conversations listlengths 2 2 |
|---|
[
{
"from": "user",
"value": "\nProblem 2. Let $n$ three-digit numbers satisfy the following properties:\n\n(1) No number contains the digit 0 .\n\n(2) The sum of the digits of each number is 9 .\n\n(3) The units digits of any two numbers are different.\n\n(4) The tens digits of any two numbers are different.... |
[
{
"from": "user",
"value": "## Problem 2\n\nLet the circles $k_{1}$ and $k_{2}$ intersect at two distinct points $A$ and $B$, and let $t$ be a common tangent of $k_{1}$ and $k_{2}$, that touches $k_{1}$ and $k_{2}$ at $M$ and $N$, respectively. If $t \\perp A M$ and $M N=2 A M$, evaluate $\\angle N M B$.\n\... |
[
{
"from": "user",
"value": "\nProblem 3. Let $A L$ and $B K$ be angle bisectors in the non-isosceles triangle $A B C$ ( $L$ lies on the side $B C, K$ lies on the side $A C$ ). The perpendicular bisector of $B K$ intersects the line $A L$ at point $M$. Point $N$ lies on the line $B K$ such that $L N$ is para... |
[
{
"from": "user",
"value": "\nProblem 4. Let $M$ be a subset of the set of 2021 integers $\\{1,2,3, \\ldots, 2021\\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$ we have $|a+b-c|>10$. Determine the largest possible number of elements of $M$.\n"
},
{
"from": "assistan... |
[
{
"from": "user",
"value": "\nProblem 1. Find all distinct prime numbers $p, q$ and $r$ such that\n\n$$\n3 p^{4}-5 q^{4}-4 r^{2}=26\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to find distinct prime numbers p, q, and r such that 3p⁴ - 5q⁴ - 4r² = 26. Hmm, primes a... |
[
{
"from": "user",
"value": "\nProblem 2. Consider an acute triangle $A B C$ with area S. Let $C D \\perp A B \\quad(D \\in A B)$, $D M \\perp A C \\quad(M \\in A C)$ and $\\quad D N \\perp B C \\quad(N \\in B C)$. Denote by $H_{1}$ and $H_{2}$ the orthocentres of the triangles $M N C$ and $M N D$ respective... |
[
{
"from": "user",
"value": "\nProblem 4. For a positive integer $n$, two players A and B play the following game: Given a pile of $s$ stones, the players take turn alternatively with A going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a multiple of $n$... |
[
{
"from": "user",
"value": "\nProblem 1. Find all pairs $(a, b)$ of positive integers such that\n\n$$\n11 a b \\leq a^{3}-b^{3} \\leq 12 a b\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all pairs of positive integers (a, b) that satisfy the inequalities 11ab ≤ a³ -... |
[
{
"from": "user",
"value": "## Problem 1.\n\nFind all pairs $(a, b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of 5 .\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all pairs of positive integers (a, b) such that both a! + b and b! + a are powers o... |
[
{
"from": "user",
"value": "\nProblem 4. Find all pairs $(p, q)$ of prime numbers such that\n\n$$\n1+\\frac{p^{q}-q^{p}}{p+q}\n$$\n\nis a prime number.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all pairs of prime numbers (p, q) such that the expression 1 + (p^q - q^... |
[
{
"from": "user",
"value": "\nProblem 1. Find all prime numbers $a, b, c$ and positive integers $k$ which satisfy the equation\n\n$$\na^{2}+b^{2}+16 \\cdot c^{2}=9 \\cdot k^{2}+1\n$$\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all prime numbers a, b, c and positive ... |
[
{
"from": "user",
"value": "\nProblem 2. Let $a, b, c$ be positive real numbers such that $a+b+c=3$. Find the minimum value of the expression\n\n$$\nA=\\frac{2-a^{3}}{a}+\\frac{2-b^{3}}{b}+\\frac{2-c^{3}}{c}\n$$\n\n![](https://cdn.mathpix.com/cropped/2024_06_05_f56efd4e6fb711c0f78eg-2.jpg?height=348&width=4... |
[
{
"from": "user",
