_{2}, \\ldots, a_{2000} \\) be real numbers in the interval \\([0,1]\\). Find the maximum possible value of\n\\[ \n\\sum_{1 \\leq i < j \\leq 2000}(j - i) \\left| a_{j} - a_{i} \\right| \n\\]"
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"value": "<think>\n\nOkay, so I n... |
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"value": "A regular polygon \\( A_{1} \\ldots A_{n} \\) is inscribed in a circle of radius \\( R \\) with center \\( O \\). Let \\( X \\) be an arbitrary point. Prove that \\( A_{1} X^{2} + \\ldots + A_{n} X^{2} = n \\left( R^{2} + d^{2} \\right) \\), where \\( d = OX \\)."
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"value": "The force exerted by the airflow on a sail can be calculated using the formula:\n\n\\[ F = \\frac{C S \\rho (v_0 - v)^2}{2}, \\]\n\nwhere \\(C\\) is the coefficient of aerodynamic force, \\(S\\) is the area of the sail (\\(S = 5 \\, \\text{m}^2\\)), \\(\\rho\\) is the air density... |
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"value": "Jitka hiked a trail. After hiking 60% of the length of the trail, she had 8 km left to go. What is the length of the trail?\n(A) 28 km\n(B) 12.8 km\n(C) 11.2 km\n(D) 13 \\(\\frac{1}{3}\\) km\n(E) 20 km"
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"value": "<think>\n\nOkay, let me try t... |
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"value": "In the expansion of \\((a+b)^n\\), there are \\(n+1\\) different terms. In the expansion of \\((a+b+c)^{10}\\), the number of different terms is:\n\n(A) 11 \n(B) 33 \n(C) 55 \n(D) 66 \n(E) 132 \n\n(Provided by the 9th American High School Mathematics Examination, 1958)"
},... |
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"value": "The sequence \\( x_{n} \\) has its first two elements as \\( x_{1}=1001 \\) and \\( x_{2}=1003 \\). For \\( n \\geq 1 \\), the recurrence relation is given by:\n\\[ x_{n+2}=\\frac{x_{n+1}-2004}{x_{n}}. \\]\nWhat is the sum of the first 2004 terms of the sequence?"
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"value": "Can some of the four notable points of a triangle coincide with others? What can be said about a triangle in which the centers of the circumscribed and inscribed circles coincide?"
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"value": "<think>\n\nOkay, so I have this geometry problem to... |
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"value": "Find any five consecutive natural numbers less than 100, whose product is divisible by 2014."
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"value": "<think>\n\nAlright, so I need to find five consecutive natural numbers less than 100 such that their product is divisible by 2014. Let me ... |
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"value": "$A, B, C, D, E$ are seated in a train that consists of 5 carriages, with each carriage only holding one person. It is known that $D$ is seated in the last carriage, $A$ is immediately behind $E$, $B$ is in a carriage before $A$, and there is at least one person between $B$ and $C... |
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"value": "Usually, two mechanisms work together to complete a certain task. Their efficiencies are not the same, and when working together, they complete the task in 30 hours. Once, the two mechanisms worked together for only 6 hours, after which the first mechanism was stopped, and the se... |
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"value": "One way to pack a 100 by 100 square with 10,000 circles, each of diameter 1, is to put them in 100 rows with 100 circles in each row. If the circles are repacked so that the centers of any three tangent circles form an equilateral triangle, what is the maximum number of additiona... |
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"value": "In the arithmetic sequence $\\left\\{a_{n}\\right\\}$, if $\\frac{a_{11}}{a_{10}} < -1$ and the sum of its first $n$ terms $S_{n}$ has a maximum value, then when $S_{n}$ takes the smallest positive value, $n = (\\quad$ ).\n(A) 1\n(B) 10\n(C) 19\n(D) 20"
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"value": "Do there exist rational numbers \\(a, b, c, d\\) that satisfy the equation\n\n$$\n(a + b \\sqrt{2})^{2n} + (c + d \\sqrt{2})^{2n} = 5 + 4 \\sqrt{2}\n$$\n\n(where \\(n\\) is a natural number)?"
