Practice library
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.36Does 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 …37Consider 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, \[ a_{i+1} - a_i \in \{2, 4\} \quad \text{for every } i. \] …38Find all natural numbers \(n\) for which the three numbers \[ n-4, \qquad 2n+2, \qquad 4n+1 \] are all perfect cubes.39Let \(m > 1\) 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 …40Let \(n \ge 3\), 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 \[ \overline{b_1b_2\ldots b_n}, \] …41Determine all triples of integers \((x, y, z)\) that satisfy \[ x^{2} + y^{2} + z^{2} = 2004\,xyz . \]42A 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 …43Find all integer solutions of the equation \[ 6x^{3} + 7y^{2} + 8z^{3} = 66\,677\,888. \]44In 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 …45Find all solutions of the equation \[ 6\left(6a^{2} + 3b^{2} + c^{2}\right) = 5d^{2} \] in integers \(a\), \(b\), \(c\), \(d\).46Solve the equation \[ 12^{x} + 10^{y} = 7102^{z} \] in the set of natural numbers.47Find all natural numbers that are powers of \(3\) and whose representation in base \(12\) contains only the digits \(6\) and \(9\).48Prove 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.49Find 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 \(1\).50Let \(k\) be a positive integer. Prove that the number \[ 2^{2k-1} + 2^{k} + 1 \] is never divisible by \(7\).51Determine the remainder left by the number \[ 3^{1000} + 4^{1000} \] when it is divided by \(13\).52Let \(x\), \(y\) 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 \(x + y + z\).53Find every pair of integers \(x\) and \(y\) satisfying \[ 2\left(x^2 + y^2\right) = 5\left(xy + 1\right) . \]54Let \(n\) be a natural number. Prove that \[ (n+1)^{3n} - n^{2n}(n+3)^n \] is divisible by \(3n+1\).55Does there exist a polynomial \(P\) with integer coefficients for which \[ \textbf{a)}\quad P(7) = 8 \ \text{ and } \ P(15) = 12; \qquad\qquad \textbf{b)}\quad P(8) = 7 \ \text{ and } \ P(12) = 15\,? \] …56Find every triple \((x, y, z)\) of positive integers for which \[ xyz + xy + xz + yz + x + y + z = 2000 . \]57Do there exist positive integers \(a\), \(b\), \(c\) such that \[ 2010 = (a + b) \cdot (b + c) \cdot (c + a) \, ? \]58Solve the equation \[ x! + 76 = y^2 \] in the set of natural numbers.59Find every natural number \(n\) for which \[ 7 \cdot 2^n + 1 \] is a perfect square, that is, the square of an integer.60Let \(a\), \(b\) 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 …61Find the smallest natural number \(n\) for which there exist natural numbers \(a\) and \(b\) whose digit sums are \(28\) and \(21\) respectively, and \[ a + b = \underbrace{11\ldots1}_{n}. \]62It is known that \[ 35! = 10333147966386144929\,ab\,6651337523200000000 , \] where the letters \(a\) and \(b\) stand for two unknown decimal digits. Determine these two digits.63Find all pairs of integers \(m\) and \(n\) satisfying \[ 2m^{2} + n^{2} = 2mn + 3n . \]64Find 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 \[ n \nmid 1 \cdot 2 \cdot 3 \cdots (n-1) . \] …65Let \(x\) and \(y\) be integers such that \(90\) divides \(x^{2} + xy + y^{2}\). Prove that then \(900\) divides \(xy\).66Find the largest positive integer that is smaller than the sum of the squares of its decimal digits.67Consider 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 \(p(m)\)?68For a natural number \(n\), let \(P(n)\) denote the product of all digits of \(n\). Find every natural number \(n\) satisfying \[ n = P(n) + 18 . \]69Determine 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 \(n\).70Prove that the number \[ \sqrt{1 + \sqrt{2 + \cdots + \sqrt{n}}} \] is irrational for every natural number \(n \geq 2\).71Decide 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 …72Determine all natural numbers \(k\), \(m\) and \(n\) for which \[ 2^k + 10^m - 10^n = 2014 . \]73Let \(a\) 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.74Find all digits \(n\) and all \(2018\)-digit positive integers \(x = \overline{a_{2017} \ldots a_2 a_1 a_0}\) for which \[ n \cdot x = \overline{(a_{2017} + n) \ldots (a_2 + n)(a_1 + n)(a_0 + n)} . \] …75Let \(S\) 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 …76Find 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.77For 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 …78For 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 \(n\).79For 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 \[ x_{12} = 149162536496481100121144 . \] …80How many pairs \((x, y)\) of rational numbers satisfy \(2x^{2} + 5y^{2} = 1\)?81Prove 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.82Integers \(x\), \(y\), \(z\) satisfy \[ x^{2}z + y^{2}x + z^{2}y = x^{2}y + y^{2}z + z^{2}x + x + y + z . \] Prove that \(27 \mid x + y + z\).83Solve the equation \[ 20^x + 2^y = 2022^z \] in the set of natural numbers.84Let \(\mathbb{N}_{0}=\mathbb{N}\cup\{0\}\). Determine