"value": "\nA2. Let $a$ and $b$ be positive real numbers such that $3 a^{2}+2 b^{2}=3 a+2 b$. Find the minimum value of\n\n$$\nA=\\sqrt{\\frac{a}{b(3 a+2)}}+\\sqrt{\\frac{b}{a(2 b+3)}}\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the minimum value... |
[
{
"from": "user",
"value": "\nC1. Consider a regular $2 n+1$-gon $P$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $S E$ contains no other points that lie on the sides of $P$ except ... |
[
{
"from": "user",
"value": "\nNT3. Find all pairs of positive integers $(x, y)$ such that $2^{x}+3^{y}$ is a perfect square.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all pairs of positive integers (x, y) such that 2^x + 3^y is a perfect square. Hmm, interesting pro... |
[
{
"from": "user",
"value": "\nNT5. Find all positive integers $n$ such that there exists a prime number $p$, such that\n\n$$\np^{n}-(p-1)^{n}\n$$\n\nis a power of 3 .\n\nNote. A power of 3 is a number of the form $3^{a}$ where $a$ is a positive integer.\n"
},
{
"from": "assistant",
"value": "<th... |
[
{
"from": "user",
"value": "\nA4. Solve the following equation for $x, y, z \\in \\mathbb{N}$\n\n$$\n\\left(1+\\frac{x}{y+z}\\right)^{2}+\\left(1+\\frac{y}{z+x}\\right)^{2}+\\left(1+\\frac{z}{x+y}\\right)^{2}=\\frac{27}{4}\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I ne... |
[
{
"from": "user",
"value": "\nA5. Find the largest positive integer $n$ for which the inequality\n\n$$\n\\frac{a+b+c}{a b c+1}+\\sqrt[n]{a b c} \\leq \\frac{5}{2}\n$$\n\nholds for all $a, b, c \\in[0,1]$. Here $\\sqrt[1]{a b c}=a b c$.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I n... |
[
{
"from": "user",
"value": "\nG4. Let $A B C$ be an acute-angled triangle with circumcircle $\\Gamma$, and let $O, H$ be the triangle's circumcenter and orthocenter respectively. Let also $A^{\\prime}$ be the point where the angle bisector of angle $B A C$ meets $\\Gamma$. If $A^{\\prime} H=A H$, find the m... |
[
{
"from": "user",
"value": "\nA1. Find all ordered triples $(x, y, z)$ of real numbers satisfying the following system of equations:\n\n$$\n\\begin{aligned}\nx^{3} & =\\frac{z}{y}-2 \\frac{y}{z} \\\\\ny^{3} & =\\frac{x}{z}-2 \\frac{z}{x} \\\\\nz^{3} & =\\frac{y}{x}-2 \\frac{x}{y}\n\\end{aligned}\n$$\n"
},... |
[
{
"from": "user",
"value": "\nA2. Find the largest possible value of the expression $\\left|\\sqrt{x^{2}+4 x+8}-\\sqrt{x^{2}+8 x+17}\\right|$ where $x$ is a real number.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the largest possible value of the expression |√(x² + 4... |
[
{
"from": "user",
"value": "\nN6. Find all ordered triples $(x, y, z)$ of integers satisfying the following system of equations:\n\n$$\n\\begin{aligned}\nx^{2}-y^{2} & =z \\\\\n3 x y+(x-y) z & =z^{2}\n\\end{aligned}\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to solve thi... |
[
{
"from": "user",
"value": "## A1 MLD\n\nLet $x, y, z$ be real numbers, satisfying the relations\n\n$$\n\\left\\{\\begin{array}{l}\nx \\geq 20 \\\\\ny \\geq 40 \\\\\nz \\geq 1675 \\\\\nx+y+z=2015\n\\end{array}\\right.\n$$\n\nFind the greatest value of the product $P=x \\cdot y \\cdot z$.\n\n"
},
{
"... |
[
{
"from": "user",
"value": "## A2 ALB\n\n3) If $x^{3}-3 \\sqrt{3} x^{2}+9 x-3 \\sqrt{3}-64=0$, find the value of $x^{6}-8 x^{5}+13 x^{4}-5 x^{3}+49 x^{2}-137 x+2015$.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's try to figure out how to solve this problem. So, we have the equation... |
[
{
"from": "user",
"value": "\nNT1 SAU\n\nWhat is the greatest number of integers that can be selected from a set of 2015 consecutive numbers so that no sum of any two selected numbers is divisible by their difference?\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure ou... |
[
{
"from": "user",
