Olympiad problems

1987 Romania Team Selection Test, 10

Tags: modular arithmetic , induction , function , algebra , polynomial , number theory

Let $a,b,c$ be integer numbers such that $(a+b+c) \mid (a^{2}+b^{2}+c^{2})$. Show that there exist infinitely many positive integers $n$ such that $(a+b+c) \mid (a^{n}+b^{n}+c^{n})$. [i]Laurentiu Panaitopol[/i]

2000 Stanford Mathematics Tournament, 23

Tags: modular arithmetic

What are the last two digits of ${7^{7^{7^7}}}$?

1979 USAMO, 1

Tags: calculus , integration , modular arithmetic

Determine all non-negative integral solutions $ (n_{1},n_{2},\dots , n_{14}) $ if any, apart from permutations, of the Diophantine Equation \[n_{1}^{4}+n_{2}^{4}+\cdots+n_{14}^{4}=1,599.\]

2013 China Team Selection Test, 2

Tags: limit , modular arithmetic , quadratic , number theory

Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying: $(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $; $(2)$ For any positive integer $n$, $a_n<1.01^n K$; $(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.

2007 Greece Junior Math Olympiad, 2

Tags: modular arithmetic , number theory

If $n$ is is an integer such that $4n+3$ is divisible by $11,$ find the form of $n$ and the remainder of $n^{4}$ upon division by $11$.

2014 Cono Sur Olympiad, 2

Tags: modular arithmetic , number theory

A pair of positive integers $(a,b)$ is called [i]charrua[/i] if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called [i]non-charrua[/i]. a) Prove that there are infinite [i]non-charrua[/i] pairs. b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is [i]charrua[/i].

2013 NIMO Problems, 2

Tags: function , modular arithmetic

Let $f$ be a function from positive integers to positive integers where $f(n) = \frac{n}{2}$ if $n$ is even and $f(n) = 3n+1$ if $n$ is odd. If $a$ is the smallest positive integer satisfying \[ \underbrace{f(f(\cdots f}_{2013\ f\text{'s}} (a)\cdots)) = 2013, \] find the remainder when $a$ is divided by $1000$. [i]Based on a proposal by Ivan Koswara[/i]

1969 IMO Shortlist, 13

Tags: modular arithmetic , number theory , divisibility , additive number theory

$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?

2003 CHKMO, 2

Tags: modular arithmetic , combinatorics

In conference there $n>2$ mathematicians. Every two mathematicians communicate in one of the $n$ offical languages of the conference. For any three different offical languages the exists three mathematicians who communicate with each other in these three languages. Find all $n$ such that this is possible.

2008 All-Russian Olympiad, 8

Tags: induction , symmetry , modular arithmetic , combinatorial geometry , combinatorics

In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

1994 Baltic Way, 9

Tags: search , modular arithmetic , number theory

Find all pairs of positive integers $(a,b)$ such that $2^a+3^b$ is the square of an integer.

2023 Romania EGMO TST, P2

Tags: function , pigeonhole principle , induction , modular arithmetic , number theory

Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.

2005 Manhattan Mathematical Olympiad, 3

Tags: modular arithmetic , number theory

Prove that for any three pairwise different integer numbers $x,y,z$ the expression $(x-y)^5 + (y-z)^5 + (z-x)^5$ is divisible by $5(x-y)(y-z)z-x)$.

2011 Putnam, A4

Tags: linear algebra , matrix , modular arithmetic , vector , polynomial , putnam matrices

For which positive integers $n$ is there an $n\times n$ matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd?

2004 Pan African, 1

Tags: number theory , modular arithmetic

Do there exist positive integers $m$ and $n$ such that: \[ 3n^2+3n+7=m^3 \]

2010 Olympic Revenge, 1

Tags: modular arithmetic , number theory

Prove that the number of ordered triples $(x, y, z)$ such that $(x+y+z)^2 \equiv axyz \mod{p}$, where $gcd(a, p) = 1$ and $p$ is prime is $p^2 + 1$.

2003 CHKMO, 4

Tags: modular arithmetic , quadratic , number theory

Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.

2014 Brazil National Olympiad, 2

Tags: modular arithmetic , number theory

Find all integers $n$, $n>1$, with the following property: for all $k$, $0\le k < n$, there exists a multiple of $n$ whose digits sum leaves a remainder of $k$ when divided by $n$.

2023 China Western Mathematical Olympiad, 4

Tags: modular arithmetic , number theory

Let ${p}$ be a prime. $a,b,c\in\mathbb Z,\gcd(a,p)=\gcd(b,p)=\gcd(c,p)=1.$ Prove that: $\exists x_1,x_2,x_3,x_4\in\mathbb Z,| x_1|,|x_2|,|x_3|,|x_4|<\sqrt p,$ satisfying $$ax_1x_2+bx_3x_4\equiv c\pmod p.$$ [i]Proposed by Wang Guangting[/i]

1995 Cono Sur Olympiad, 3

Tags: floor function , function , modular arithmetic , inequalities , number theory , relatively prime

Let $n$ be a natural number and $f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor$($\lfloor$ $\rfloor$ denotes the floor function). 1. Show that if for some integer $r$: $f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times), then $n$ is multiple of $1995$. 2. Show that if $n$ is multiple of 1995, then there exists r such that:$f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times). Determine $r$ if $n=1995.500=997500$

2001 National Olympiad First Round, 3

Tags: modular arithmetic , quadratic

How many primes $p$ are there such that $2p^4-7p^2+1$ is equal to square of an integer? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None of the preceding} $

2006 Italy TST, 2

Tags: modular arithmetic , euler , number theory

Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$. a) Find all $n$ such that $A_{n}\neq \emptyset$ b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero. c) Is there $n$ such that $|{A_{n}}| = 130$?

2002 China Team Selection Test, 3

Tags: modular arithmetic , induction , number theory

Find all groups of positive integers $ (a,x,y,n,m)$ that satisfy $ a(x^n \minus{} x^m) \equal{} (ax^m \minus{} 4) y^2$ and $ m \equiv n \pmod{2}$ and $ ax$ is odd.

2006 Balkan MO, 4

Tags: modular arithmetic , function , algebra

Let $m$ be a positive integer and $\{a_n\}_{n\geq 0}$ be a sequence given by $a_0 = a \in \mathbb N$, and \[ a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + m & \textrm{ otherwise. } \end{cases} \] Find all values of $a$ such that the sequence is periodical (starting from the beginning).

PEN E Problems, 7

Tags: modular arithmetic , euler

Show that there exists a positive integer $ k$ such that $ k \cdot 2^{n} \plus{} 1$ is composite for all $ n \in \mathbb{N}_{0}$.