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"value": "\n92.1. Determine all real numbers $x>1, y>1$, and $z>1$, satisfying the equation\n\n$$\n\\begin{aligned}\nx+y+z+\\frac{3}{x-1} & +\\frac{3}{y-1}+\\frac{3}{z-1} \\\\\n& =2(\\sqrt{x+2}+\\sqrt{y+2}+\\sqrt{z+2})\n\\end{aligned}\n$$\n"
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"value": "... |
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"value": "\n92.4. Peter has many squares of equal side. Some of the squares are black, some are white. Peter wants to assemble a big square, with side equal to $n$ sides of the small squares, so that the big square has no rectangle formed by the small squares such that all the squares in t... |
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"value": "\n93.2. A hexagon is inscribed in a circle of radius $r$. Two of the sides of the hexagon have length 1, two have length 2 and two have length 3. Show that $r$ satisfies the equation\n\n$$\n2 r^{3}-7 r-3=0\n$$\n\n![](https://cdn.mathpix.com/cropped/2024_06_06_f01d327b68eb688a4496... |
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"value": "\n93.3. Find all solutions of the system of equations\n\n$$\n\\left\\{\\begin{aligned}\ns(x)+s(y) & =x \\\\\nx+y+s(z) & =z \\\\\ns(x)+s(y)+s(z) & =y-4\n\\end{aligned}\\right.\n$$\n\nwhere $x, y$, and $z$ are positive integers, and $s(x), s(y)$, and $s(z)$ are the numbers of digit... |
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"value": "\n94.4. Determine all positive integers $n<200$, such that $n^{2}+(n+1)^{2}$ is the square of an integer.\n"
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"value": "<think>\nOkay, let's try to figure out how to solve this problem. We need to find all positive integers n less than 200 suc... |
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"value": "\n95.2. Messages are coded using sequences consisting of zeroes and ones only. Only sequences with at most two consecutive ones or zeroes are allowed. (For instance the sequence 011001 is allowed, but 011101 is not.) Determine the number of sequences consisting of exactly 12 numb... |
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"value": "\n96.4. The real-valued function $f$ is defined for positive integers, and the positive integer a satisfies\n\n$$\n\\begin{gathered}\nf(a)=f(1995), \\quad f(a+1)=f(1996), \\quad f(a+2)=f(1997) \\\\\nf(n+a)=\\frac{f(n)-1}{f(n)+1} \\quad \\text { for all positive integers } n\n\\en... |
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"value": "\n97.1. Let A be a set of seven positive numbers. Determine the maximal number of triples $(x, y, z)$ of elements of A satisfying $x<y$ and $x+y=z$.\n"
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"value": "<think>\nOkay, so I need to figure out the maximal number of triples (x, y, z) i... |
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"value": "\n99.2. Consider 7-gons inscribed in a circle such that all sides of the 7-gon are of different length. Determine the maximal number of $120^{\\circ}$ angles in this kind of a 7-gon.\n"
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"value": "<think>\nAlright, let's tackle this problem ab... |
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"value": "\n00.1. In how many ways can the number 2000 be written as a sum of three positive, not necessarily different integers? (Sums like $1+2+3$ and $3+1+2$ etc. are the same.)\n"
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"value": "<think>\nOkay, so I have this problem here: \"In how many ... |
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"value": "\n01.3. Determine the number of real roots of the equation\n\n$$\nx^{8}-x^{7}+2 x^{6}-2 x^{5}+3 x^{4}-3 x^{3}+4 x^{2}-4 x+\\frac{5}{2}=0\n$$\n"
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"value": "<think>\nOkay, so I need to figure out how many real roots this equation has: \n\nx⁸ - x... |
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"value": "\n02.1. The trapezium $A B C D$, where $A B$ and $C D$ are parallel and $A D<C D$, is inscribed in the circle c. Let DP be a chord of the circle, parallel to AC. Assume that the tangent to $c$ at $D$ meets the line $A B$ at $E$ and that $P B$ and $D C$ meet at $Q$. Show that $E Q... |
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"value": "\n02.2. In two bowls there are in total $N$ balls, numbered from 1 to $N$. One ball is moved from one of the bowls to the other. The average of the numbers in the bowls is increased in both of the bowls by the same amount, $x$. Determine the largest possible value of $x$.\n"