"value": "\nC2 Consider 50 points in the plane, no three of them belonging to the same line. The points have been colored into four colors. Prove that there are at least 130 scalene triangles whose vertices are colored in the same color.\n\n"
},
{
"from": "assistant",
"value": ... |
[
{
"from": "user",
"value": "\nG2 Let $A B C D$ be a convex quadrilateral with $\\varangle D A C=\\varangle B D C=36^{\\circ}, \\varangle C B D=18^{\\circ}$ and $\\varangle B A C=72^{\\circ}$. If $P$ is the point of intersection of the diagonals $A C$ and $B D$, find the measure of $\\varangle A P D$.\n\n"
... |
[
{
"from": "user",
"value": "## A2\n\nLet $a, b$ and $c$ be positive real numbers such that abc $=\\frac{1}{8}$. Prove the inequality\n\n$$\na^{2}+b^{2}+c^{2}+a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2} \\geq \\frac{15}{16}\n$$\n\nWhen does equality hold?\n"
},
{
"from": "assistant",
"value": "<think>\nA... |
[
{
"from": "user",
"value": "## C2\n\nIn a country with $n$ cities, all direct airlines are two-way. There are $r>2014$ routes between pairs of different cities that include no more than one intermediate stop (the direction of each route matters). Find the least possible $n$ and the least possible $r$ for th... |
[
{
"from": "user",
"value": "## C3\n\nFor a given positive integer n, two players $A$ and B play the following game: Given is pile of $\\boldsymbol{\\Omega}$ stones. The players take turn alternatively with A going first. On each turn the player is allowed to take one stone, a prime number of stones, or a mu... |
[
{
"from": "user",
"value": "## G3\n\nLet $C D \\perp A B(D \\in A B), D M \\perp A C(M \\in A C)$ and $D N \\perp B C(N \\in B C)$ for an acute triangle ABC with area $S$. If $H_{1}$ and $H_{2}$ are the orthocentres of the triangles $M N C$ and MND respectively. Evaluate the area of the quadrilateral $\\mat... |
[
{
"from": "user",
"value": "\nN3\n\nFind the integer solutions of the equation\n\n$$\nx^{2}=y^{2}\\left(x+y^{4}+2 y^{2}\\right)\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all integer solutions to the equation \\( x^{2} = y^{2}(x + y^{4} + 2 y^{2}) \\). Hmm, let's... |
[
{
"from": "user",
"value": "\nA3. Let $A$ and $B$ be two non-empty subsets of $X=\\{1,2, \\ldots, 11\\}$ with $A \\cup B=X$. Let $P_{A}$ be the product of all elements of $A$ and let $P_{B}$ be the product of all elements of $B$. Find the minimum and maximum possible value of $P_{A}+P_{B}$ and find all poss... |
[
{
"from": "user",
"value": "\nC4. We have a group of $n$ kids. For each pair of kids, at least one has sent a message to the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly $25 \\%$ have sent a message to $A$. How many possible two-digit values of $n$ are there?\n"
},
... |
[
{
"from": "user",
"value": "\nN2. Find all triples $(p, q, r)$ of prime numbers such that all of the following numbers are integers\n\n$$\n\\frac{p^{2}+2 q}{q+r}, \\quad \\frac{q^{2}+9 r}{r+p}, \\quad \\frac{r^{2}+3 p}{p+q}\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nAlright, so I have t... |
[
{
"from": "user",
"value": "\nN4. Find all integers $x, y$ such that\n\n$$\nx^{3}(y+1)+y^{3}(x+1)=19\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nAlright, so I need to find all integer solutions (x, y) for the equation x³(y + 1) + y³(x + 1) = 19. Hmm, okay. Let me start by trying to under... |
[
{
"from": "user",
"value": "\nN5. Find all positive integers $x, y, z$ such that\n\n$$\n45^{x}-6^{y}=2019^{z}\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, let me try to figure out this problem. So, we need to find all positive integers x, y, z such that 45^x - 6^y = 2019^z. Hmm, tha... |
[
{
"from": "user",