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"value": "<think>\n\nOkay, let me try to figure o... |
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"value": "Given a positive integer \\( m \\), for a positive integer \\( l \\), let \\( A_{l} = (4l + 1)(4l + 2) \\cdots (4(5^{m} + 1)l) \\). Prove that there exist infinitely many positive integers \\( l \\) such that \\( 5^{5ml} \\mid A_{l} \\) and \\( 5^{5ml + 1} + A_{l} \\) is also an ... |
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"value": "Given that \\(2^{96} - 1\\) is divisible by two integers between 60 and 70, what are these two numbers?\n(A) 61, 63\n(B) 61, 65\n(C) 63, 65\n(D) 63, 67"
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"value": "<think>\n\nOkay, let me try to figure out this problem. The question says that ... |
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"value": "Every day, Patrick the dog chews up one slipper from the available stock at home. There is a strict 0.5 probability that Patrick wants to chew on the left slipper and a 0.5 probability that he wants to chew on the right slipper. If the desired slipper is not available, Patrick ge... |
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"value": "\nCalculate the limit of the function:\n\n$$\n\\lim _{x \\rightarrow 0} \\frac{\\arcsin 3x}{\\sqrt{2+x}-\\sqrt{2}}\n$$"
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"value": "<think>\n\nOkay, so I need to find the limit as x approaches 0 of (arcsin(3x)) divided by (sqrt(2 + x) - sqrt(2)... |
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"value": "List, in increasing order, each multiple of 3 whose sum with 1 is a perfect square:\n\n$$\n3, 15, 24, 48, \\ldots\n$$\n\na) What is the next number that will appear in this sequence after 48?\n\nb) What is the eighth number in this sequence?\n\nc) What number will appear in the 2... |
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"value": "Sofia was taking some sweets to her grandmother: 7 blackberry sweets, 6 coconut sweets, and 3 chocolate sweets. On the way, the greedy Sofia eats 2 sweets. Which of the following situations is possible?\n\n(A) Grandmother did not receive any chocolate sweets.\n\n(B) Grandmother r... |
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"value": "A group of children are arranged into two equal rows. Every child in the back row is taller than the child standing in front of him in the other row. Prove that this remains true if each row is rearranged so that the children increase in height from left to right."
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"... |
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"value": "Arrange the numbers $1,2,3, \\cdots, 37$ in a sequence $a_{1}, a_{2}, \\cdots, a_{37}$, where $a_{1}=37$ and $a_{2}=1$, such that $a_{1}+a_{2}+\\cdots+a_{k}$ is divisible by $a_{k+1}$ for $k=1,2, \\cdots, 36$. Find $a_{37}$ and $a_{3}$."
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"val... |
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"value": "Prove that if \\( \\frac{a}{b} = \\frac{b}{c} \\), then \\( a^{2} + c^{2} \\geq 2 b^{2} \\)."
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"value": "<think>\n\nOkay, so I need to prove that if a/b = b/c, then a² + c² is greater than or equal to 2b². Hmm, let's start by understanding the... |
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"value": "We consider the two sequences $\\left(a_{n}\\right)_{n \\geq 0}$ and $\\left(b_{n}\\right)_{n \\geq 0}$ of integers, defined by $a_{0}=b_{0}=2$ and $a_{1}=b_{1}=14$ and by\n\n$$\n\\begin{aligned}\na_{n} & =14 a_{n-1}+a_{n-2} \\\\\nb_{n} & =6 b_{n-1}-b_{n-2}\n\\end{aligned}\n$$\n\... |
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"value": "What is the value of the expression \\(x^{2000} + x^{1999} + x^{1998} + 1000x^{1000} + 1000x^{999} + 1000x^{998} + 2000x^{3} + 2000x^{2} + 2000x + 3000\\) (where \\(x\\) is a real number), if \\(x^{2} + x + 1 = 0\\)? Vasya calculated the answer to be 3000. Is Vasya correct?"
