Practice library
Problems
1Find every natural number with the following property: when the sum of its digits is added to the number itself, the result is \(313\).Open2Find 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.3Let \(d\) 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 \(m\).4Find the smallest three-digit number with the property that every digit of its triple is even.5Let \(n\) 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) …6Show that the number \(7^{2018} + 9^{2020n}\) is divisible by \(5\) for every natural number \(n\).7Determine every pair of integers \((x, y)\) for which \[ \frac{1}{x} + \frac{1}{y} = \frac{1}{2} . \]8Find all natural numbers \(m\) and \(n\) that satisfy \[ \frac{3}{m} + \frac{5}{n} = 1 . \]9Jana 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 …10A 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 …11Find 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 …12Find all pairs of integers \(x\) and \(y\) that satisfy the equation \[ x^2 + xy + y^2 = 1 . \]13Find all pairs of natural numbers \(a\) and \(b\) that satisfy the equation \[ a^2 - 5ab + 24 = 0 . \]14Nika 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 …15Find all prime numbers \(p\), \(q\) and \(r\) for which \[ 15p + 7pq + qr = pqr . \]16Show that the number \[ 2^{\,2n+3} + 3^{\,n+2}\cdot 7^{\,n} \] is divisible by \(17\) for every natural number \(n\).17Find all integers \(n\) for which the number \[ \frac{9n+8}{n+7} \] is also an integer.18Let \(a\) 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 \(a + b\)?19Find all pairs of integers \(a\) and \(b\) satisfying \[ 4a - 2b + 22ab^2 - 11b^3 = 2024 . \]20The 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 …21Find all rational numbers \(r\) and all integers \(k\) that satisfy \[ r\bigl(5k - 7r\bigr) = 3 . \]22Find every integer \(n\) that can be written in the form \[ n = \frac{m+2021}{2021-m}, \] where \(m\) is an integer.23Find 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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