"value": "\nN7. Find all perfect squares $n$ such that if the positive integer $a \\geqslant 15$ is some divisor of $n$ then $a+15$ is a prime power.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all perfect squares n such that for every positive integ... |
[
{
"from": "user",
"value": "\nNT1 Solve in positive integers the equation $1005^{x}+2011^{y}=1006^{z}$.\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to solve the equation 1005^x + 2011^y = 1006^z where x, y, z are positive integers. Hmm, let's see. First, I should check if t... |
[
{
"from": "user",
"value": "\nNT2 Find all prime numbers $p$ such that there exist positive integers $x, y$ that satisfy the relation $x\\left(y^{2}-p\\right)+y\\left(x^{2}-p\\right)=5 p$.\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all prime numbers p such that the... |
[
{
"from": "user",
"value": "\nNT4 Find all prime positive integers $p, q$ such that $2 p^{3}-q^{2}=2(p+q)^{2}$.\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to solve this problem: Find all prime positive integers p and q such that 2p³ - q² = 2(p + q)². Hmm, primes... That sh... |
[
{
"from": "user",
"value": "\nG2 Let $A D, B F$ and $C E$ be the altitudes of $\\triangle A B C$. A line passing through $D$ and parallel to $A B$ intersects the line $E F$ at the point $G$. If $H$ is the orthocenter of $\\triangle A B C$, find the angle $\\widehat{C G H}$.\n\n"
},
{
"from": "assist... |
[
{
"from": "user",
"value": "\nG3 Let $A B C$ be a triangle in which ( $B L$ is the angle bisector of $\\widehat{A B C}(L \\in A C), A H$ is an altitude of $\\triangle A B C(H \\in B C)$ and $M$ is the midpoint of the side $[A B]$. It is known that the midpoints of the segments $[B L]$ and $[M H]$ coincides.... |
[
{
"from": "user",
"value": "\nNT2. Find all four digit numbers A such that\n\n$$\n\\frac{1}{3} A+2000=\\frac{2}{3} \\bar{A}\n$$\n\nwhere $\\bar{A}$ is the number with the same digits as $A$, but written in opposite order. (For example, $\\overline{1234}=4321$.)\n"
},
{
"from": "assistant",
"valu... |
[
{
"from": "user",
"value": "\nC2. Given $m \\times n$ table, each cell signed with \"-\". The following operations are\n\n(i) to change all the signs in entire row to the opposite, i. e. every \"-\" to \"+\", and every \"+\" to \"-\";\n\n(ii) to change all the signs in entire column to the opposite, i. e. e... |
[
{
"from": "user",
"value": "\nProblem A2. Determine all four digit numbers $\\overline{a b c d}$ such that\n\n$$\na(a+b+c+d)\\left(a^{2}+b^{2}+c^{2}+d^{2}\\right)\\left(a^{6}+2 b^{6}+3 c^{6}+4 d^{6}\\right)=\\overline{a b c d}\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need t... |
[
{
"from": "user",
"value": "\nProblem A3. Find all pairs $(x, y)$ of real numbers such that $|x|+|y|=1340$ and $x^{3}+y^{3}+2010 x y=670^{3}$.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to solve this system of equations:\n\n1. |x| + |y| = 1340\n2. x³ + y³ + 2010xy = 670³\n\n... |
[
{
"from": "user",
"value": "\nProblem G2. Consider a triangle $A B C$ and let $M$ be the midpoint of the side $B C$. Suppose $\\angle M A C=\\angle A B C$ and $\\angle B A M=105^{\\circ}$. Find the measure of $\\angle A B C$.\n"
},
{
"from": "assistant",
"value": "<think>\nAlright, let's tackle ... |
[
{
"from": "user",
"value": "\nProblem G4. Let $A L$ and $B K$ be angle bisectors in the non-isosceles triangle $A B C(L \\in B C, K \\in A C)$. The perpendicular bisector of $B K$ intersects the line $A L$ at point $M$. Point $N$ lies on the line $B K$ such that $L N \\| M K$. Prove that $L N=N A$.\n\n![](h... |
[
{
"from": "user",
"value": "\nProblem N1. Find all positive integers $n$ such that $n 2^{n+1}+1$ is a perfect square.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all positive integers n where the expression n times 2^{n+1} plus 1 is a perfect square. Let me start by u... |