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"value": "\n03.2. Find all triples of integers $(x, y, z)$ satisfying\n\n$$\nx^{3}+y^{3}+z^{3}-3 x y z=2003\n$$\n"
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"value": "<think>\nOkay, so I have this problem here: find all triples of integers (x, y, z) that satisfy the equation x³ + y³ + z³ - 3xy... |
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"value": "\n03.4. Let $\\mathbb{R}^{*}=\\mathbb{R} \\backslash\\{0\\}$ be the set of non-zero real numbers. Find all functions $f: \\mathbb{R}^{*} \\rightarrow \\mathbb{R}^{*}$ satisfying\n\n$$\nf(x)+f(y)=f(x y f(x+y))\n$$\n\nfor $x, y \\in \\mathbb{R}^{*}$ and $x+y \\neq 0$.\n"
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"value": "\n04.1. 27 balls, labelled by numbers from 1 to 27, are in a red, blue or yellow bowl. Find the possible numbers of balls in the red bowl, if the averages of the labels in the red, blue, and yellow bowl are 15, 3 ja 18, respectively.\n"
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"value": "\n06.3. A sequence of positive integers $\\left\\{a_{n}\\right\\}$ is given by\n\n$$\na_{0}=m \\quad \\text { and } \\quad a_{n+1}=a_{n}^{5}+487\n$$\n\nfor all $n \\geq 0$. Determine all values of $m$ for which the sequence contains as many square numbers as possible.\n"
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"value": "\n07.1. Find one solution in positive integers to the equation\n\n$$\nx^{2}-2 x-2007 y^{2}=0\n$$\n"
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"value": "<think>\nOkay, so I need to find a solution in positive integers for the equation \\( x^2 - 2x - 2007y^2 = 0 \\). Hmm, let's start b... |
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"value": "\n07.4. A line through a point $A$ intersects a circle in two points, $B$ and $C$, in such a way that $B$ lies between $A$ and $C$. From the point $A$ draw the two tangents to the circle, meeting the circle at points $S$ and $T$. Let $P$ be the intersection of the lines $S T$ and... |
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"value": "\n08.2. Assume that $n \\geq 3$ people with different names sit around a round table. We call any unordered pair of them, say $M$ and $N$, dominating, if\n\n(i) $M$ and $N$ do not sit on adjacent seats, and\n\n(ii) on one (or both) of the arcs connecting $M$ and $N$ along the tab... |
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"value": "\n08.3. Let $A B C$ be a triangle and let $D$ and $E$ be points on $B C$ and $C A$, respectively, such that $A D$ and $B E$ are angle bisectors of $A B C$. Let $F$ and $G$ be points on the circumcircle of $A B C$ such that $A F$ and $D E$ are parallel and $F G$ and $B C$ are para... |
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"value": "\n09.2. On a faded piece of paper it is possible, with some effort, to discern the following:\n\n$$\n\\left(x^{2}+x+a\\right)\\left(x^{15}-\\ldots\\right)=x^{17}+x^{13}+x^{5}-90 x^{4}+x-90\n$$\n\nSome parts have got lost, partly the constant term of the first factor of the left s... |
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"value": "\n10.3. Laura has 2010 lamps connected with 2010 buttons in front of her. For each button, she wants to know the corresponding lamp. In order to do this, she observes which lamps are lit when Richard presses a selection of buttons. (Not pressing anything is also a possible select... |
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"value": "\n10.4. A positive integer is called simple if its ordinary decimal representation consists entirely of zeroes and ones. Find the least positive integer $k$ such that each positive integer $n$ can be written as $n=a_{1} \\pm a_{2} \\pm a_{3} \\pm \\cdots \\pm a_{k}$, where $a_{1}... |
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"value": "## Problem 2\n\nLet $A B C D$ be a cyclic quadrilateral satisfying $A B=A D$ and $A B+B C=C D$.\n\nDetermine $\\angle C D A$.\n"
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"value": "<think>\nOkay, so I need to solve this problem about a cyclic quadrilateral ABCD where AB equals AD, an... |
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"value": "## Problem 1\n\nFind all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ such that\n\n$$\nf(f(x) f(1-x))=f(x) \\quad \\text { and } \\quad f(f(x))=1-f(x)\n$$\n\nfor all real $x$.\n"