},... |
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"value": "Ken is the best sugar cube retailer in the nation. Trevor, who loves sugar, is coming over to make an order. Ken knows Trevor cannot afford more than 127 sugar cubes, but might ask for any number of cubes less than or equal to that. Ken prepares seven cups of cubes, with which he... |
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"value": "You have infinitely many boxes, and you randomly put 3 balls into them. The boxes are labeled \\(1, 2, \\ldots\\). Each ball has a probability \\(\\frac{1}{2^n}\\) of being put into box \\(n\\). The balls are placed independently of each other. What is the probability that some b... |
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"value": "Professor Antônio discovered an interesting property related to the integer $x$ that represents his age. He told his students that $x^{2}=\\overline{a b a c}$ and that $x=\\overline{a b}+\\overline{a c}$. What is the professor's age?\n\nNote: We are using a bar to distinguish the... |
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"value": "The faces of a cube are inscribed with six natural numbers (possibly repeating) such that the numbers on adjacent faces differ by more than 1. What is the smallest possible sum of these six numbers?"
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"value": "<think>\n\nOkay, so I need to fi... |
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"value": "Let \\(a, b,\\) and \\(c\\) be strictly positive real numbers such that \\(abc = 1\\). Show that\n\n$$\n\\left(a+\\frac{1}{b}\\right)^{2}+\\left(b+\\frac{1}{c}\\right)^{2}+\\left(c+\\frac{1}{a}\\right)^{2} \\geq 3(a+b+c+1)\n$$"
},
{
"from": "assistant",
"value": "<thi... |
[
{
"from": "user",
"value": "Determine the nearest integer to\n(a) $\\frac{19}{15}+\\frac{19}{3}$\n(b) $\\frac{85}{42}+\\frac{43}{21}+\\frac{29}{14}+\\frac{15}{7}$\n(c) $-\\frac{11}{10}-\\frac{1}{2}-\\frac{7}{5}+\\frac{2}{3}$"
},
{
"from": "assistant",
"value": "<think>\n\nOkay, let me try to wor... |
[
{
"from": "user",
"value": "Let \\( p \\) be a given prime, and \\( a_{1}, \\cdots, a_{k} \\) be \\( k \\) integers (\\( k \\geq 3 \\)) that are not divisible by \\( p \\) and are pairwise distinct modulo \\( p \\). Define\n\\[\nS = \\{n \\mid 1 \\leq n \\leq p-1, \\left(n a_{1}\\right)_{p} < \\cdots < \\le... |
[
{
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"value": "\nIn a mini-football tournament, bets are accepted on four teams. Bets on the first team have odds of $1: 5$ (if the first team wins, the player receives the amount they bet on this team plus five times that amount, i.e., six times the bet amount, and if they lose, the money is n... |
[
{
"from": "user",
"value": "(a) Determine the largest possible value $M$ that $x + y + z$ can take if $x$, $y$, and $z$ are positive real numbers satisfying\n\n$$\n16 x y z = (x + y)^2 (x + z)^2\n$$\n\n(b) Show that there are infinitely many triples $(x, y, z)$ of positive rational numbers such that\n\n$$\n... |
[
{
"from": "user",
"value": "Find all pairs of positive numbers \\((x, y)\\) that satisfy the system of equations:\n\\[\n\\left\\{\n\\begin{array}{l}\n2 x - \\sqrt{x y} - 4 \\sqrt{\\frac{x}{y}} + 2 = 0 \\\\\n2 x^{2} + x^{2} y^{4} = 18 y^{2}.\n\\end{array}\n\\right.\n\\]"
},
{
"from": "assistant",
... |
[
{
"from": "user",
"value": "\\(1-\\cos (2 \\alpha - \\pi) + \\cos (4 \\alpha - 2 \\pi) = 4 \\cos 2 \\alpha \\cos \\left(\\frac{\\pi}{6} + \\alpha\\right) \\cos \\left(\\frac{\\pi}{6} - \\alpha\\right)\\)."