[
{
"from": "user",
"value": "\nProblem N2. Find all positive integers $n$ such that $36^{n}-6$ is a product of two or more consecutive positive integers.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all positive integers n such that 36^n - 6 is a product of two or more ... |
[
{
"from": "user",
"value": "\nA1. Let $a, b, c$ be positive real numbers such that $a b c=8$. Prove that\n\n$$\n\\frac{a b+4}{a+2}+\\frac{b c+4}{b+2}+\\frac{c a+4}{c+2} \\geq 6\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to prove that for positive real numbers a, b, c wit... |
[
{
"from": "user",
"value": "\nA3. Determine the number of pairs of integers $(m, n)$ such that\n\n$$\n\\sqrt{n+\\sqrt{2016}}+\\sqrt{m-\\sqrt{2016}} \\in \\mathbb{Q}\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. The problem is to determine the number of pairs of integers (m... |
[
{
"from": "user",
"value": "\nC1. Let $S_{n}$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13}=\\frac{1}{1}+\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}+\\frac{1}{6}+\\frac{1}{7}+\\frac{1}{8}+\\frac{1}{9}+\\frac{1}{1}+\\frac{... |
[
{
"from": "user",
"value": "\nC2. The natural numbers from 1 to 50 are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. The probl... |
[
{
"from": "user",
"value": "\nN1. Determine the largest positive integer $n$ that divides $p^{6}-1$ for all primes $p>7$.\n"
},
{
"from": "assistant",
"value": "<think>\nAlright, so the problem is asking for the largest positive integer n that divides p⁶ - 1 for all primes p > 7. Hmm, okay. Let ... |
[
{
"from": "user",
"value": "\nN2. Find the maximum number of natural numbers $x_{1}, x_{2}, \\ldots, x_{m}$ satisfying the conditions:\n\na) No $x_{i}-x_{j}, 1 \\leq i<j \\leq m$ is divisible by 11 ; and\n\nb) The sum $x_{2} x_{3} \\ldots x_{m}+x_{1} x_{3} \\ldots x_{m}+\\cdots+x_{1} x_{2} \\ldots x_{m-1}$ ... |
[
{
"from": "user",
"value": "\nNT 3. Find the largest integer $k(k \\geq 2)$, for which there exists an integer $n(n \\geq k)$ such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions:\n\n1. each chosen number is not divisible... |
[
{
"from": "user",
"value": "\nNT 4. Find all prime numbers $p$ such that\n\n$$\n(x+y)^{19}-x^{19}-y^{19}\n$$\n\nis a multiple of $p$ for any positive integers $x, y$.\n"
},
{
"from": "assistant",
"value": "<think>\nAlright, let me try to figure out this problem. So, we need to find all prime num... |
[
{
"from": "user",
"value": "\nNT 8. Find all pairs $(p, q)$ of prime numbers such that\n\n$$\n1+\\frac{p^{q}-q^{p}}{p+q}\n$$\n\nis a prime number.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all pairs of prime numbers (p, q) such that the expression 1 + (p^q - q^p)/(p... |
[
{
"from": "user",
"value": "\nA 2. Find the maximum positive integer $k$ such that for any positive integers $m, n$ such that $m^{3}+n^{3}>$ $(m+n)^{2}$, we have\n\n$$\nm^{3}+n^{3} \\geq(m+n)^{2}+k\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the maximum positive i... |
[
{
"from": "user",
"value": "\nA 4. Let $k>1, n>2018$ be positive integers, and let $n$ be odd. The nonzero rational numbers $x_{1}$, $x_{2}, \\ldots, x_{n}$ are not all equal and satisfy\n\n$$\nx_{1}+\\frac{k}{x_{2}}=x_{2}+\\frac{k}{x_{3}}=x_{3}+\\frac{k}{x_{4}}=\\cdots=x_{n-1}+\\frac{k}{x_{n}}=x_{n}+\\frac... |
[
{
"from": "user",
"value": "\nA 5. Let $a, b, c, d$ and $x, y, z, t$ be real numbers such that\n\n$$\n0 \\leq a, b, c, d \\leq 1, \\quad x, y, z, t \\geq 1 \\text { and } a+b+c+d+x+y+z+t=8\n$$\n\nProve that\n\n$$\na^{2}+b^{2}+c^{2}+d^{2}+x^{2}+y^{2}+z^{2}+t^{2} \\leq 28\n$$\n\nWhen does the equality hold?\n... |
[
{
"from": "user",