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"value": "<think>\nOkay, so I need to find all functions... |
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"value": "## Problem 2\n\nIn Wonderland, the towns are connected by roads, and whenever there is a direct road between two towns there is also a route between these two towns that does not use that road. (There is at most one direct road between any two towns.) The Queen\nof Hearts ordered... |
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"value": "\nPROBLEM 1. The real numbers $a, b, c$ are such that $a^{2}+b^{2}=2 c^{2}$, and also such that $a \\neq b, c \\neq-a, c \\neq-b$. Show that\n\n$$\n\\frac{(a+b+2 c)\\left(2 a^{2}-b^{2}-c^{2}\\right)}{(a-b)(a+c)(b+c)}\n$$\n\nis an integer.\n"
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"value": "\nProblem 3. Find the smallest positive integer $n$, such that there exist $n$ integers $x_{1}, x_{2}, \\ldots, x_{n}$ (not necessarily different), with $1 \\leq x_{k} \\leq n, 1 \\leq k \\leq n$, and such that\n\n$$\nx_{1}+x_{2}+\\cdots+x_{n}=\\frac{n(n+1)}{2}, \\quad \\text { a... |
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"value": "\nProblem 4. The number 1 is written on the blackboard. After that a sequence of numbers is created as follows: at each step each number $a$ on the blackboard is replaced by the numbers $a-1$ and $a+1$; if the number 0 occurs, it is erased immediately; if a number occurs more tha... |
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"value": "## Problem 2.\n\nFind the primes $p, q, r$, given that one of the numbers $p q r$ and $p+q+r$ is 101 times the other.\n"
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"value": "<think>\nOkay, so I need to find three prime numbers p, q, and r such that either the product pqr is 101 times ... |
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"value": "## Problem 4.\n\nAn encyclopedia consists of 2000 numbered volumes. The volumes are stacked in order with number 1 on top and 2000 in the bottom. One may perform two operations with the stack:\n\n(i) For $n$ even, one may take the top $n$ volumes and put them in the bottom of the... |
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"value": "\nProblem 2. In a football tournament there are $n$ teams, with $n \\geq 4$, and each pair of teams meets exactly once. Suppose that, at the end of the tournament, the final scores form an arithmetic sequence where each team scores 1 more point than the following team on the scor... |
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"value": "XXI OM - II - Task 5\n\nGiven is the polynomial $ P(x) = \\frac{1}{2} - \\frac{1}{3}x + \\frac{1}{6}x^2 $. Let $ Q(x) = \\sum_{k=0}^{m} b_k x^k $ be a polynomial defined by the formula \n\n\n\n\n\n\nCalculate $ \\sum_{k=0}^m |b_k| $."
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"value": "XXIV OM - I - Problem 8\n\nFind a polynomial with integer coefficients of the lowest possible degree, for which $ \\sqrt{2} + \\sqrt{3} $ is a root."
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"value": "<think>\nAlright, so I need to find a polynomial with integer coefficients of the ... |
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"value": "XLVII OM - I - Problem 2\n\nA palindromic number is defined as a natural number whose decimal representation read from left to right is the same as when read from right to left. Let $ (x_n) $ be the increasing sequence of all palindromic numbers. Determine all prime numbers that ... |
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"value": "LVII OM - II - Problem 6\n\nGiven a prime number $ p $ and an integer $ n $, where $ p \\geq n \\geq 3 $. The set $ A $ consists of $ n $-term sequences with terms from the set $ \\{0,1,2,\\dots ,p-1\\} $ and has the following property:\nFor any two sequences $ (x_1,x_2,\\dots ,x... |
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"value": "Given an integer $ c \\geq 1 $. To each subset $ A $ of the set $ \\{1,2, \\ldots ,n\\} $, we assign a number $ w(A) $ from the set $ \\{1,2, \\ldots ,c\\} $ such that the following condition is satisfied:\n\nLet $ a(n) $ be the number of such assignments. Calculate $ \\lim_{n\\t... |
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"value": "XXIV OM - I - Problem 10\n\nFind the smallest natural number $ n > 1 $ with the following property: there exists a set $ Z $ consisting of $ n $ points in the plane such that every line $ AB $ ($ A, B \\in Z $) is parallel to some other line $ CD $ ($ C, D \\in Z $)."
