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

2006 IberoAmerican Olympiad For University Students, 1
Tags: modular arithmetic , number theory
Let $m,n$ be positive integers greater than $1$. We define the sets $P_m=\left\{\frac{1}{m},\frac{2}{m},\cdots,\frac{m-1}{m}\right\}$ and $P_n=\left\{\frac{1}{n},\frac{2}{n},\cdots,\frac{n-1}{n}\right\}$. Find the distance between $P_m$ and $P_n$, that is defined as \[\min\{|a-b|:a\in P_m,b\in P_n\}\]

2001 India National Olympiad, 4
Tags: modular arithmetic , combinatorics
Show that given any nine integers, we can find four, $a, b, c, d$ such that $a + b - c - d$is divisible by $20$. Show that this is not always true for eight integers.

2011 ELMO Shortlist, 1
Tags: function , modular arithmetic , combinatorics
Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that a) If $A\subseteq S$, then $A\in F$ if and only if $S\setminus A\notin F$; b) If $A\subseteq B\subseteq S$ and $B\in F$, then $A\in F$. Determine if there must exist a function $f:S\to\mathbb{R}$ such that for every $A\subseteq S$, $A\in F$ if and only if \[\sum_{s\in A}f(s)<\sum_{s\in S\setminus A}f(s).\] [i]Evan O'Dorney.[/i]

1973 IMO Longlists, 9
Tags: modular arithmetic
Prove that $2^{147} - 1$ is divisible by $343$.

2013 AMC 10, 25
Tags: search , modular arithmetic
Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N=749$, Bernardo writes the numbers 10,444 and 3,245, and LeRoy obtains the sum $S=13,689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$? ${ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D}}\ 20\qquad\textbf{(E)}\ 25 $

2018 CHKMO, 3
Tags: number theory , relatively prime , modular arithmetic
Let $k$ be a positive integer. Prove that there exists a positive integer $\ell$ with the following property: if $m$ and $n$ are positive integers relatively prime to $\ell$ such that $m^m\equiv n^n \pmod{\ell}$, then $m\equiv n \pmod k$.

2018 China Girls Math Olympiad, 7
Tags: combinatorics , number theory , modular arithmetic
Given $2018 \times 4$ grids and tint them with red and blue. So that each row and each column has the same number of red and blue grids, respectively. Suppose there're $M$ ways to tint the grids with the mentioned requirement. Determine $M \pmod {2018}$.

PEN M Problems, 15
Tags: modular arithmetic , recursive sequences , logarithm
For a given positive integer $k$ denote the square of the sum of its digits by $f_{1}(k)$ and let $f_{n+1}(k)=f_{1}(f_{n}(k))$. Determine the value of $f_{1991}(2^{1990})$.

PEN D Problems, 17
Tags: congruence , modular arithmetic
Determine all positive integers $n$ such that $ xy+1 \equiv 0 \; \pmod{n} $ implies that $ x+y \equiv 0 \; \pmod{n}$.

1982 IMO Longlists, 52
Tags: invariant , modular arithmetic , combinatorics
We are given $2n$ natural numbers \[1, 1, 2, 2, 3, 3, \ldots, n - 1, n - 1, n, n.\] Find all $n$ for which these numbers can be arranged in a row such that for each $k \leq n$, there are exactly $k$ numbers between the two numbers $k$.

2012 India Regional Mathematical Olympiad, 3
Tags: modular arithmetic
Find all natural numbers $x,y,z$ such that \[(2^x-1)(2^y-1)=2^{2^z}+1.\]

2009 India Regional Mathematical Olympiad, 2
Tags: modular arithmetic , number theory
Show that there is no integer $ a$ such that $ a^2 \minus{} 3a \minus{} 19$ is divisible by $ 289$.

1985 Dutch Mathematical Olympiad, 2
Tags: modular arithmetic , floor function , number theory
Among the numbers $ 11n \plus{} 10^{10}$, where $ 1 \le n \le 10^{10}$ is an integer, how many are squares?

2013 Harvard-MIT Mathematics Tournament, 10
Tags: hmmt , modular arithmetic
Let $N$ be a positive integer whose decimal representation contains $11235$ as a contiguous substring, and let $k$ be a positive integer such that $10^k>N$. Find the minimum possible value of \[\dfrac{10^k-1}{\gcd(N,10^k-1)}.\]

2013 Online Math Open Problems, 25
Tags: modular arithmetic , calculus , integration
Positive integers $x,y,z \le 100$ satisfy \begin{align*} 1099x+901y+1110z &= 59800 \\ 109x+991y+101z &= 44556 \end{align*} Compute $10000x+100y+z$. [i]Evan Chen[/i]

1974 IMO Longlists, 13
Tags: modular arithmetic
Prove that $2^{147} - 1$ is divisible by $343.$

2014 Math Hour Olympiad, 8-10.7
Tags: floor function , modular arithmetic
If $a$ is any number, $\lfloor a \rfloor$ is $a$ rounded down to the nearest integer. For example, $\lfloor \pi \rfloor =$ $3$. Show that the sequence $\lfloor \frac{2^{1}}{17} \rfloor$, $\lfloor \frac{2^{2}}{17} \rfloor$, $\lfloor \frac{2^{3}}{17} \rfloor$, $\dots$ contains infinitely many odd numbers.

1985 ITAMO, 5
Tags: modular arithmetic
A sequence of integers $a_1$, $a_2$, $a_3$, $\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?

2009 Turkey MO (2nd round), 1
Tags: modular arithmetic , number theory
Find all prime numbers $p$ for which $p^3-4p+9$ is a perfect square.

2012 Romanian Masters In Mathematics, 1
Tags: induction , modular arithmetic , graph theory , combinatorics
Given a finite number of boys and girls, a [i]sociable set of boys[/i] is a set of boys such that every girl knows at least one boy in that set; and a [i]sociable set of girls[/i] is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.) [i](Poland) Marek Cygan[/i]

1978 Romania Team Selection Test, 1
Tags: number theory , modular arithmetic
Show that for every natural number $ a\ge 3, $ there are infinitely many natural numbers $ n $ such that $ a^n\equiv 1\pmod n . $ Does this hold for $ n=2? $

2003 China Team Selection Test, 2
Tags: quadratic , modular arithmetic , combinatorics
Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.

2014 NIMO Problems, 4
Tags: pigeonhole principle , modular arithmetic
Prove that there exist integers $a$, $b$, $c$ with $1 \le a < b < c \le 25$ and \[ S(a^6+2014) = S(b^6+2014) = S(c^6+2014) \] where $S(n)$ denotes the sum of the decimal digits of $n$. [i]Proposed by Evan Chen[/i]

2005 National Olympiad First Round, 18
Tags: modular arithmetic , quadratic
How many integers $0\leq x < 121$ are there such that $x^5+5x^2 + x + 1 \equiv 0 \pmod{121}$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

2013 Turkey Team Selection Test, 1
Tags: polynomial , greatest common divisor , modular arithmetic , number theory , algebra
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.