"value": "\nA 6. Let $a, b, c$ be positive numbers such that $a b+b c+c a=3$. Prove that\n\n$$\n\\frac{a}{\\sqrt{a^{3}+5}}+\\frac{b}{\\sqrt{b^{3}+5}}+\\frac{c}{\\sqrt{c^{3}+5}} \\leq \\frac{\\sqrt{6}}{2}\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I nee... |
[
{
"from": "user",
"value": "\nC 1. A set $S$ is called neighbouring if it has the following two properties:\na) $S$ has exactly four elements\n\nb) for every element $x$ of $S$, at least one of the numbers $x-1$ or $x+1$ belongs to $S$.\n\nFind the number of all neighbouring subsets of the set $\\{1,2, \\ld... |
[
{
"from": "user",
"value": "\nNT 1. Find all the integers pairs $(x, y)$ which satisfy the equation\n\n$$\nx^{5}-y^{5}=16 x y\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all integer pairs (x, y) that satisfy the equation x⁵ - y⁵ = 16xy. Hmm, this looks like a Diop... |
[
{
"from": "user",
"value": "\nA2 Find the maximum value of $z+x$, if $(x, y, z, t)$ satisfies the conditions:\n\n$$\n\\left\\{\\begin{array}{l}\nx^{2}+y^{2}=4 \\\\\nz^{2}+t^{2}=9 \\\\\nx t+y z \\geq 6\n\\end{array}\\right.\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to fi... |
[
{
"from": "user",
"value": "\nG2 In a right trapezoid $A B C D(A B \\| C D)$ the angle at vertex $B$ measures $75^{\\circ}$. Point $H$ is the foot of the perpendicular from point $A$ to the line $B C$. If $B H=D C$ and $A D+A H=8$, find the area of $A B C D$.\n"
},
{
"from": "assistant",
"value"... |
[
{
"from": "user",
"value": "\nNT1 Determine all positive integer numbers $k$ for which the numbers $k+9$ are perfect squares and the only prime factors of $k$ are 2 and 3 .\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's tackle this problem. We need to find all positive integers \\( ... |
[
{
"from": "user",
"value": "\nNT2 A group of $n>1$ pirates of different age owned total of 2009 coins. Initially each pirate (except for the youngest one) had one coin more than the next younger.\n\na) Find all possible values of $n$.\n\nb) Every day a pirate was chosen. The chosen pirate gave a coin to eac... |
[
{
"from": "user",
"value": "\nNT3 Find all pairs $(x, y)$ of integers which satisfy the equation\n\n$$\n(x+y)^{2}\\left(x^{2}+y^{2}\\right)=2009^{2}\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all integer pairs (x, y) that satisfy the equation (x + y)^2 (x^2 + y^2... |
[
{
"from": "user",
"value": "\nC4 In a group of $n$ people, each one had a different ball. They performed a sequence of swaps; in each swap, two people swapped the ball they had at that moment. Each pair of people performed at least one swap. In the end each person had the ball he/she had at the start. Find ... |
[
{
"from": "user",
"value": "\nA4. Find all the integer solutions of the equation\n\n$$\n9 x^{2} y^{2}+9 x y^{2}+6 x^{2} y+18 x y+x^{2}+2 y^{2}+5 x+7 y+6=0\n$$\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all integer solutions (x, y) to this equation:\n\n9x²y² + 9xy² ... |
[
{
"from": "user",
"value": "\nG3. Let $A B C D E F$ be a regular hexagon. The points $\\mathrm{M}$ and $\\mathrm{N}$ are internal points of the sides $\\mathrm{DE}$ and $\\mathrm{DC}$ respectively, such that $\\angle A M N=90^{\\circ}$ and $A N=\\sqrt{2} \\cdot C M$. Find the measure of the angle $\\angle B... |
[
{
"from": "user",
"value": "\nG6. A point $O$ and the circles $k_{1}$ with center $O$ and radius $3, k_{2}$ with center $O$ and radius 5, are given. Let $A$ be a point on $k_{1}$ and $B$ be a point on $k_{2}$. If $A B C$ is equilateral triangle, find the maximum value of the distance $O C$.\n\n"
},
{
... |
[
{
"from": "user",
"value": "\nNT2. Find all natural numbers $n$ such that $5^{n}+12^{n}$ is perfect square.\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all natural numbers n where 5^n + 12^n is a perfect square. Let me start by understanding what the problem is aski... |