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"value": "XXV - I - Task 1\n\nDuring World War I, a battle took place near a certain castle. One of the shells destroyed a statue of a knight with a spear standing at the entrance to the castle. This happened on the last day of the month. The product of the day of the month, the month numb... |
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"value": "XXIX OM - I - Problem 7\n\nFor a fixed natural number $ n>2 $, we define: $ x_1 = n $, $ y_1=1 $, $ x_{i+1} = \\left[\\frac{1}{2}(x_i+y_i)\\right] $, $ y_{i+1} = \\left[ \\frac{n}{x_{i+1}}\\right] $. Prove that the smallest of the numbers $ x_1, x_2, \\ldots, x_n $ is equal to $ ... |
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"value": "V OM - I - Problem 7\n\nIn the plane, a line $ p $ and points $ A $ and $ B $ are given. Find a point $ M $ on the line $ p $ such that the sum of the squares $ AM^2 + BM^2 $ is minimized."
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"value": "<think>\nOkay, so I need to find a point M... |
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"value": "LVII OM - I - Problem 4\n\nParticipants in a mathematics competition solved six problems, each graded with one of the scores 6, 5, 2, 0. It turned out that\nfor every pair of participants $ A, B $, there are two problems such that in each of them $ A $ received a different score ... |
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"value": "XXIII OM - I - Problem 7\n\nA broken line contained in a square with a side length of 50 has the property that the distance from any point of this square to it is less than 1. Prove that the length of this broken line is greater than 1248."
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"value": "LVII OM - III - Problem 2\n\nDetermine all positive integers $ k $ for which the number $ 3^k+5^k $ is a power of an integer with an exponent greater than 1."
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"value": "<think>\nOkay, so I need to find all positive integers k such that 3^k + ... |
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"value": "XXIV OM - I - Problem 12\n\nIn a class of n students, a Secret Santa event was organized. Each student draws the name of the person for whom they are to buy a gift, so student $ A_1 $ buys a gift for student $ A_2 $, $ A_2 $ buys a gift for $ A_3 $, ..., $ A_k $ buys a gift for $... |
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"value": "VIII OM - I - Task 6\n\nFind a four-digit number, whose first two digits are the same, the last two digits are the same, and which is a square of an integer."
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"value": "<think>\nOkay, let's see. I need to find a four-digit number where the fi... |
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"value": "XXXIV OM - I - Problem 1\n\n$ A $ tosses a coin $ n $ times, $ B $ tosses it $ n+1 $ times. What is the probability that $ B $ will get more heads than $ A $?"
