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: 2008

PEN O Problems, 19
Tags: modular arithmetic
Let $m, n \ge 2$ be positive integers, and let $a_{1}, a_{2}, \cdots,a_{n}$ be integers, none of which is a multiple of $m^{n-1}$. Show that there exist integers $e_{1}, e_{2}, \cdots, e_{n}$, not all zero, with $\vert e_i \vert<m$ for all $i$, such that $e_{1}a_{1}+e_{2}a_{2}+ \cdots +e_{n}a_{n}$ is a multiple of $m^n$.

1971 Bundeswettbewerb Mathematik, 2
Tags: modular arithmetic
You are given a piece of paper. You can cut the paper into $8$ or $12$ pieces. Then you can do so for any of the new pieces or let them uncut and so on. Can you get exactly $60$ pieces¿ Show that you can get every number of pieces greater than $60$.

2012 Bosnia Herzegovina Team Selection Test, 3
Tags: quadratic , prime number , modular arithmetic , calculus , number theory , integration
Prove that for all odd prime numbers $p$ there exist a natural number $m<p$ and integers $x_1, x_2, x_3$ such that: \[mp=x_1^2+x_2^2+x_3^2.\]

2004 Kazakhstan National Olympiad, 3
Tags: number theory , modular arithmetic
Does there exist a sequence $\{a_n\}$ of positive integers satisfying the following conditions: $a)$ every natural number occurs in this sequence and exactly once; $b)$ $a_1 + a_2 +... + a_n$ is divisible by $n^n$ for each $n = 1,2,3, ...$ ?

2009 Canada National Olympiad, 4
Tags: number theory , greatest common divisor , modular arithmetic
Find all ordered pairs of integers $(a,b)$ such that $3^a + 7^b$ is a perfect square.

1995 All-Russian Olympiad, 3
Tags: number theory , algorithm , modular arithmetic
Does there exist a sequence of natural numbers in which every natural number occurs exactly once, such that for each $k = 1, 2, 3, \dots$ the sum of the first $k$ terms of the sequence is divisible by $k$? [i]A. Shapovalov[/i]

1972 AMC 12/AHSME, 31
Tags: relatively prime , modular arithmetic , number theory
When the number $2^{1000}$ is divided by $13$, the remainder in the division is $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }7\qquad \textbf{(E) }11$

2013 ELMO Shortlist, 2
Tags: 3d geometry , modular arithmetic , polynomial , algebra , geometry , number theory
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

1995 IMO, 6
Tags: counting , modular arithmetic , prime , number theory
Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $ p$?

1980 IMO, 3
Tags: decimal representation , digit , modular arithmetic , combinatorics
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

2010 IMO Shortlist, 2
Tags: equation , number theory , lte lemma , modular arithmetic
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2011 Indonesia TST, 4
Tags: number theory , induction , modular arithmetic
A prime number $p$ is a [b]moderate[/b] number if for every $2$ positive integers $k > 1$ and $m$, there exists k positive integers $n_1, n_2, ..., n_k $ such that \[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \] If $q$ is the smallest [b]moderate[/b] number, then determine the smallest prime $r$ which is not moderate and $q < r$.

PEN D Problems, 7
Tags: congruence , modular arithmetic , number theory
Somebody incorrectly remembered Fermat's little theorem as saying that the congruence $a^{n+1} \equiv a \; \pmod{n}$ holds for all $a$ if $n$ is prime. Describe the set of integers $n$ for which this property is in fact true.

2003 China Team Selection Test, 2
Tags: quadratic , modular arithmetic , number theory
Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.

2006 Federal Competition For Advanced Students, Part 2, 1
Tags: relatively prime , function , modular arithmetic , floor function , number theory
Let $ N$ be a positive integer. How many non-negative integers $ n \le N$ are there that have an integer multiple, that only uses the digits $ 2$ and $ 6$ in decimal representation?

2013 Princeton University Math Competition, 4
Tags: modular arithmetic
Compute the smallest integer $n\geq 4$ such that $\textstyle\binom n4$ ends in $4$ or more zeroes (i.e. the rightmost four digits of $\textstyle\binom n4$ are $0000$).

2021 Science ON all problems, 1
Tags: modular arithmetic , abstract algebra , number theory
Are there any integers $a,b$ and $c$, not all of them $0$, such that $$a^2=2021b^2+2022c^2~~?$$ [i] (Cosmin Gavrilă)[/i]

2014 Contests, 3
Tags: modular arithmetic , number theory
Find all nonnegative integer numbers such that $7^x- 2 \cdot 5^y = -1$

1975 IMO Shortlist, 11
Tags: sequence , additive number theory , modular arithmetic , number theory
Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$

1970 Miklós Schweitzer, 4
Tags: number theory , quadratic , absolute value , modular arithmetic , function
If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\] J. Suranyi

2007 Brazil National Olympiad, 2
Tags: number theory , quadratic , induction , modular arithmetic
Find the number of integers $ c$ such that $ \minus{}2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 \plus{} c$ is a multiple of $ 2^{2007}$.

1997 India National Olympiad, 2
Tags: special factorizations , number theory , quadratic , diophantine equation , modular arithmetic
Show that there do not exist positive integers $m$ and $n$ such that \[ \dfrac{m}{n} + \dfrac{n+1}{m} = 4 . \]

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

1993 Iran MO (2nd round), 1
Tags: divisibility theory , modular arithmetic
Suppose that $p$ is a prime number and is greater than $3$. Prove that $7^{p}-6^{p}-1$ is divisible by $43$.