},
{
"from": "assistant",
"value": "<think>\n\nAlright, so I need to prove this trigonomet... |
[
{
"from": "user",
"value": "There are 45 students in a certain exam in the Zhonghuan Cup. Among them, 35 students are good at English, 31 students are good at Chinese, and 24 students are good at both subjects. How many students are not good at either subject?"
},
{
"from": "assistant",
"value":... |
[
{
"from": "user",
"value": "Denis has identical ten-ruble coins, identical two-ruble coins, and identical one-ruble coins (more than 20 coins of each type). In how many ways can Denis pay exactly 16 rubles for a pie without receiving change? It is not necessary to use coins of each type."
},
{
"from... |
[
{
"from": "user",
"value": "For each value of the parameter \\( k \\), solve the equation:\n\n\\[ 2+\\cos ^{2} x+\\cos 4 x+\\cos 2 x+2 \\sin 3 x \\cdot \\sin 7 x+\\sin ^{2} 7 x=\\cos ^{2}\\left(\\frac{\\pi k}{2022}\\right) \\]"
},
{
"from": "assistant",
"value": "<think>\n\nOkay, so I have this ... |
[
{
"from": "user",
"value": "Given real numbers \\( x \\) and \\( y \\) that satisfy \\[ \\frac{x^{2}}{9}+\\frac{y^{2}}{4}=1 \\], find the maximum value of the function \\( U = x + y \\)."
},
{
"from": "assistant",
"value": "<think>\n\nOkay, so I need to find the maximum value of U = x + y given ... |
[
{
"from": "user",
"value": "Given a circle \\( \\odot O \\), point \\( A \\) lies outside the circle, and two tangents \\( AM \\) and \\( AN \\) are drawn from \\( A \\) to touch the circle at points \\( M \\) and \\( N \\) respectively. Point \\( P \\) is chosen on the extension of segment \\( OM \\) such ... |
[
{
"from": "user",
"value": "It is known that all 'krakozyabrs' have either horns or wings (or both). From the results of a global census of 'krakozyabrs,' it was found that 20% of the 'krakozyabrs' with horns also have wings, and 25% of the 'krakozyabrs' with wings also have horns. How many 'krakozyabrs' ar... |
[
{
"from": "user",
"value": "Show that there are no seven lines in the plane such that there are at least six points which lie on the intersection of just three lines and at least four points which lie on the intersection of just two lines."
},
{
"from": "assistant",
"value": "<think>\n\nOkay, so... |
[
{
"from": "user",
"value": "\nDetermine how much money the Romanov family will save by using a multi-tariff meter over three years.\n\nThe cost of the meter is 3500 rubles. The installation cost is 1100 rubles. On average, the family's electricity consumption is 300 kWh per month, with 230 kWh used from 23:... |
[
{
"from": "user",
"value": "Given that \\(\\theta_{1}, \\theta_{2}, \\cdots, \\theta_{n}\\) are all non-negative real numbers and satisfy the equation\n\\[\n\\theta_{1} + \\theta_{2} + \\cdots + \\theta_{n} = \\pi.\n\\]\n\nFind the maximum value of \\(\\sin^2 \\theta_{1} + \\sin^2 \\theta_{2} + \\cdots + \\... |
[
{
"from": "user",
"value": "In the expression \\((x+y+z)^{2026}+(x-y-z)^{2026}\\), the brackets were expanded and like terms combined. How many terms of the form \\(x^{a} y^{b} z^{c}\\) with a non-zero coefficient resulted?"