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"value": "<think>\nOkay, let me try to figure out this probability problem. So, th... |
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"value": "LI OM - II - Problem 4\n\nPoint $ I $ is the center of the circle inscribed in triangle $ ABC $, where $ AB \\neq AC $. Lines $ BI $ and $ CI $ intersect sides $ AC $ and $ AB $ at points $ D $ and $ E $, respectively. Determine all possible measures of angle $ BAC $ for which th... |
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"value": "XXXIV OM - II - Problem 6\n\nFor a given number $ n $, let $ p_n $ denote the probability that when a pair of integers $ k, m $ satisfying the conditions $ 0 \\leq k \\leq m \\leq 2^n $ is chosen at random (each pair is equally likely), the number $ \\binom{m}{k} $ is even. Calcu... |
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"value": "LVII OM - III - Problem 6\n\nDetermine all pairs of integers $ a $, $ b $, for which there exists a polynomial $ P(x) $ with integer coefficients, such that the product $ (x^2 + ax + b)\\cdot P(x) $ is a polynomial of the form\n\n\n\nwhere each of the numbers $ c_0,c_1,\\dots ,c_... |
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"value": "XXIX OM - II - Problem 4\n\nFrom the vertices of a regular $2n$-gon, 3 different points are chosen randomly. Let $p_n$ be the probability that the triangle with vertices at the chosen points is acute. Calculate $\\lim_{n\\to \\infty} p_n$.\nNote. We assume that all choices of thr... |
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"value": "IX OM - II - Task 2\n\nSix equal disks are placed on a plane in such a way that their centers lie at the vertices of a regular hexagon with a side equal to the diameter of the disks. How many rotations will a seventh disk of the same size make while rolling externally on the same... |
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"value": "XXXVIII OM - III - Zadanie 5\n\nWyznaczyć najmniejszą liczbę naturalną $ n $, dla której liczba $ n^2-n+11 $ jest iloczynem czterech liczb pierwszych (niekoniecznie różnych).\n"
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"value": "<think>\nOkay, so I have this problem here where I nee... |
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"value": "LIX OM - I - Task 7\n\nIn an $ n $-person association, there are $ 2n-1 $ committees (any non-empty set of association members\nforms a committee). A chairperson must be selected in each committee. The following condition must be met: If\ncommittee $ C $ is the union $ C = A\\cup... |
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"value": "XXXII - I - Problem 10\n\nDetermine all functions $ f $ mapping the set of all rational numbers $ \\mathbb{Q} $ to itself that satisfy the following conditions:\na) $ f(1)=2 $,\nb) $ f(xy) = f(x)f(y)-f(x+y)+1 $ for $ x, y \\in \\mathbb{Q} $."
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"value": "XXXI - III - Task 1\n\nCalculate the area of an octagon inscribed in a circle, knowing that each of four consecutive sides of this octagon has a length of 1, and each of the remaining four has a length of 2."
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"value": "<think>\nOkay, let's tr... |
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"value": "XII OM - II - Task 4\n\nFind the last four digits of the number $ 5^{5555} $."
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"value": "<think>\nOkay, so I need to find the last four digits of 5^5555. Hmm, the last four digits... That means I need to compute 5^5555 modulo 10000, right? Be... |
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"value": "XLVII OM - III - Problem 1\n\nDetermine all pairs $ (n,r) $, where $ n $ is a positive integer and $ r $ is a real number, for which the polynomial $ (x + 1)^n - r $ is divisible by the polynomial $ 2x^2 + 2x + 1 $."
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"value": "<think>\nOkay, ... |
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"value": "XXI OM - III - Task 5\n\nIn how many ways can a set consisting of twelve elements be divided into six disjoint two-element sets?"
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"value": "<think>\nOkay, so the problem is asking: In how many ways can a set consisting of twelve elements be d... |
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"value": "XXXVIII OM - III - Problem 2\n\nA regular $ n $-gon is inscribed in a circle of length 1. From among the arcs of the circle having endpoints at the vertices of this polygon, we randomly draw five arcs $ L_1, \\ldots, L_5 $ with replacement, where the probability of drawing each a... |
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"value": "XLII OM - III - Problem 2\n\nLet $ X $ be the set of points in the plane $ (x.y) $ with both coordinates being integers. A path of length $ n $ is any sequence $ (P_0, P_1,\\ldots, P_n) $ of points in the set $ X $ such that $ |P_{i-1}P_{i}|=1 $ for $ i \\in \\{1,2,\\ldots,n\\} $... |
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"value": "LI OM - I - Task 5\n\nDetermine all pairs $ (a,b) $ of natural numbers for which the numbers $ a^3 + 6ab + 1 $ and $ b^3 + 6ab + 1 $ are cubes of natural numbers."
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"value": "<think>\nOkay, let's tackle this problem where we need to find all p... |
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"value": "XXXV OM - I - Problem 9\n\nThree events satisfy the conditions:\na) their probabilities are equal,\nb) any two of them are independent,\nc) they do not occur simultaneously.\nDetermine the maximum value of the probability of each of these events."
