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Problems
1For a natural number \(n\), let \(f(n)\) be the number written with the same digits taken in the opposite order (that is, read from right to left) whenever \(n\) is not divisible by \(10\); if \(10 \mid n\), …Open2It is known that the number \[ 21982145917308330487013369 \] is equal to \(n^{13}\) for some natural number \(n\). Determine \(n\).3Determine the smallest natural number which, when divided by \(4\), \(6\), \(8\), \(10\) and \(12\), leaves the remainders \(2\), \(4\), \(6\), \(8\) and \(10\) respectively.4Let \(p\) be a number such that \(p\) and \(p^{2} + 2\) are both prime. Prove that \(p^{3} + 2\) is prime as well.5Let \(x\) and \(y\) be integers. Prove that if \(6x + 11y\) is divisible by \(31\), then \(x + 7y\) is divisible by \(31\) as well.6Let \(q\) be an odd integer. Prove that the equation \[ x^{3} + 3x + q = 0 \] has no solutions in integers.7Determine whether the number \[ 10^{5^{10^{5^{10}}}} + 5^{10^{5^{10^{5}}}} \] is divisible by \(11\).8Determine the smallest six-digit number whose digits are all different and which is divisible by \(11\).9Prove that no integers \(m\) and \(n\) satisfy \[ (m + n + 2)^2 = 3(mn + 1). \]10Determine the greatest common divisor of the numbers \(2^{2006}-1\) and \(2^{2004}-1\).11A natural number \(n\) leaves the remainder \(35\) on division by \(2009\), and also leaves the remainder \(35\) on division by \(2010\). What remainder does \(n\) leave on division by \(42\)?12Let \[ N = 1^{n} + 2^{n} + 3^{n} + 4^{n}, \qquad n \in \mathbb{N}. \] What is the greatest number of zeros in which the number \(N\) can end?13Let \(a_{1}, a_{2}, \ldots, a_{n}\) be pairwise distinct numbers from the set \(\{1, 2, \ldots, n\}\), where \(n \in \mathbb{N}\). Prove that the number \[ (a_{1} - 1) + (a_{2} - 2)^{2} + \cdots + (a_{n} - n)^{n} \] …14Find all pairs \((n, m)\) of integers for which \[ 3n^2 + 2nm + 3 = m^2 + 10. \]15Let \(n\) be a natural number. Prove that \(3n^2 + 3n + 7\) is not the cube of any natural number.16Find the greatest common divisor of the two numbers \[ \underbrace{11111111}_{8}, \qquad \underbrace{11\ldots11}_{100}, \] written with eight and with one hundred digits \(1\), respectively.17Determine in how many ways the number \(441000\) can be written as a product of two factors \(m\) and \(n\) with \[ m > 1, \qquad n > 1, \qquad \gcd(m,n) = 1 , \] where the order of the factors is irrelevant, …18Find all triples of pairwise distinct nonzero decimal digits \(a\), \(b\), \(c\) for which the fractions \[ \frac{\overline{ab}}{\overline{bc}} \qquad \text{and} \qquad \frac{a}{c} \] have the same value. …19The sum of \(49\) natural numbers equals \(999\). Find the largest possible value of their greatest common divisor.20In the course of a five-year programme of study a student passed \(31\) exams in total. In every year he passed more exams than in the year before, and in the fifth year he passed three times as many exams …21Determine the smallest natural number the product of whose digits equals \(75600\).22Let \(a\), \(b\), \(c\) be positive integers such that all three of the numbers \[ p = b^{c} + a, \qquad q = a^{b} + c, \qquad r = c^{a} + b \] are prime. Prove that two of the numbers \(p\), \(q\), \(r\) …23Find all prime numbers \(p\), \(q\), \(r\), not necessarily different from one another, and all positive integers \(n\), for which \[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{1}{n}. \]24Natural numbers \(a\), \(b\) and \(c\) satisfy \[ a + \cfrac{1}{b + \cfrac{1}{c}} = \frac{4016}{2007} . \] Prove that \[ \cfrac{1}{c + \cfrac{1}{b + \cfrac{1}{a}}} = \frac{2007}{4016} . \]25Does the number \[ 2010^{2010} + 10^{2011} \] have more digits in its decimal representation than the number \(2010^{2010}\)?26For a natural number \(k\), let \(S(k)\) denote the sum of its digits. Do there exist natural numbers \(n\) and \(m\) such that \[ S(n) \cdot S(n+1) \cdot \ldots \cdot S(n+m) = 2011^{2010}\,? \]27Determine all natural numbers \(n\) for which the number of positive divisors of \(n^{3}\) is exactly \(2011\) greater than the number of positive divisors of \(n\).28Determine all pairs of prime numbers \(p\) and \(q\) for which \((p^3 + 1)^q\) is the square of a natural number.29Let \(a\) and \(b\) be arbitrary natural numbers, let \(M\) be their least common multiple and \(D\) their greatest common divisor. Prove that \[ a^n + b^n \le M^n + D^n \] holds for every natural number …30Find a five-digit natural number whose half is the square of a natural number and whose third is the cube of a natural number.31Prove that a natural number of the form \(4n + 1\) can be represented as a sum of two squares if and only if the number \(8n + 2\) can be represented as a sum of two squares.32The decimal representation of a positive integer \(n\) uses only the digits \(1\), \(3\), \(7\) and \(9\), and each of these four digits appears at least once. Prove that the digits of \(n\) can be rearranged …33Let \(n\) be a natural number. Prove that \(n^2 + 3n + 5\) is never divisible by \(121\).34a) Suppose the ordered quadruple \((x, y, z, w)\) is a solution of the equation \[ x^2 + y^2 + z^2 + w^2 = xyzw . \] Prove that \((yzw - x,\, y,\, z,\, w)\) is a solution of the same equation. b) Prove …35Determine all natural numbers \(n\) for which the number \[ n^2 + 7n + 2 \] is equal to a product of several (at least two) consecutive natural numbers.36Find every natural number with the following property: when the sum of its digits is added to the number itself, the result is \(313\).37Find the smallest natural number \(n\) for which the sum \[ n + 2n + 3n + \cdots + 9n \] is a number whose decimal representation has all of its digits equal.38Does there exist a natural number \(n\) for which the decimal expansion of \(n!