[
{
"from": "user",
"value": "\nC4. Let $p_{1}, p_{2}, \\ldots, p_{2005}$ be different prime numbers. Let $\\mathrm{S}$ be a set of natural numbers which elements have the property that their simple divisors are some of the numbers $p_{1}, p_{2}, \\ldots, p_{2005}$ and product of any two elements from $\\math... |
[
{
"from": "user",
"value": "\nA4 Let $x, y$ be positive real numbers such that $x^{3}+y^{3} \\leq x^{2}+y^{2}$. Find the greatest possible value of the product $x y$.\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the greatest possible value of the product xy, given th... |
[
{
"from": "user",
"value": "\nA6 Let $x_{i}>1$, for all $i \\in\\{1,2,3, \\ldots, 2011\\}$. Prove the inequality $\\sum_{i=1}^{2011} \\frac{x_{i}^{2}}{x_{i+1}-1} \\geq 8044$ where $x_{2012}=x_{1}$. When does equality hold?\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this in... |
[
{
"from": "user",
"value": "\nA7 Let $a, b, c$ be positive real numbers with $a b c=1$. Prove the inequality:\n\n$$\n\\frac{2 a^{2}+\\frac{1}{a}}{b+\\frac{1}{a}+1}+\\frac{2 b^{2}+\\frac{1}{b}}{c+\\frac{1}{b}+1}+\\frac{2 c^{2}+\\frac{1}{c}}{a+\\frac{1}{c}+1} \\geq 3\n$$\n\n"
},
{
"from": "assistant",... |
[
{
"from": "user",
"value": "\nA2 Find all real numbers $a, b, c, d$ such that $a+b+c+d=20$ and $a b+a c+a d+b c+b d+c d=$ 150 .\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all real numbers a, b, c, d such that their sum is 20 and the sum of their pairwise products i... |
[
{
"from": "user",
"value": "\nA4 Find all triples $(x, y, z)$ of real numbers that satisfy the system\n\n$$\n\\left\\{\\begin{array}{l}\nx+y+z=2008 \\\\\nx^{2}+y^{2}+z^{2}=6024^{2} \\\\\n\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}=\\frac{1}{2008}\n\\end{array}\\right.\n$$\n\n"
},
{
"from": "assistant",
... |
[
{
"from": "user",
"value": "\nA5 Find all triples $(x, y, z)$ of real positive numbers, which satisfy the system\n\n$$\n\\left\\{\\begin{array}{l}\n\\frac{1}{x}+\\frac{4}{y}+\\frac{9}{z}=3 \\\\\nx+y+z \\leq 12\n\\end{array}\\right.\n$$\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I... |
[
{
"from": "user",
"value": "\nA9 Consider an integer $n \\geq 4$ and a sequence of real numbers $x_{1}, x_{2}, x_{3}, \\ldots, x_{n}$. An operation consists in eliminating all numbers not having the rank of the form $4 k+3$, thus leaving only the numbers $x_{3}, x_{7}, x_{11}, \\ldots$ (for example, the seq... |
[
{
"from": "user",
"value": "\nG3 The vertices $A$ and $B$ of an equilateral $\\triangle A B C$ lie on a circle $k$ of radius 1 , and the vertex $C$ is inside $k$. The point $D \\neq B$ lies on $k, A D=A B$ and the line $D C$ intersects $k$ for the second time in point $E$. Find the length of the segment $C ... |
[
{
"from": "user",
"value": "\nG7 Let $A B C$ be an isosceles triangle with $A C=B C$. The point $D$ lies on the side $A B$ such that the semicircle with diameter $[B D]$ and center $O$ is tangent to the side $A C$ in the point $P$ and intersects the side $B C$ at the point $Q$. The radius $O P$ intersects t... |
[
{
"from": "user",
"value": "\nNT1 Find all the positive integers $x$ and $y$ that satisfy the equation\n\n$$\nx(x-y)=8 y-7\n$$\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all positive integers x and y that satisfy the equation x(x - y) = 8y - 7. Let me try to figure... |
[
{
"from": "user",
"value": "\nNT3 Let $s(a)$ denote the sum of digits of a given positive integer $a$. The sequence $a_{1}, a_{2}, \\ldots a_{n}, \\ldots$ of positive integers is such that $a_{n+1}=a_{n}+s\\left(a_{n}\\right)$ for each positive integer $n$. Find the greatest possible $n$ for which it is pos... |