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"value": "XXXVII OM - III - Problem 2\n\nDetermine the maximum volume of tetrahedra having three faces with an area of 1."
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"value": "<think>\nOkay, so I need to figure out the maximum volume of a tetrahedron that has three faces each with an area of 1.... |
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"value": "L OM - I - Task 3\n\nIn an isosceles triangle $ ABC $, angle $ BAC $ is a right angle. Point $ D $ lies on side $ BC $, such that $ BD = 2 \\cdot CD $. Point $ E $ is the orthogonal projection of point $ B $ onto line $ AD $. Determine the measure of angle $ CED $."
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"value": "XXXVIII OM - I - Problem 4\n\nFor a given natural number $ n $, we consider the family $ W $ of all sets $ A $ of pairs of natural numbers not greater than $ n $, such that if $ (a, b) \\in A $, then $ (x, y) \\in A $ for $ 1 \\leq x \\leq a $, $ 1 \\leq y \\leq b $.\nProve that ... |
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"value": "XXXI - I - Task 1\n\nDetermine for which values of the parameter $ a $ a rhombus with side length $ a $ is a cross-section of a cube with edge length 2 by a plane passing through the center of the cube."
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"value": "<think>\nAlright, so I need ... |
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"value": "LV OM - I - Task 7\n\nFind all solutions to the equation $ a^2+b^2=c^2 $ in positive integers such that the numbers $ a $ and $ c $ are prime, and the number $ b $ is the product of at most four prime numbers."
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"value": "<think>\nAlright, let... |
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"value": "XXX OM - I - Task 10\n\nGiven are sequences of polynomials $ (P_n) $, $ (Q_n) $ defined as follows:\n\n\n\n\n\n\nLet $ a_n $ be the number of coefficients of the polynomial $ P_n $ equal to 1. Prove that the limit $ \\lim_{n\\to \\infty} \\frac{a_n}{2^n} $ exists and calculate it... |
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"value": "XXI OM - II - Problem 6\n\nIf $ A $ is a subset of set $ X $, then we define $ A^1 = A $, $ A^{-1} = X - A $. Subsets $ A_1, A_2, \\ldots, A_k $ are called mutually independent if the product $ A_1^{\\varepsilon_1} \\cap A_2^{\\varepsilon_2} \\ldots A_k^{\\varepsilon_k} $ is non-... |
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"value": "XIII OM - III - Task 4\n\nIn how many ways can a set of $ n $ items be divided into two sets?"
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"value": "<think>\nOkay, so the problem is asking: In how many ways can a set of n items be divided into two sets? Hmm, let me think. First, I need... |
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"value": "LV OM - III - Task 5\n\nDetermine the maximum number of lines in space passing through a fixed point and such that any two intersect at the same angle."
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"value": "<think>\nOkay, so I have this geometry problem here: Determine the maximum numb... |
[
{
"from": "user",
"value": "LIII OM - II - Problem 2\n\nIn a convex quadrilateral $ABCD$, the following equalities hold:\n\n\n\n\n\nProve that $AD = CD$."
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I have this problem from a Polish Olympiad, LIII OM - II - Problem 2. The probl... |
[
{
"from": "user",
"value": "XLIV OM - II - Problem 3\n\nOn the edge $ |OA_1| $ of the tetrahedron $ |OA_1B_1C_1| $, points $ A_2 $, $ A_3 $ are chosen such that $ |OA_1| > |OA_2| > |OA_3| > O $. Let $ B_2 $, $ B_3 $ be points on the edge $ |OB_1| $, and $ C_2 $, $ C_3 $ be points on the edge $ |OC_1| $, suc... |
[
{
"from": "user",
"value": "LVII OM - I - Problem 1\n\nDetermine all non-negative integers $ n $ for which the number\n$ 2^n +105 $ is a perfect square of an integer."
