This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 15460

2019 Moldova Team Selection Test, 8
Tags: number theory
For any positive integer $k$ denote by $S(k)$ the number of solutions $(x,y)\in \mathbb{Z}_+ \times \mathbb{Z}_+$ of the system $$\begin{cases} \left\lceil\frac{x\cdot d}{y}\right\rceil\cdot \frac{x}{d}=\left\lceil\left(\sqrt{y}+1\right)^2\right\rceil \\ \mid x-y\mid =k , \end{cases}$$ where $d$ is the greatest common divisor of positive integers $x$ and $y.$ Determine $S(k)$ as a function of $k$. (Here $\lceil z\rceil$ denotes the smalles integer number which is bigger or equal than $z.$)

1992 IberoAmerican, 1
Tags: number theory
For every positive integer $n$ we define $a_{n}$ as the last digit of the sum $1+2+\cdots+n$. Compute $a_{1}+a_{2}+\cdots+a_{1992}$.

1974 Chisinau City MO, 81
Tags: number theory
Determine which number each letter denotes in the equalities $(YX)^Y=BYX$ and $(AA)^H = AHHA$, if different (identical) letters correspond to different (identical) numbers.

1999 Croatia National Olympiad, Problem 4
Tags: number theory
A triple of numbers $(a_1,a_2,a_3)=(3,4,12)$ is given. The following operation is performed a finite number of times: choose two numbers $a,b$ from the triple and replace them by $0.6x-0.8y$ and $0.8x+0.6y$. Is it possible to obtain the (unordered) triple $(2,8,10)$?

2021 ISI Entrance Examination, 3
Tags: power series , algebra , number theory
Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$ where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.

2009 USA Team Selection Test, 3
Tags: number theory , least common multiple , algebra , greatest common divisor , geometry , geometric transformation , polynomial
For each positive integer $ n$, let $ c(n)$ be the largest real number such that \[ c(n) \le \left| \frac {f(a) \minus{} f(b)}{a \minus{} b}\right|\] for all triples $ (f, a, b)$ such that --$ f$ is a polynomial of degree $ n$ taking integers to integers, and --$ a, b$ are integers with $ f(a) \neq f(b)$. Find $ c(n)$. [i]Shaunak Kishore.[/i]

2020 Balkan MO, 4
Tags: number theory
Let $a_1=2$ and, for every positive integer $n$, let $a_{n+1}$ be the smallest integer strictly greater than $a_n$ that has more positive divisors than $a_n$. Prove that $2a_{n+1}=3a_n$ only for finitely many indicies $n$. [i] Proposed by Ilija Jovčevski, North Macedonia[/i]

2018 BMT Spring, Tie 3
Tags: number theory
Let $f : Z^2 \to C$ be a function such that $f(x+11, y) = f(x, y+11) = f(x, y)$, and $f(x, y)f(z,w) = f(xz - yw,xw + yz)$. How many possible values can $f(1, 1)$ have?

1987 Polish MO Finals, 5
Tags: prime , number theory , product , min
Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).

1996 Baltic Way, 6
Tags: number theory
Let $a,b,c,d$ be positive integers such that $ab\equal{}cd$. Prove that $a\plus{}b\plus{}c\plus{}d$ is a composite number.

2009 Brazil Team Selection Test, 4
Tags: number theory , sequence , greatest common divisor
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

1971 IMO Longlists, 11
Tags: modular arithmetic , number theory
Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.

2010 IberoAmerican Olympiad For University Students, 6
Tags: polynomial , search , number theory , algebra
Prove that, for all integer $a>1$, the prime divisors of $5a^4-5a^2+1$ have the form $20k\pm1,k\in\mathbb{Z}$. [i]Proposed by Géza Kós.[/i]

1979 IMO Longlists, 49
Tags: inequalities , number theory
Let there be given two sequences of integers $f_i(1), f_i(2), \cdots (i = 1, 2)$ satisfying: $(i) f_i(nm) = f_i(n)f_i(m)$ if $\gcd(n,m) = 1$; $(ii)$ for every prime $P$ and all $k = 2, 3, 4, \cdots$, $f_i(P^k) = f_i(P)f_i(P^{k-1}) - P^2f(P^{k-2}).$ Moreover, for every prime $P$: $(iii) f_1(P) = 2P,$ $(iv) f_2(P) < 2P.$ Prove that $|f_2(n)| < f_1(n)$ for all $n$.

