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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 …3Prove 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.4Find 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 \(1\).5Let \(k\) be a positive integer. Prove that the number \[ 2^{2k-1} + 2^{k} + 1 \] is never divisible by \(7\).6Determine the remainder left by the number \[ 3^{1000} + 4^{1000} \] when it is divided by \(13\).7Let \(x\), \(y\) 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 \(x + y + z\).8Find every pair of integers \(x\) and \(y\) satisfying \[ 2\left(x^2 + y^2\right) = 5\left(xy + 1\right) . \]9Let \(n\) be a natural number. Prove that \[ (n+1)^{3n} - n^{2n}(n+3)^n \] is divisible by \(3n+1\).10Does there exist a polynomial \(P\) with integer coefficients for which \[ \textbf{a)}\quad P(7) = 8 \ \text{ and } \ P(15) = 12; \qquad\qquad \textbf{b)}\quad P(8) = 7 \ \text{ and } \ P(12) = 15\,? \] …11Find every triple \((x, y, z)\) of positive integers for which \[ xyz + xy + xz + yz + x + y + z = 2000 . \]12Do there exist positive integers \(a\), \(b\), \(c\) such that \[ 2010 = (a + b) \cdot (b + c) \cdot (c + a) \, ? \]13Solve the equation \[ x! + 76 = y^2 \] in the set of natural numbers.14Find every natural number \(n\) for which \[ 7 \cdot 2^n + 1 \] is a perfect square, that is, the square of an integer.15Let \(a\), \(b\) 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 …16Find the smallest natural number \(n\) for which there exist natural numbers \(a\) and \(b\) whose digit sums are \(28\) and \(21\) respectively, and \[ a + b = \underbrace{11\ldots1}_{n}. \]17It is known that \[ 35! = 10333147966386144929\,ab\,6651337523200000000 , \] where the letters \(a\) and \(b\) stand for two unknown decimal digits. Determine these two digits.18Find all pairs of integers \(m\) and \(n\) satisfying \[ 2m^{2} + n^{2} = 2mn + 3n . \]19Find 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 \[ n \nmid 1 \cdot 2 \cdot 3 \cdots (n-1) . \] …20Let \(x\) and \(y\) be integers such that \(90\) divides \(x^{2} + xy + y^{2}\). Prove that then \(900\) divides \(xy\).21Find the largest positive integer that is smaller than the sum of the squares of its decimal digits.22Consider 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 \(p(m)\)?23For a natural number \(n\), let \(P(n)\) denote the product of all digits of \(n\). Find every natural number \(n\) satisfying \[ n = P(n) + 18 . \]24Determine 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 \(n\).25Prove that the number \[ \sqrt{1 + \sqrt{2 + \cdots + \sqrt{n}}} \] is irrational for every natural number \(n \geq 2\).26Decide 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 …27Determine all natural numbers \(k\), \(m\) and \(n\) for which \[ 2^k + 10^m - 10^n = 2014 . \]28Let \(a\) 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.29Find all digits \(n\) and all \(2018\)-digit positive integers \(x = \overline{a_{2017} \ldots a_2 a_1 a_0}\) for which \[ n \cdot x = \overline{(a_{2017} + n) \ldots (a_2 + n)(a_1 + n)(a_0 + n)} . \] …30Let \(S\) 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 …31Find 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.32For 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 …33For 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 \(n\).34For 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 \[ x_{12} = 149162536496481100121144 . \] …35How many pairs \((x, y)\) of rational numbers satisfy \(2x^{2} + 5y^{2} = 1\)?36Prove 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.37Integers \(x\), \(y\), \(z\) satisfy \[ x^{2}z + y^{2}x + z^{2}y = x^{2}y + y^{2}z + z^{2}x + x + y + z . \] Prove that \(27 \mid x + y + z\).38Solve the equation \[ 20^x + 2^y = 2022^z \] in the set of natural numbers.39Let \(\mathbb{N}_{0}=\mathbb{N}\cup\{0\}\). Determine all pairs \((a,b)\in\mathbb{N}_{0}\times\mathbb{N}_{0}\) for which \[ 1+3^{a}+2025^{b}=2027^{b}. \]40Determine 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 …41A 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 …42Find all natural numbers \(n\) for which the number \[ n \cdot 2^{n} + 4 \] is the square of an integer.
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