all pairs \((a,b)\in\mathbb{N}_{0}\times\mathbb{N}_{0}\) for which \[ 1+3^{a}+2025^{b}=2027^{b}. \]85(a) Factor the expression \(x^{4}+x^{2}y^{2}+y^{4}\) into a product of polynomials of lower degree. (b) Decide whether the number \(9^{1998}+3^{1998}+1\) is prime.86Let \(a_1, a_2, \dots, a_n\) and \(b_1, b_2, \dots, b_n\), where \(n > 1\), be positive integers satisfying \[ \frac{a_1}{b_1} = \frac{a_2}{b_2} = \cdots = \frac{a_n}{b_n}. \] Prove that \(a_1 + a_2 + \cdots + a_n + b_1 + b_2 + \cdots + b_n\) …87Determine 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 …88A 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 …89Find all natural numbers \(n\) for which the number \[ n \cdot 2^{n} + 4 \] is the square of an integer.90Can the numbers \(1, 2, \dots, 100\) be distributed into three groups so that the sum of the numbers in the first group is divisible by \(102\), the sum of the numbers in the second group is divisible …91Let \(a = 123456789\) and \(b = 987654321\). (1) Find \(\gcd(a, b)\). (2) Find the remainder left by \(\operatorname{lcm}(a, b)\) on division by \(11\).92Let \(m\) be an arbitrary integer. Prove that there is at least one pair \((x, y)\) of integers for which \[ 2x^2 + 11xy + 12y^2 + 4x + 5y + 6 = 2m . \]93Find 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 …94A natural number \(n \geqslant 2\) is divided by each of the natural numbers \(1, 2, \ldots, n-1\) in turn, and all the remainders obtained are written down. Find every \(n\) for which the sum of the distinct …95For a natural number \(n\), let \(S(n)\) be the sum of its decimal digits and \(P(n)\) the product of its decimal digits. Find all natural numbers \(n\) for which \[ S(n) + P(n) = n . \]96Find every triple of integers \(x, y, z\) with \[ 3x^2 + 3y^2 + 3z^2 + 2x + 2y + 2z = 2004 . \]97Prove that there is a natural number \(n\) for which the number \[ 2p^{n} + 3 \] is composite for every prime number \(p\).98For 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 …99Does there exist a natural number which is a perfect square and whose sum of digits equals \(2008^{2009}\)?100Let \(a\), \(b\), \(c\) 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 …101For 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 \(f \colon \mathbb{N} \to \mathbb{N}_0\) …102Prove 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 \(n^2\).103Let \(n\) be a natural number and let \(d\) be a positive divisor of \(2n^2\). Can \(n^2 + d\) be a perfect square?104Let \(k > 3\) and consider the number \(2^k\). Prove that no rearrangement of the decimal digits of \(2^k\) can produce the number \(2^n\) with \(n > k\).105In how many ways can natural numbers \(a\), \(b\), \(c\) be chosen so that all of the following hold? \[ 1^{\circ} \quad a < b < c < 52 ; \] \[ 2^{\circ} \quad a \mid c ; \qquad 3^{\circ} \quad b \mid c ; \] …106Call 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 …107Let \(P(x)\) 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 \(P(p+q) = 2010\).108It 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 \(a \neq 0\). …109Let \(p\) be a prime number. Suppose that for some \(k \in \mathbb{N}\) the number \[ k^3 + pk^2 \] is a perfect cube. Prove that \(3 \mid p - 1\).110The 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 \(m^{n^{k}}\) …111Determine 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.112How many triples \((a, b, c)\) of positive integers are there such that \(2a + 1\) is divisible by \(b\), \(2b + 1\) is divisible by \(c\), and \(2c + 1\) is divisible by \(a\)?113For some integer \(n > 3\), all the digits of the number \[ 1 + 2 + \cdots + n \] are equal to one another. Which digit can this be?114Does there exist a polynomial \(P(x)\) whose coefficients are not all integers, such that \[ P(0)=0 \qquad\text{and}\qquad \frac{P(a)-P(b)}{a-b}\ \text{ is an integer for every pair of distinct integers } a, b\ ? \] …115The product of the binomial coefficient \(\binom{64}{21}\) and an unknown odd number equals \[ 5{\ast}\,6{\ast}0\,{\ast}8{\ast}\,862\,{\ast}1{\ast}\,7{\ast}7\,{\ast}4{\ast}\,4{\ast}5\,12{\ast}\,9{\ast}{\ast} \, , \] …116For 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 \[ f(n) < f(n+1) < f(n+2) < f(n+3) . \]117Suppose that pairwise different positive integers \(a_1, a_2, \dots, a_{2024}\) satisfy \[ [a_1, a_2] + (a_2, a_3) + [a_3, a_4] + (a_4, a_5) + \dots + [a_{2023}, a_{2024}] + (a_{2024}, a_1) = a_1 + a_2 + \dots + a_{2024} . \] …118For a positive integer \(x\), let \(S(x)\) denote the sum of the decimal digits of \(x\). (a) Determine the smallest element of the set \(\{\, S(11n^2 + n + 1) \mid n \text{ a positive integer} \,\}\). …119Determine all nonnegative integers \(n\) and all digits \(a\), \(b\), \(c\) for which the number \[ M = \overline{1\,\underbrace{0 \dots 0}_{n}\,a\,\underbrace{0 \dots 0}_{n}\,b\,\underbrace{0 \dots 0}_{n}\,c} \] …
Showing 119 of 651 - problem statements are free for everyone.