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Problems
1Find all integer solutions of the equation \[ 6x^{3} + 7y^{2} + 8z^{3} = 66\,677\,888. \]Open2In a school there are \(2023\) pupils and \(2023\) lockers, the lockers bearing the numbers \(1, 2, \ldots, 2023\). At the start every locker is closed. The pupils file past the lockers one after another …3One digit of the seven-digit number \(2345678\) is to be deleted, so that the six-digit number left behind is divisible by \(9\). Which digit must it be? A \(8\) B \(7\) C \(6\) D \(5\) E \(4\)4Prove that a positive integer whose decimal representation uses no digits other than \(2\) and \(6\) cannot be written as a difference of the squares of two integers.5Find all natural numbers \(n\) for which the fraction \[ \frac{2n+3}{5n+7} \] can be reduced, that is, for which its numerator and denominator have a common divisor greater than \(1\).6Let \(k\) be a positive integer. Prove that the number \[ 2^{2k-1} + 2^{k} + 1 \] is never divisible by \(7\).7Determine the remainder left by the number \[ 3^{1000} + 4^{1000} \] when it is divided by \(13\).8Let \(x\), \(y\) and \(z\) be positive integers satisfying both \[ x^{3} - y^{3} - z^{3} = 3xyz \qquad \text{and} \qquad x^{2} = 2(y + z). \] Determine the value of \(x + y + z\).9Lili noticed that the digits of the year \(2015\) have average \(2\), because \(\frac{2+0+1+5}{4} = 2\). How many years of the 21st century after \(2015\) have the same digit average as \(2015\)? A \(1\) …10Find every pair of integers \(x\) and \(y\) satisfying \[ 2\left(x^2 + y^2\right) = 5\left(xy + 1\right) . \]11Let \(n\) be a natural number. Prove that \[ (n+1)^{3n} - n^{2n}(n+3)^n \] is divisible by \(3n+1\).12Does there exist a polynomial \(P\) with integer coefficients for which \[ \textbf{a)}\quad P(7) = 8 \ \text{ and } \ P(15) = 12; \qquad\qquad \textbf{b)}\quad P(8) = 7 \ \text{ and } \ P(12) = 15\,? \] …13There are \(17\) girls and \(12\) boys on a playground. At least how many more children must arrive so that everybody present can then be divided into two groups of the same size, in such a way that each …14Find every triple \((x, y, z)\) of positive integers for which \[ xyz + xy + xz + yz + x + y + z = 2000 . \]15Do there exist positive integers \(a\), \(b\), \(c\) such that \[ 2010 = (a + b) \cdot (b + c) \cdot (c + a) \, ? \]16Solve the equation \[ x! + 76 = y^2 \] in the set of natural numbers.17Find every natural number \(n\) for which \[ 7 \cdot 2^n + 1 \] is a perfect square, that is, the square of an integer.18Let \(a\), \(b\) and \(c\) be positive integers for which both of the numbers \[ 24^{a} + 2^{b} + 2018^{c} \qquad \text{and} \qquad 10^{c} + 3^{a} + 2018^{b} \] are divisible by \(7\). Prove that the number …19Find the smallest natural number \(n\) for which there exist natural numbers \(a\) and \(b\) whose digit sums are \(28\) and \(21\) respectively, and \[ a + b = \underbrace{11\ldots1}_{n}. \]20It is known that \[ 35! = 10333147966386144929\,ab\,6651337523200000000 , \] where the letters \(a\) and \(b\) stand for two unknown decimal digits. Determine these two digits.21Find all pairs of integers \(m\) and \(n\) satisfying \[ 2m^{2} + n^{2} = 2mn + 3n . \]22Find all composite natural numbers \(n\) which do not divide the product of all natural numbers smaller than \(n\), that is, all composite \(n\) for which \[ n \nmid 1 \cdot 2 \cdot 3 \cdots (n-1) . \] …23Let \(x\) and \(y\) be integers such that \(90\) divides \(x^{2} + xy + y^{2}\). Prove that then \(900\) divides \(xy\).24Find the largest positive integer that is smaller than the sum of the squares of its decimal digits.25Consider the polynomials \[ p(x) = x^3 + x^2 + x + 2 , \qquad q(x) = x^3 - x + 3 . \] Does there exist an integer \(m\) such that \(q(m)\) divides \(p(m)\)?26For a natural number \(n\), let \(P(n)\) denote the product of all digits of \(n\). Find every natural number \(n\) satisfying \[ n = P(n) + 18 . \]27Find all integers \(x\) and \(y\) that satisfy \[ 3xy + 2x + y = 12 . \]28Find every natural number \(n\) with the following property: there is a rectangle whose side lengths are natural numbers, whose perimeter equals \(n\), and whose area is numerically equal to \(n\) as well. …29Determine all natural numbers \(n\) for which the following assertion is true: a natural number \(x\) is divisible by \(n\) if and only if the sum of the digits of \(x\) is divisible by \(n\).30Prove that the number \[ \sqrt{1 + \sqrt{2 + \cdots + \sqrt{n}}} \] is irrational for every natural number \(n \geq 2\).31Decide whether the following claim is true, and prove your answer. For every positive integer \(n\) there exists a positive integer \(x\) such that all three of the following hold: \(x\) is divisible by …32Determine all natural numbers \(k\), \(m\) and \(n\) for which \[ 2^k + 10^m - 10^n = 2014 . \]33Let \(a\) and \(b\) be natural numbers. Prove that natural numbers \(c\) and \(d\) with \[ a^2 + b^2 + c^2 = d^2 \] exist if and only if at least one of the numbers \(a\) and \(b\) is even.34Find all digits \(n\) and all \(2018\)-digit positive integers \(x = \overline{a_{2017} \ldots a_2 a_1 a_0}\) for which \[ n \cdot x = \overline{(a_{2017} + n) \ldots (a_2 + n)(a_1 + n)(a_0 + n)} . \] …35Let \(S\) be a finite set of natural numbers with the property that for every two elements \(x\) and \(y\) of \(S\) there exists an element \(z \in S\) such that \(z \mid x - y\). Prove that \(S\) contains …36Find all prime numbers \(p\), \(q\) and \(r\) for which the number \[ p^{\,q+r} + q^{\,p+r} + r^{\,q+p} \] is the square of an odd natural number.37For every natural number, Perica computed the remainder that this number leaves on division by the sum of its digits in the decimal system, and wrote that remainder on the board. Has Perica in this way …38Jure wrote all the natural numbers from \(1\) to \(2015\) on a board. Urska then went through the numbers on the board from the smallest to the largest and rubbed out every one that was not divisible by …39Find all natural numbers \(n\) and all primes \(p\) for which \[ \sqrt{\,n + \frac{p}{n}\,} \] is a natural number.40For a natural number \(n\), determine the greatest common divisor of the two numbers \[ n^{2} + 1 \qquad \text{and} \qquad (n+1)^{2} + 1 , \] expressed in terms of \(n\).41For every natural number \(n\), let \(x_n\) be the number obtained by writing the squares of the first \(n\) natural numbers one after another, in increasing order; for example \[ x_{12} = 149162536496481100121144 . \] …42How many pairs \((x, y)\) of rational numbers satisfy \(2x^{2} + 5y^{2} = 1\)?43Prove that for every integer \(n \geqslant 2\) one can find \(n\) pairwise distinct positive integers whose squares add up to the square of a positive integer.44Integers \(x\), \(y\), \(z\) satisfy \[ x^{2}z + y^{2}x + z^{2}y = x^{2}y + y^{2}z + z^{2}x + x + y + z . \] Prove that \(27 \mid x + y + z\).45Solve the equation \[ 20^x + 2^y = 2022^z \] in the set of natural numbers.46Let \(\mathbb{N}_{0}=\mathbb{N}\cup\{0\}\). Determine all pairs \((a,b)\in\mathbb{N}_{0}\times\mathbb{N}_{0}\) for which \[ 1+3^{a}+2025^{b}=2027^{b}. \]47Determine all natural numbers \(n\) with all of the following properties: \(n\) is divisible by \(2\) but not by \(4\); the sum of the digits of \(n\) equals \(6\); the number of divisors of \(n\) equals …48A collection of \(2020\) consecutive positive integers is split into two subsets of \(1010\) numbers each. Can the least common multiple of all the numbers in the first subset be equal to the least common …49Find all natural numbers \(n\) for which the number \[ n \cdot 2^{n} + 4 \] is the square of an integer.50Benjamin was working out the sum \(1 + 2 + 3 + \dots + 2012\). He left out a few of the terms, and the wrong total he ended up with was divisible by \(2011\). Anika was working out the sum \[ A = 1 + 2 + 3 + \dots + 2013 , \] …
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