[
{
"from": "user",
"value": "\nNT4 Find all integers $n$ such that $n^{4}+8 n+11$ is a product of two or more consecutive integers.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all integers n where n^4 + 8n + 11 is a product of two or more consecutive integers. Hmm, let... |
[
{
"from": "user",
"value": "\nNT6 Let $f: \\mathbb{N} \\rightarrow \\mathbb{R}$ be a function, satisfying the following condition:\n\nfor every integer $n>1$, there exists a prime divisor $p$ of $n$ such that $f(n)=f\\left(\\frac{n}{p}\\right)-f(p)$. If\n\n$$\nf\\left(2^{2007}\\right)+f\\left(3^{2008}\\righ... |
[
{
"from": "user",
"value": "\nNT7 Determine the minimal prime number $p>3$ for which no natural number $n$ satisfies\n\n$$\n2^{n}+3^{n} \\equiv 0(\\bmod p)\n$$\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the minimal prime number p greater than 3 such that there's no... |
[
{
"from": "user",
"value": "\nNT8 Let $a, b, c, d, e, f$ are nonzero digits such that the natural numbers $\\overline{a b c}, \\overline{d e f}$ and $\\overline{a b c d e f}$ are squares.\n\na) Prove that $\\overline{a b c d e f}$ can be represented in two different ways as a sum of three squares of natural... |
[
{
"from": "user",
"value": "\nALG 5. Let $A B C$ be a scalene triangle with $B C=a, A C=b$ and $A B=c$, where $a_{r} b, c$ are positive integers. Prove that\n\n$$\n\\left|a b^{2}+b c^{2}+c a^{2}-a^{2} b-b^{2} c-c^{2} a\\right| \\geq 2\n$$\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so ... |
[
{
"from": "user",
"value": "\nCOM 2 Natural numbers 1,2,3, .., 2003 are written in an arbitrary sequence $a_{1}, a_{2}, a_{3}, \\ldots a_{2003}$. Let $b_{1}=1 a_{1}, b_{2}=2 a_{2}, b_{3}=3 a_{3}, \\ldots, b_{2003}=2003 a_{2003}$, and $B$ be the maximum of the numbers $b_{1}, b_{2}, b_{3}, \\ldots, b_{2003}$... |
[
{
"from": "user",
"value": "\n87.3. Let $f$ be a strictly increasing function defined in the set of natural numbers satisfying the conditions $f(2)=a>2$ and $f(m n)=f(m) f(n)$ for all natural numbers $m$ and $n$. Determine the smallest possible value of $a$.\n"
},
{
"from": "assistant",
"value":... |
[
{
"from": "user",
"value": "\n88.1. The positive integer $n$ has the following property: if the three last digits of $n$ are removed, the number $\\sqrt[3]{n}$ remains. Find $n$.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find a positive integer n such that if I remove th... |
[
{
"from": "user",
"value": "\n88.3. Two concentric spheres have radii $r$ and $R, r<R$. We try to select points $A, B$ and $C$ on the surface of the larger sphere such that all sides of the triangle $A B C$ would be tangent to the surface of the smaller sphere. Show that the points can be selected if and on... |
[
{
"from": "user",
"value": "\n88.4. Let $m_{n}$ be the smallest value of the function\n\n$$\nf_{n}(x)=\\sum_{k=0}^{2 n} x^{k}\n$$\n\nShow that $m_{n} \\rightarrow \\frac{1}{2}$, as $n \\rightarrow \\infty$.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to show that the minimum ... |
[
{
"from": "user",
"value": "\n89.2. Three sides of a tetrahedron are right-angled triangles having the right angle at their common vertex. The areas of these sides are $A, B$, and $C$. Find the total surface area of the tetrahedron.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so there'... |
[
{
"from": "user",
"value": "\n91.1. Determine the last two digits of the number\n\n$$\n2^{5}+2^{5^{2}}+2^{5^{3}}+\\cdots+2^{5^{1991}}\n$$\n\nwritten in decimal notation.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the last two digits of the sum 2^5 + 2^(5^2) + 2^(5^3)... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.