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 \[ a^2 - 5ab + 24 = 0 . \]14One 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 \(4\)15Nika 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 …16Lili 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 \(1\) …17Find all prime numbers \(p\), \(q\) and \(r\) for which \[ 15p + 7pq + qr = pqr . \]18Show that the number \[ 2^{\,2n+3} + 3^{\,n+2}\cdot 7^{\,n} \] is divisible by \(17\) for every natural number \(n\).19Find all integers \(n\) for which the number \[ \frac{9n+8}{n+7} \] is also an integer.20Let \(a\) 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 \(a + b\)?21There 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 …22Find all pairs of integers \(a\) and \(b\) satisfying \[ 4a - 2b + 22ab^2 - 11b^3 = 2024 . \]23The 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 …24Find all rational numbers \(r\) and all integers \(k\) that satisfy \[ r\bigl(5k - 7r\bigr) = 3 . \]25Find every integer \(n\) that can be written in the form \[ n = \frac{m+2021}{2021-m}, \] where \(m\) is an integer.26Find 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.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. …29Jure 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 …30Find all natural numbers \(n\) and all primes \(p\) for which \[ \sqrt{\,n + \frac{p}{n}\,} \] is a natural number.31The value of the expression \(10^{2016} - 10^{15}\) is a natural number. What is the sum of the digits of that number? A \(1\) B \(17\) C \(2001\) D \(18\,000\) E \(18\,009\)32The digits of a five-digit number add up to \(44\). What is the product of the digits of that number? A \(2^{3} \cdot 3^{8}\) B \(2^{3} \cdot 9^{3}\) C \(8 \cdot 4^{9}\) D \(8 \cdot 3^{4}\) E None of the …33Let \(x\), \(y\), \(z\) and \(w\) be natural numbers. At most how many of the six sums \[ x+y, \quad x+z, \quad x+w, \quad y+z, \quad y+w, \quad z+w \] can be odd? A \(2\) B \(3\) C \(4\) D \(5\) E \(6\) …34For how many integers \(k\) is the number \(k + 6\) an integer multiple of the number \(k - 6\)? A \(0\) B \(4\) C \(6\) D \(8\) E \(12\)35What is the largest natural number \(n\) with the property that, when \(n\) is divided by \(20\), the remainder is equal to the quotient? A \(21\) B \(92\) C \(231\) D \(399\) E \(440\)36Let \(N\) be the number whose decimal expansion consists of \(2025\) copies of the digit \(3\): \[ N = \underbrace{33\ldots3}_{2025} . \] What remainder does \(N\) leave when it is divided by \(12\)? A …37Let \(k\), \(m\) and \(n\) be natural numbers, none of which is divisible by \(5\). Prove that at least one of the three numbers \(k^2 - m^2, \qquad m^2 - n^2, \qquad n^2 - k^2\) is divisible by \(5\). …38The smallest natural number whose square ends in three fours is \(38\), because \(38^2 = 1444\). Which natural number is the next smallest one with this property?39A prime number \(p\) is given. Find all pairs of natural numbers \(x\) and \(y\) (positive integers) that satisfy \[ p\,(x - 5) = x\,y . \]40Is there a natural number \(n\) with the following property: if \(n\) is multiplied by the sum of its own digits, then the digits of the resulting product add up to \(3\)?41Find all pairs of prime numbers \(p\) and \(q\) for which the number \[ 2p^{2}q + 45pq^{2} \] is a perfect square.42The integers \(a\), \(b\), \(c\), \(d\) satisfy \(a > b > c > d\) and \[ (1 - a)(1 - b)(1 - c)(1 - d) = 10 . \] Which values can the expression \(a + b - c - d\) take?43Determine all prime numbers \(p\) and \(q\) for which the number \[ 2^{2} + p^{2} + q^{2} \] is also prime.44Find every natural number \(n \ge 10\) all of whose decimal digits are nonzero and which has the following property: deleting any single digit of \(n\) leaves a number that divides \(n\).45Find all triples of prime numbers \(p\), \(q\), \(r\) for which \[ p + q^{2} = r^{4} . \]46Eva, Igor, Marko and Marusa each wrote one natural number on a sheet of paper. Deleting the last digit of Eva's number gives Igor's number; deleting the last digit of Igor's number gives Marko's number; …47Tine collects stamps. For his birthday he received a new album with room for plenty of them, so he took \(2002\) tolars out of his money box and decided to spend all of it on stamps. A friend offered him …48Oliver rolled an ordinary die \(100\) times and multiplied together all \(100\) numbers that came up on top. The product he obtained was \(6^{70}\). What is the smallest possible number of rolls on which …49Benjamin 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 \[ A = 1 + 2 + 3 + \dots + 2013 , \] …50Find all primes \(p\), \(q\), \(r\) and \(s\) for which \[ p + q = r \qquad\text{and}\qquad q + r = s^{2} . \]51A large rectangle is cut into seven smaller rectangles, each of which has both of its side lengths equal to a whole number of metres. Five of the seven pieces have their areas written in them, as shown …52Find every prime \(p\) for which the number \(p^2 + 11\) has fewer than \(11\) positive divisors.53Determine every pair of coprime natural numbers \(m\) and \(n\) for which \[ \frac{5m - n}{m + n} \] is itself a natural number. (Here the natural numbers are the positive integers \(1, 2, 3, \ldots\).) …54Find all pairs of natural numbers \(m\) and \(n\) whose sum equals \(2007\) and whose product is divisible by \(2007\).55Find the smallest natural number \(n\) that is divisible by \(20\) and for which \(n^2\) is a perfect cube and \(n^3\) is a perfect square.56Let \(m\) and \(n\) be positive integers such that \(5m+n\) divides \(5n+m\). Prove that \(m\) divides \(n\).57A kangaroo called Pythagoras likes exactly those natural numbers that are divisible by \(4\), have digit sum \(3\), and have exactly five digits equal to \(0\) in their decimal representation. How many …58Find all prime numbers \(p\), \(q\) and \(r\) that satisfy \[ r^{4} = pq + 4 . \]59Find all natural numbers \(n\) whose cube equals the sum of the squares of three divisors of \(n\), where the three divisors need not be different from one another.60Prove that there are no natural numbers \(a\) and \(b\) satisfying \[ \sqrt{a} + \sqrt{b} = \sqrt{2021} . \]61Find all pairs of natural numbers \(a\) and \(b\) for which \[ v = ab - 2a - 4b , \] where \(v\) denotes the least common multiple of \(a\) and \(b\).62Determine the smallest possible value of \[ \left|25^{m} - 36^{n}\right| \] where \(m\) and \(n\) are positive integers.63Ana picked the eight digits \(1, 2, 3, 4, 5, 6, 7\) and \(9\). She then forms groups of four two-digit primes, each group using all of her chosen digits. What is the sum of the four primes in one such …64Find all pairs of coprime integers \(x\) and \(y\) that satisfy the equation \[ 4x^{3} + y^{3} = 3xy^{2} . \]65Determine all quintuples of primes \(p_{1} \le p_{2} \le p_{3} \le p_{4} \le p_{5}\) with the property that each of the five primes divides the sum of the remaining four.66A teacher handed Matej four sheets of paper, each carrying one nonzero digit. Matej laid the sheets in a row and so formed a four-digit number. He then interchanged two of the sheets, without flipping …67We want to choose a set \(P\) of \(k\) primes and a set \(N\) of \(n\) consecutive positive integers in such a way that every number \(a \in N\) is divisible by at least one prime \(p \in P\). (a) Determine …
Showing 67 of 651 - problem statements are free for everyone.