Practice library
Problems
1The 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 \(18\,009\)Open2The 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 …3Let \(x\), \(y\), \(z\) 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 \(6\) …4For 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 \(12\)5What 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 \(440\)6Let \(N\) 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 …7Let \(k\), \(m\) 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 \(5\). …8The 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?9A prime number \(p\) is given. Find all pairs of natural numbers \(x\) and \(y\) (positive integers) that satisfy \[ p\,(x - 5) = x\,y . \]10Is 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 \(3\)?11Find all pairs of prime numbers \(p\) and \(q\) for which the number \[ 2p^{2}q + 45pq^{2} \] is a perfect square.12The 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?13Determine all prime numbers \(p\) and \(q\) for which the number \[ 2^{2} + p^{2} + q^{2} \] is also prime.14Find 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 \(n\).15Find all triples of prime numbers \(p\), \(q\), \(r\) for which \[ p + q^{2} = r^{4} . \]16Eva, 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; …17Tine 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 …18Oliver 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 …19Find all primes \(p\), \(q\), \(r\) and \(s\) for which \[ p + q = r \qquad\text{and}\qquad q + r = s^{2} . \]20A 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 …21Find every three-digit number \(\overline{abc}\) whose digits \(a\), \(b\), \(c\) are all nonzero and which satisfies \[ \overline{abc} = 3 \cdot a! + 2 \cdot b! + c! . \] Here \(\overline{abc}\) is the …22Find every prime \(p\) for which the number \(p^2 + 11\) has fewer than \(11\) positive divisors.23Determine 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 \(1, 2, 3, \ldots\).) …24Find all pairs of natural numbers \(m\) and \(n\) whose sum equals \(2007\) and whose product is divisible by \(2007\).25Find 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.26Let \(m\) and \(n\) be positive integers such that \(5m+n\) divides \(5n+m\). Prove that \(m\) divides \(n\).27A 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 …28Find all prime numbers \(p\), \(q\) and \(r\) that satisfy \[ r^{4} = pq + 4 . \]29Find 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.30Prove that there are no natural numbers \(a\) and \(b\) satisfying \[ \sqrt{a} + \sqrt{b} = \sqrt{2021} . \]31Find all pairs of natural numbers \(a\) and \(b\) for which \[ v = ab - 2a - 4b , \] where \(v\) denotes the least common multiple of \(a\) and \(b\).32Determine the smallest possible value of \[ \left|25^{m} - 36^{n}\right| \] where \(m\) and \(n\) are positive integers.33Ana 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 …34Prove that there is a natural number \(n\) for which the number \[ 2p^{n} + 3 \] is composite for every prime number \(p\).35For a natural number \(n\), let \(x_n\) be the number obtained by writing the natural numbers from \(1\) to \(n\) one after another, for example \[ x_{15} = 123456789101112131415 . \] Find all natural …36Does there exist a natural number which is a perfect square and whose sum of digits equals \(2008^{2009}\)?37Let \(a\), \(b\), \(c\) be arbitrary positive integers. Prove the inequality \[ \gcd(a,\,b-1)\cdot\gcd(b,\,c-1)\cdot\gcd(c,\,a-1) \;\le\; ab+bc+ca-a-b-c+1 , \] and prove that equality is attained for infinitely …38For a positive integer \(n\), let \(x_n\) be the number obtained by writing the decimal representations of \(1, 2, \dots, n\) one after another; for example \(x_{14} = 1234567891011121314\). Define \(f \colon \mathbb{N} \to \mathbb{N}_0\) …39Prove that there exist infinitely many pairs \((m, n)\) of distinct positive integers such that the sum of all positive divisors of \(m^2\) is equal to the sum of all positive divisors of \(n^2\).40Find all pairs of coprime integers \(x\) and \(y\) that satisfy the equation \[ 4x^{3} + y^{3} = 3xy^{2} . \]41In how many ways can natural numbers \(a\), \(b\), \(c\) be chosen so that all of the following hold? \[ 1^{\circ} \quad a < b < c < 52 ; \] \[ 2^{\circ} \quad a \mid c ; \qquad 3^{\circ} \quad b \mid c ; \] …42Call a positive integer symmetric if its decimal representation reads the same from left to right as from right to left. Prove that there are infinitely many positive integers \(n\) for which the numbers …43Let \(P(x)\) be a polynomial with integer coefficients for which there exist prime numbers \(p < q < r\) with \[ \{P(p),\, P(q),\, P(r)\} = \{20,\, 3,\, 2010\} . \] Prove that \(P(p+q) = 2010\).44It is known that for some positive integers \(x\) and \(y\), \[ 23^{x} \cdot 111^{y} = \overline{aab3dc6902b2c74d456b} , \] where \(a, b, c, d\) are digits, not necessarily different, and \(a \neq 0\). …45Let \(p\) be a prime number. Suppose that for some \(k \in \mathbb{N}\) the number \[ k^3 + pk^2 \] is a perfect cube. Prove that \(3 \mid p - 1\).46The numbers \(a, b, c, x, y, z\) satisfy \[ \{a, b, c\} = \{x, y, z\} = \{15, 3, 2014\} . \] Must the number \[ a^{b^{c}} + x^{y^{z}} \] be composite? (For \(m, n, k \in \mathbb{N}\), the symbol \(m^{n^{k}}\) …47Determine all pairs of positive integers \(a\) and \(b\) for which the number \[ a^4 b + 3b - 2a^2 b^2 - a^2 - 3b^3 \] is a power of two.48Determine 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.49Does there exist a polynomial \(P(x)\) whose coefficients are not all integers, such that \[ P(0)=0 \qquad\text{and}\qquad \frac{P(a)-P(b)}{a-b}\ \text{ is an integer for every pair of distinct integers } a, b\ ? \] …50The product of the binomial coefficient \(\binom{64}{21}\) and an unknown odd number equals \[ 5{\ast}\,6{\ast}0\,{\ast}8{\ast}\,862\,{\ast}1{\ast}\,7{\ast}7\,{\ast}4{\ast}\,4{\ast}5\,12{\ast}\,9{\ast}{\ast} \, , \] …51For a positive integer \(n\), let \(f(n)\) denote the least common multiple of the numbers \(1, 2, \ldots, n\). Find all positive integers \(n\) for which \[ f(n) < f(n+1) < f(n+2) < f(n+3) . \]52Suppose that pairwise different positive integers \(a_1, a_2, \dots, a_{2024}\) satisfy \[ [a_1, a_2] + (a_2, a_3) + [a_3, a_4] + (a_4, a_5) + \dots + [a_{2023}, a_{2024}] + (a_{2024}, a_1) = a_1 + a_2 + \dots + a_{2024} . \] …53For a positive integer \(x\), let \(S(x)\) denote the sum of the decimal digits of \(x\). (a) Determine the smallest element of the set \(\{\, S(11n^2 + n + 1) \mid n \text{ a positive integer} \,\}\). …54A 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 …55We 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 …56Determine all nonnegative integers \(n\) and all digits \(a\), \(b\), \(c\) for which the number \[ M = \overline{1\,\underbrace{0 \dots 0}_{n}\,a\,\underbrace{0 \dots 0}_{n}\,b\,\underbrace{0 \dots 0}_{n}\,c} \] …
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