\) has the form \[ n! = \ldots 2012\,\underbrace{00\ldots 0}_{k}, \] that is, ends in the digit block \(2012\) followed by …39Let \(d\) denote the greatest common divisor and \(v\) the least common multiple of the natural numbers \(m\) and \(n\). Prove that if \[ 3m + n = 3v + d , \] then \(n\) divides \(m\).40Find the smallest three-digit number with the property that every digit of its triple is even.41Consider strictly increasing sequences \(a_1, a_2, a_3, \dots\) of prime numbers in which any two consecutive terms differ by \(2\) or by \(4\); that is, \[ a_{i+1} - a_i \in \{2, 4\} \quad \text{for every } i. \] …42Find all natural numbers \(n\) for which the three numbers \[ n-4, \qquad 2n+2, \qquad 4n+1 \] are all perfect cubes.43Let \(m > 1\) be a natural number. Prove that there is no sequence of \(2^{m}\) consecutive natural numbers all of which have exactly \(m\) prime factors, counted with multiplicity. (For example, the number …44Let \(n \ge 3\), and suppose \(n\) consecutive odd three-digit numbers are given. Prove that these \(n\) numbers can be arranged into a sequence \(b_1, b_2, \ldots, b_n\) so that the number \[ \overline{b_1b_2\ldots b_n}, \] …45Let \(n\) be the number \(100\ldots001\) whose decimal expansion consists of the digit \(1\), then \(2017\) digits \(0\), then the digit \(1\) again. Decide, with proof, whether \(n\) is divisible by (a) …46Show that the number \(7^{2018} + 9^{2020n}\) is divisible by \(5\) for every natural number \(n\).47Determine every pair of integers \((x, y)\) for which \[ \frac{1}{x} + \frac{1}{y} = \frac{1}{2} . \]48Find all natural numbers \(m\) and \(n\) that satisfy \[ \frac{3}{m} + \frac{5}{n} = 1 . \]49Jana and Zana each wrote down her own age. Both ages turned out to be two-digit numbers written with the same two digits, only in the opposite order. Five years from now, Jana will be exactly twice as …50A competition paper had \(20\) problems. Each problem answered correctly was worth \(8\) points, each problem answered incorrectly cost \(5\) points, and a problem left blank scored \(0\). Tine handed …51Find the smallest three-digit number with the following property: if its digits are written in the reverse order and the number so obtained is added to the original number, then every digit of the sum …52Find all pairs of integers \(x\) and \(y\) that satisfy the equation \[ x^2 + xy + y^2 = 1 . \]53Determine all triples of integers \((x, y, z)\) that satisfy \[ x^{2} + y^{2} + z^{2} = 2004\,xyz . \]54A clock has three hands, each turning at its own constant speed: the second hand completes a full circle in one minute, the minute hand in one hour, and the hour hand in twelve hours. At midnight all three …55Find all solutions of the equation \[ 6\left(6a^{2} + 3b^{2} + c^{2}\right) = 5d^{2} \] in integers \(a\), \(b\), \(c\), \(d\).56Solve the equation \[ 12^{x} + 10^{y} = 7102^{z} \] in the set of natural numbers.57Find all natural numbers that are powers of \(3\) and whose representation in base \(12\) contains only the digits \(6\) and \(9\).58Find all pairs of natural numbers \(a\) and \(b\) that satisfy the equation \[ a^2 - 5ab + 24 = 0 . \]59Nika and Tim played a series of card games. A draw was impossible. They agreed in advance that after each game the winner receives more points than the loser and that the loser receives a positive number …60Find all prime numbers \(p\), \(q\) and \(r\) for which \[ 15p + 7pq + qr = pqr . \]61Show that the number \[ 2^{\,2n+3} + 3^{\,n+2}\cdot 7^{\,n} \] is divisible by \(17\) for every natural number \(n\).62Find all integers \(n\) for which the number \[ \frac{9n+8}{n+7} \] is also an integer.63Let \(a\) and \(b\) be natural numbers, both greater than \(1\), such that \[ \sqrt{a\sqrt{a\sqrt{a}}} = b . \] What is the smallest possible value of the sum \(a + b\)?64Find all pairs of integers \(a\) and \(b\) satisfying \[ 4a - 2b + 22ab^2 - 11b^3 = 2024 . \]65The edge of a wooden cube is a natural number \(a > 2\). The cube is painted all over and then cut into unit cubes. It turns out that the number of unit cubes with exactly two painted faces divides the …66Find all rational numbers \(r\) and all integers \(k\) that satisfy \[ r\bigl(5k - 7r\bigr) = 3 . \]67Find every integer \(n\) that can be written in the form \[ n = \frac{m+2021}{2021-m}, \] where \(m\) is an integer.68Find all three-digit numbers \(\overline{abc}\) that are divisible by \(9\) and satisfy \[ \overline{abc} = a^6 + b^2 + c^3, \] where \(a\), \(b\), \(c\) denote the digits of the number.
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