1999 Kazakhstan National Olympiad, 6
Tags: remainder , number theory with sequences , sequence , divisible , number theory
In a sequence of natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_ {1999} $, $ a_n-a_ {n-1} -a_ {n-2} $ is divisible by $ 100 (3 \leq n \leq 1999) $. It is known that $ a_1 = 19$ and $ a_2 = 99$. Find the remainder of $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $ by $8$.

2007 Stars of Mathematics, 4
Tags: number theory , gcd , greatest common divisor
Show that any subset of $ A=\{ 1,2,...,2007\} $ having $ 27 $ elements contains three distinct numbers such that the greatest common divisor of two of them divides the other one. [i]Dan Schwarz[/i]

2002 Belarusian National Olympiad, 2
Tags: integer , inequalities , number theory
Given rational numbers $a_1,...,a_n$ such that $\sum_{i=1}^n \{ka_i\}<\frac{n}{2}$ for any positive integer $k$. a) Prove that at least one of $a_1,...,a_n$ is integer. b) Is the previous statement true, if the number $\frac{n}{2}$ is replaced by the greater number? (Here $\{x\}$ means a fractional part of $x$.) (N. Selinger)

2023 LMT Fall, 4
Tags: number theory
Fred chooses a positive two-digit number with distinct nonzero digits. Laura takes Fred’s number and swaps its digits. She notices that the sum of her number and Fred’s number is a perfect square and the positive difference between them is a perfect cube. Find the greater of the two numbers.

1980 Yugoslav Team Selection Test, Problem 2
Tags: polynomial , algebra , number theory
Let $a,b,c,m$ be integers, where $m>1$. Prove that if $$a^n+bn+c\equiv0\pmod m$$for each natural number $n$, then $b^2\equiv0\pmod m$. Must $b\equiv0\pmod m$ also hold?

2014 Greece National Olympiad, 2
Tags: fraction , number theory
Find all the integers $n$ for which $\frac{8n-25}{n+5}$ is cube of a rational number.

2024 Junior Balkan Team Selection Tests - Moldova, 5
Tags: number theory
Prove that a number of the form $80\dots01$ (there is at least 1 zero) can't be a perfect square.

2009 Argentina National Olympiad, 6
Tags: sum of digits , digit , recurrence relation , number theory
A sequence $a_0,a_1,a_2,...,a_n,...$ is such that $a_0=1$ and, for each $n\ge 0$ , $a_{n+1}=m \cdot a_n$ , where $m$ is an integer between $2$ and $9$ inclusive. Also, every integer between $2$ and $9$ has even been used at least once to get $a_{n+1} $ from $a_n$ . Let $Sn$ the sum of the digits of $a_n$ , $n=0,1,2,...$ . Prove that $S_n \ge S_{n+1}$ for infinite values ​​of $n$.

2025 Malaysian IMO Training Camp, 4
Tags: number theory
For each positive integer $k$, find all positive integer $n$ such that there exists a permutation $a_1,\ldots,a_n$ of $1,2,\ldots,n$ satisfying $$a_1a_2\ldots a_i\equiv i^k \pmod n$$ for each $1\le i\le n$. [i](Proposed by Tan Rui Xuen and Ivan Chan Guan Yu)[/i]

2010 Brazil National Olympiad, 3
Tags: greatest common divisor , modular arithmetic , search , number theory
Find all pairs $(a, b)$ of positive integers such that \[ 3^a = 2b^2 + 1. \]

2021 Purple Comet Problems, 11
Tags: number theory
Find the minimum possible value of |m -n|, where $m$ and $n$ are integers satisfying $m + n = mn - 2021$.