},
{
"from": "assistant",
"value": "<think>\nAlright, let me try to figure out this problem. So, the question is asking for all ... |
[
{
"from": "user",
"value": "LVI OM - I - Problem 11\n\nA circle with center $ I $ is inscribed in a convex quadrilateral $ ABCD $, with the point $ I $ not lying on the line $ AC $. The diagonals $ AC $ and $ BD $ intersect at point $ E $. The line passing through point $ E $ and perpendicular to the line $... |
[
{
"from": "user",
"value": "XVI OM - II - Task 4\n\nFind all prime numbers $ p $ such that $ 4p^2 +1 $ and $ 6p^2 + 1 $ are also prime numbers."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this problem: Find all prime numbers p such that 4p² + 1 and 6p² + 1 are also prime number... |
[
{
"from": "user",
"value": "XXXVI OM - I - Zadanie 9\n\nW urnie jest 1985 kartek z napisanymi liczbami 1,2,3,..., 1985, każda lczba na innej kartce. Losujemy bez zwracania 100 kartek. Znaleźć wartość oczekiwaną sumy liczb napisanych na wylosowanych kartkach.\n"
},
{
"from": "assistant",
"value":... |
[
{
"from": "user",
"value": "VI OM - II - Task 2\n\nFind the natural number $ n $ knowing that the sum\n\n\n\n\n\n\nis a three-digit number with identical digits."
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. The problem is to find a natural number n such that when we sum up some... |
[
{
"from": "user",
"value": "XXXIX OM - I - Problem 1\n\nFor each positive number $ a $, determine the number of roots of the polynomial $ x^3+(a+2)x^2-x-3a $."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this problem here where I need to determine the number of roots of the poly... |
[
{
"from": "user",
"value": "LVIII OM - I - Problem 2\n\nDetermine all pairs of positive integers $ k $, $ m $, for which each of the numbers $ {k^2+4m} $, $ {m^2+5k} $ is a perfect square."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all pairs of positive integers k and ... |
[
{
"from": "user",
"value": "Given an integer $ n \\geqslant 1 $. To each non-empty subset $ A $ of the set $ \\{1, 2,\\dots ,n\\} $\nwe assign a number $ w(A) $ as follows: If $ a_1 > a_2 > \\dots > a_k $ are all the elements\nof the set $ A $, then\n\n\n\nObtain the sum of all $ 2n-1 $ numbers $ w(A) $."
... |
[
{
"from": "user",
"value": "XX OM - II - Task 2\n\nFind all four-digit numbers in which the thousands digit is equal to the hundreds digit, and the tens digit is equal to the units digit, and which are squares of integers."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all... |
[
{
"from": "user",
"value": "XXXV OM - II - Problem 1\n\nFor a given natural number $ n $, find the number of solutions to the equation $ \\sqrt{x} + \\sqrt{y} = n $ in natural numbers $ x, y $."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the number of solutions in natur... |
[
{
"from": "user",
"value": "XXXII - II - Problem 6\n\nThe areas of the bases of a given truncated triangular pyramid are $ B_1 $ and $ B_2 $. This pyramid can be cut by a plane parallel to the bases such that a sphere can be inscribed in each of the resulting parts. Prove that the lateral surface area of th... |
[
{
"from": "user",
"value": "XXXV OM - III - Problem 1\n\nDetermine the number of functions $ f $ mapping an $ n $-element set to itself such that $ f^{n-1} $ is a constant function, while $ f^{n-2} $ is not a constant function, where $ f^k = f\\circ f \\circ \\ldots \\circ f $, and $ n $ is a fixed natural ... |
[
{
"from": "user",
"value": "XXIX OM - III - Problem 4\n\nLet $ X $ be an $ n $-element set. Prove that the sum of the number of elements in the sets $ A \\cap B $ extended over all ordered pairs $ (A, B) $ of subsets of $ X $ equals $ n \\cdot 4^{n-1} $."
