Olympiad problems

2009 Putnam, A4

Let $ S$ be a set of rational numbers such that (a) $ 0\in S;$ (b) If $ x\in S$ then $ x\plus{}1\in S$ and $ x\minus{}1\in S;$ and (c) If $ x\in S$ and $ x\notin\{0,1\},$ then $ \frac{1}{x(x\minus{}1)}\in S.$ Must $ S$ contain all rational numbers?

2011 ELMO Shortlist, 4

Tags: modular arithmetic , number theory , quadratic

Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and \[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\] [i]Alex Zhu.[/i] [hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]

1977 Germany Team Selection Test, 4

Tags: modular arithmetic , divisibility , digit , decimal representation , digit sum

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

2006 Team Selection Test For CSMO, 1

Tags: relatively prime , modular arithmetic , number theory , quadratic , difference of squares , algebra

Find all the pairs of positive numbers such that the last digit of their sum is 3, their difference is a primer number and their product is a perfect square.

1977 Germany Team Selection Test, 3

Tags: number theory , modular arithmetic , sequence , additive 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 $p \neq q.$

2001 Junior Balkan MO, 1

Tags: modular arithmetic , geometry

Solve the equation $a^3+b^3+c^3=2001$ in positive integers. [i]Mircea Becheanu, Romania[/i]

2005 Germany Team Selection Test, 3

Tags: algorithm , modular arithmetic , induction , number theory , inequalities

We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.

2012 AIME Problems, 11

Tags: analytic geometry , geometry , slope , modular arithmetic , invariant , graphing lines , rotation

A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n=(x_n,y_n)$, the frog jumps to $P_{n+1}$, which may be any of the points $(x_n+7, y_n+2)$, $(x_n+2,y_n+7)$, $(x_n-5, y_n-10)$, or $(x_n-10,y_n-5)$. There are $M$ points $(x,y)$ with $|x|+|y| \le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000$.

2006 India IMO Training Camp, 3

Tags: modular arithmetic , combinatorics , invariant

There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

PEN A Problems, 41

Tags: modular arithmetic , divisibility theory

Show that there are infinitely many composite numbers $n$ such that $3^{n-1}-2^{n-1}$ is divisible by $n$.

2007 Junior Balkan Team Selection Tests - Romania, 3

Tags: modular arithmetic , combinatorics , induction

Consider a $n$x$n$ table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a $2$x$2$ square which contains an odd number of unit squares white colored.

2007 Pan African, 1

Tags: modular arithmetic , number theory

Find all natural numbers $N$ consisting of exactly $1112$ digits (in decimal notation) such that: (a) The sum of the digits of $N$ is divisible by $2000$; (b) The sum of the digits of $N+1$ is divisible by $2000$; (c) $1$ is a digit of $N$.

2009 Romania Team Selection Test, 2

Tags: number theory , modular arithmetic

Let $a$ and $n$ be two integers greater than $1$. Prove that if $n$ divides $(a-1)^k$ for some integer $k\geq 2$, then $n$ also divides $a^{n-1}+a^{n-2}+\cdots+a+1$.

2014 Contests, 3

Tags: combinatorial geometry , modular arithmetic , combinatorics , euler , discrete intermediate value theorem

For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\frac{S_1}{S_2}=\frac{39}{64}.$

2005 National Olympiad First Round, 6

Tags: modular arithmetic

Which of the following divides $3^{3n+1} + 5^{3n+2}+7^{3n+3}$ for every positive integer $n$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 53 $

1970 IMO Shortlist, 4

Tags: modular arithmetic , product , partition , number theory

Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.

1988 IMO Shortlist, 1

Tags: modular arithmetic , linear algebra , algebra , linear recurrence , sequence , divisibility

An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.

2010 Contests, 1

Tags: modular arithmetic , number theory

[b]a) [/b]Is the number $ 1111\cdots11$ (with $ 2010$ ones) a prime number? [b]b)[/b] Prove that every prime factor of $ 1111\cdots11$ (with $ 2011$ ones) is of the form $ 4022j\plus{}1$ where $ j$ is a natural number.

2009 India Regional Mathematical Olympiad, 2

Tags: number theory , modular arithmetic

Show that there is no integer $ a$ such that $ a^2 \minus{} 3a \minus{} 19$ is divisible by $ 289$.

2014 NIMO Problems, 4

Tags: logarithm , modular arithmetic , floor function , euler

Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer. Compute the remainder when $3^n$ is divided by $1000$.

1992 China Team Selection Test, 3

Tags: modular arithmetic , number theory , quadratic

For any prime $p$, prove that there exists integer $x_0$ such that $p | (x^2_0 - x_0 + 3)$ $\Leftrightarrow$ there exists integer $y_0$ such that $p | (y^2_0 - y_0 + 25).$

2004 Tournament Of Towns, 4

Tags: modular arithmetic , number theory

A positive integer $a > 1$ is given (in decimal notation). We copy it twice and obtain a number $b = \overline{aa}$ which happens to be a multiple of $a^2$. Find all possible values of $b/a^2$.

1994 Dutch Mathematical Olympiad, 3

Tags: modular arithmetic , number theory , geometry , 3d geometry

$ (a)$ Prove that every multiple of $ 6$ can be written as a sum of four cubes. $ (b)$ Prove that every integer can be written as a sum of five cubes.

2010 Kazakhstan National Olympiad, 1

Tags: modular arithmetic , number theory , geometry

It is given that for some $n \in \mathbb{N}$ there exists a natural number $a$, such that $a^{n-1} \equiv 1 \pmod{n}$ and that for any prime divisor $p$ of $n-1$ we have $a^{\frac{n-1}{p}} \not \equiv 1 \pmod{n}$. Prove that $n$ is a prime.

1978 IMO Longlists, 52

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

Let $p$ be a prime and $A = \{a_1, \ldots , a_{p-1} \}$ an arbitrary subset of the set of natural numbers such that none of its elements is divisible by $p$. Let us define a mapping $f$ from $\mathcal P(A)$ (the set of all subsets of $A$) to the set $P = \{0, 1, \ldots, p - 1\}$ in the following way: $(i)$ if $B = \{a_{i_{1}}, \ldots , a_{i_{k}} \} \subset A$ and $\sum_{j=1}^k a_{i_{j}} \equiv n \pmod p$, then $f(B) = n,$ $(ii)$ $f(\emptyset) = 0$, $\emptyset$ being the empty set. Prove that for each $n \in P$ there exists $B \subset A$ such that $f(B) = n.$