Olympiad problems

2014 Online Math Open Problems, 30

Let $p = 2^{16}+1$ be an odd prime. Define $H_n = 1+ \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$. Compute the remainder when \[ (p-1)! \sum_{n = 1}^{p-1} H_n \cdot 4^n \cdot \binom{2p-2n}{p-n} \] is divided by $p$. [i]Proposed by Yang Liu[/i]

2004 France Team Selection Test, 1

Tags: number theory , quadratic , modular arithmetic

If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$. Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?

2012 EGMO, 5

Tags: number theory , modular arithmetic , equation , gcd , prime

The numbers $p$ and $q$ are prime and satisfy \[\frac{p}{{p + 1}} + \frac{{q + 1}}{q} = \frac{{2n}}{{n + 2}}\] for some positive integer $n$. Find all possible values of $q-p$. [i]Luxembourg (Pierre Haas)[/i]

2012 All-Russian Olympiad, 1

Tags: modular arithmetic , number theory

Let $a_1,\ldots ,a_{10}$ be distinct positive integers, all at least $3$ and with sum $678$. Does there exist a positive integer $n$ such that the sum of the $20$ remainders of $n$ after division by $a_1,a_2,\ldots ,a_{10},2a_1,2a_2,\ldots ,2a_{10}$ is $2012$?

2008 Germany Team Selection Test, 1

Tags: floor function , modular arithmetic , inequalities

Show that there is a digit unequal to 2 in the decimal represesentation of $ \sqrt [3]{3}$ between the $ 1000000$-th und $ 3141592$-th position after decimal point.

2005 AMC 10, 16

Tags: modular arithmetic

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is $ 6$. How many two-digit numbers have this property? $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 9\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 19$

2017 Serbia National Math Olympiad, 1

Tags: modular arithmetic

Let $a$ be a positive integer.Suppose that $\forall n$ ,$\exists d$, $d\not =1$, $d\equiv 1\pmod n$ ,$d\mid n^2a-1$.Prove that $a$ is a perfect square.

2013 Iran Team Selection Test, 4

Tags: analytic geometry , modular arithmetic , geometry , parallelogram , vector , number theory , combinatorics

$m$ and $n$ are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows: Starting from one of the hexagons, the Donkey moves $m$ cells in one of the $6$ directions, then it turns $60$ degrees clockwise and after that moves $n$ cells in this new direction until it reaches it's final cell. At most how many cells are in the Philosopher's chessboard such that one cannot go from anyone of them to the other with a finite number of movements of the Donkey? [i]Proposed by Shayan Dashmiz[/i]

2001 Putnam, 5

Tags: modular arithmetic

Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.

2013 Taiwan TST Round 1, 5

Tags: modular arithmetic , number theory , equation

An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers. [b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$. [b]b)[/b] Decide whether $a=2$ is friendly.

2005 Korea - Final Round, 5

Tags: modular arithmetic , number theory

Find all positive integers $m$ and $n$ such that both $3^{m}+1$ and $3^{n}+1$ are divisible by $mn$.

PEN O Problems, 44

Tags: modular arithmetic

A set $C$ of positive integers is called good if for every integer $k$ there exist distinct $a, b \in C$ such that the numbers $a+k$ and $b+k$ are not relatively prime. Prove that if the sum of the elements of a good set $C$ equals $2003$, then there exists $c \in C$ such that the set $C-\{c\}$ is good.

2011 Middle European Mathematical Olympiad, 8

Tags: modular arithmetic , greatest common divisor , arithmetic sequence , relatively prime , algebra , system of equations , number theory

We call a positive integer $n$ [i]amazing[/i] if there exist positive integers $a, b, c$ such that the equality \[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\] holds. Prove that there exist $2011$ consecutive positive integers which are [i]amazing[/i]. [b]Note.[/b] By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.

2008 Harvard-MIT Mathematics Tournament, 22

Tags: function , modular arithmetic

For a positive integer $ n$, let $ \theta(n)$ denote the number of integers $ 0 \leq x < 2010$ such that $ x^2 \minus{} n$ is divisible by $ 2010$. Determine the remainder when $ \displaystyle \sum_{n \equal{} 0}^{2009} n \cdot \theta(n)$ is divided by $ 2010$.

2011 Iran MO (3rd Round), 3

Tags: modular arithmetic , number theory

Let $k$ be a natural number such that $k\ge 7$. How many $(x,y)$ such that $0\le x,y<2^k$ satisfy the equation $73^{73^x}\equiv 9^{9^y} \pmod {2^k}$? [i]Proposed by Mahyar Sefidgaran[/i]

2017 Azerbaijan EGMO TST, 4

Tags: calculus , integration , induction , modular arithmetic , number theory

Find all natural numbers a, b such that $ a^{n}\plus{} b^{n} \equal{} c^{n\plus{}1}$ where c and n are naturals.

2017 Iran Team Selection Test, 4

Tags: number theory , prime number , modular arithmetic

We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$. Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$. Is there always a number $x$ that satisfies all the equations? [i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]

2003 AIME Problems, 13

Tags: probability , counting , distinguishability , modular arithmetic , function , limit , vector

A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

2007 AMC 12/AHSME, 24

Tags: trigonometry , floor function , modular arithmetic , algebra , function , domain , arithmetic sequence

For each integer $ n > 1,$ let $ F(n)$ be the number of solutions of the equation $ \sin x \equal{} \sin nx$ on the interval $ [0,\pi].$ What is $ \sum_{n \equal{} 2}^{2007}F(n)?$ $ \textbf{(A)}\ 2,014,524 \qquad \textbf{(B)}\ 2,015,028 \qquad \textbf{(C)}\ 2,015,033 \qquad \textbf{(D)}\ 2,016,532 \qquad \textbf{(E)}\ 2,017,033$

2009 Germany Team Selection Test, 2

Tags: function , number theory , modular arithmetic , divisor , functional equation

For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$. [*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list] [i]Proposed by Bruno Le Floch, France[/i]

2014 European Mathematical Cup, 1

Tags: modular arithmetic , prime factorization , number theory

Prove that there exist infinitely many positive integers which cannot be written in form $a^{d(a)}+b^{d(b)}$ for some positive integers $a$ and $b$ For positive integer $d(a)$ denotes number of positive divisors of $a$ [i]Proposed by Borna Vukorepa[/i]

1973 Bundeswettbewerb Mathematik, 3

Tags: modular arithmetic , combinatorics

For covering the floor of a rectangular room rectangular tiles of sizes $2 \times 2$ and $4 \times 1$ were used. Show that it's not possible to cover the floor if there is one plate less of one type and one more of the other type.

2007 Romania Team Selection Test, 4

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

i) Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, the $p$-th term of the progression is also prime. ii) Find all polynomials $f(X) \in \mathbb{Z}[X]$, such that there exist $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, $| f(p) |$ is also prime. [i]Dan Schwarz[/i]

2003 Canada National Olympiad, 2

Tags: modular arithmetic , euler , function , number theory

Find the last three digits of the number $2003^{{2002}^{2001}}$.

2008 Iran Team Selection Test, 11

Tags: function , modular arithmetic , number theory , relatively prime , functional equation

$ k$ is a given natural number. Find all functions $ f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $ m,n\in\mathbb{N}$ the following holds: \[ f(m)\plus{}f(n)\mid (m\plus{}n)^k\]