},
{
"from": "assistant",
"value": "<thi... |
[
{
"from": "user",
"value": "XLVIII OM - I - Problem 9\n\nDetermine all functions $ f: \\langle 1,\\ \\infty) \\to \\langle 1,\\ \\infty) $ that satisfy the following conditions:\n\n[(i)] $ f(x+1) = \\frac{(f(x))^2-1}{x} $ for $ x \\geq 1 $;\n[(ii)] the function $ g(x)=\\frac{f(x)}{x} $ is bounded."
},
{... |
[
{
"from": "user",
"value": "XXXI - I - Problem 11\n\nProve that $ \\tan \\frac{\\pi}{7} \\tan \\frac{2\\pi}{7} \\tan \\frac{3\\pi}{7} = \\sqrt{7} $."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to prove that the product of the tangents of π/7, 2π/7, and 3π/7 is equal to the squa... |
[
{
"from": "user",
"value": "LVI OM - I - Task 2\n\nDetermine all natural numbers $ n>1 $ for which the value of the sum $ 2^{2}+3^{2}+\\ldots+n^{2} $ is a power of a prime number with a natural exponent."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all natural numbers n ... |
[
{
"from": "user",
"value": "XL OM - II - Task 3\n\nGiven is a trihedral angle $ OABC $ with vertex $ O $ and a point $ P $ inside it. Let $ V $ be the volume of the parallelepiped with two vertices at points $ O $ and $ P $, whose three edges are contained in the rays $ OA^{\\rightarrow} $, $ OB^{\\rightarr... |
[
{
"from": "user",
"value": "LIX OM - I - Problem 11\n\nPoints $ P_1, P_2, P_3, P_4, P_5, P_6, P_7 $ lie on the sides $ BC, CA, AB, BC, CA, AB, BC $ of triangle $ ABC $, respectively, and the following equalities are satisfied:\n\n\n\n\n\n\nProve that $ P_1 = P_7 $."
},
{
"from": "assistant",
"va... |
[
{
"from": "user",
"value": "XLVIII OM - I - Problem 7\n\nCalculate the upper limit of the volume of tetrahedra contained within a sphere of a given radius $ R $, one of whose edges is the diameter of the sphere."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out the maxi... |
[
{
"from": "user",
"value": "XXI OM - I - Problem 10\n\nFind the largest natural number $ k $ with the following property: there exist $ k $ different subsets of an $ n $-element set, such that any two of them have a non-empty intersection."
},
{
"from": "assistant",
"value": "<think>\nAlright, l... |
[
{
"from": "user",
"value": "LIX OM - II - Task 1\n\nDetermine the maximum possible length of a sequence of consecutive integers, each of which can be expressed in the form $ x^3 + 2y^2 $ for some integers $ x, y $."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out the m... |
[
{
"from": "user",
"value": "XV OM - I - Problem 11\n\nIn triangle $ ABC $, angle $ A $ is $ 20^\\circ $, $ AB = AC $. On sides $ AB $ and $ AC $, points $ D $ and $ E $ are chosen such that $ \\measuredangle DCB = 60^\\circ $ and $ \\measuredangle EBC = 50^\\circ $. Calculate the angle $ EDC $."
},
{
... |
[
{
"from": "user",
"value": "XLIII OM - III - Zadanie 2\n\nWyznaczyć wszystkie funkcje $ f $ określone na zbiorze liczb wymiernych dodatnich, o wartościach w tym samym zbiorze, spełniające dla wszystkich dodatnich liczb wymiernych warunki\n\n\n\n\n\n\n"
},
{
"from": "assistant",
"value": "<think>... |
[
{
"from": "user",
"value": "XXVIII - II - Task 3\n\nIn a hat, there are 7 slips of paper. On the $ n $-th slip, the number $ 2^n-1 $ is written ($ n = 1, 2, \\ldots, 7 $). We draw slips randomly until the sum exceeds 124. What is the most likely value of this sum?"
},
{
"from": "assistant",
"val... |
[
{
"from": "user",
"value": "IV OM - III - Task 3\n\nThrough each vertex of a tetrahedron of a given volume $ V $, a plane parallel to the opposite face of the tetrahedron has been drawn. Calculate the volume of the tetrahedron formed by these planes."
},
{
"from": "assistant",
"value": "<think>\... |
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