},
{
"from": "assistant",
"value": "<think>\n\nOkay, so I have this pro... |
[
{
"from": "user",
"value": "Using a compass and a ruler, construct a point such that the tangents drawn from it to two given circles are equal to the given segments."
},
{
"from": "assistant",
"value": "<think>\n\nOkay, so I need to figure out how to construct a point such that the tangents from... |
[
{
"from": "user",
"value": "Find the value of the expression \\(\\frac{a^{2}}{b c}+\\frac{b^{2}}{a c}+\\frac{c^{2}}{a b}\\) if \\(a + b + c = 0\\)."
},
{
"from": "assistant",
"value": "<think>\n\nOkay, let's see. The problem is to find the value of the expression \\(\\frac{a^{2}}{b c}+\\frac{b^{... |
[
{
"from": "user",
"value": "It is known that none of the numbers \\(a\\), \\(b\\), \\(c\\) are integers. Is it possible that each of the products \\(ab\\), \\(bc\\), \\(ca\\), and \\(abc\\) is an integer?"
},
{
"from": "assistant",
"value": "<think>\n\nOkay, so the problem is asking whether it's... |
[
{
"from": "user",
"value": "Form a sequence from the consecutive positive integers from 1 to $n$ (where $n > 1$) in such a way that the sum of any two adjacent terms in the sequence is a perfect square. What is the smallest value of the positive integer $n$?"
},
{
"from": "assistant",
"value": "... |
[
{
"from": "user",
"value": "How many rational terms are in the expansion of\n\na) $(\\sqrt{2}+\\sqrt[4]{3})^{100}$\n\nb) $(\\sqrt{2}+\\sqrt[3]{3})^{300}$?"
},
{
"from": "assistant",
"value": "<think>\n\nOkay, so I need to figure out how many rational terms are in the expansions of these two expr... |
[
{
"from": "user",
"value": "Assume there is a gathering with $n$ couples attending. Each individual during a certain period can either rest alone or join a chatting group, referred to as a team. Each couple is not in the same team, and every other pair of individuals is in exactly one team together. Prove t... |
[
{
"from": "user",
"value": "Determine the value of\n$$\n\\frac{3}{\\sin ^{2} 20^{\\circ}}-\\frac{1}{\\cos ^{2} 20^{\\circ}}+64 \\sin ^{2} 20^{\\circ}.\n$$"
},
{
"from": "assistant",
"value": "<think>\n\nOkay, let's try to figure out the value of this expression: 3 over sine squared 20 degrees mi... |
[
{
"from": "user",
"value": "Given a non-negative arithmetic sequence $\\{a_{n}\\}$ with common difference $d \\neq 0$, let $S_{n}$ represent the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$. Prove:\n\n1. If $m, n, p \\in \\mathbf{N}^{*}$ and $m+n=2p$, then $\\frac{1}{S_{m}} + \\frac{1}{S_{n}} \\... |
[
{
"from": "user",
"value": "We placed a cube on the center of a potter's wheel, with a natural number written on each of its faces. Just before we spun the wheel, from our point of view, we could see three faces of the cube with numbers totaling 42. After rotating the potter's wheel by $90^{\\circ}$, we obs... |
[
{
"from": "user",
"value": "Find the integer part of the expression\n\n\\[a=\\sqrt{1981+\\sqrt{1981+\\sqrt{1981+\\ldots+\\sqrt{1981+\\sqrt{1981}}}}},\\]\n\nif the number 1981 appears $n$ times ($n \\geq 2$)."
},
{
"from": "assistant",
"value": "<think>\n\nOkay, so I need to find the integer part... |
[
{
"from": "user",
"value": "Twenty cities are connected by 172 airline routes. Prove that, using these airline routes, it is possible to travel from any city to any other city (possibly with layovers)."
},
{
"from": "assistant",
"value": "<think>\n\nOkay, let's try to figure out how to prove tha... |
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