Olympiad problems

2011 Mongolia Team Selection Test, 3

Tags: modular arithmetic , algebra , polynomial , Vieta , number theory unsolved , number theory

Let $m$ and $n$ be positive integers such that $m>n$ and $m \equiv n \pmod{2}$. If $(m^2-n^2+1) \mid n^2-1$, then prove that $m^2-n^2+1$ is a perfect square. (proposed by G. Batzaya, folklore)

2002 All-Russian Olympiad, 4

Tags: floor function , pigeonhole principle , inequalities , number theory unsolved , number theory

From the interval $(2^{2n},2^{3n})$ are selected $2^{2n-1}+1$ odd numbers. Prove that there are two among the selected numbers, none of which divides the square of the other.

2007 Estonia Math Open Senior Contests, 4

Tags: algebra , polynomial , number theory unsolved , number theory

The Fibonacci sequence is determined by conditions $ F_0 \equal{} 0, F1 \equal{} 1$, and $ F_k\equal{}F_{k\minus{}1}\plus{}F_{k\minus{}2}$ for all $ k \ge 2$. Let $ n$ be a positive integer and let $ P(x) \equal{} a_mx^m \plus{}. . .\plus{} a_1x\plus{} a_0$ be a polynomial that satisfies the following two conditions: (1) $ P(F_n) \equal{} F_{n}^{2}$ ; (2) $ P(F_k) \equal{} P(F_{k\minus{}1}) \plus{} P(F_{k\minus{}2}$ for all $ k \ge 2$. Find the sum of the coefficients of P.

2006 USA Team Selection Test, 4

Tags: induction , modular arithmetic , number theory unsolved , number theory

Let $n$ be a positive integer. Find, with proof, the least positive integer $d_{n}$ which cannot be expressed in the form \[\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}},\] where $a_{i}$ and $b_{i}$ are nonnegative integers for each $i.$

2014 European Mathematical Cup, 1

Tags: modular arithmetic , number theory , prime factorization , number theory unsolved

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]

1997 Federal Competition For Advanced Students, P2, 1

Tags: linear algebra , matrix , number theory unsolved , number theory

Let $ a$ be a fixed integer. Find all integer solutions $ x,y,z$ of the system: $ 5x\plus{}(a\plus{}2)y\plus{}(a\plus{}2)z\equal{}a,$ $ (2a\plus{}4)x\plus{}(a^2\plus{}3)y\plus{}(2a\plus{}2)z\equal{}3a\minus{}1,$ $ (2a\plus{}4)x\plus{}(2a\plus{}2)y\plus{}(a^2\plus{}3)z\equal{}a\plus{}1.$

2010 South africa National Olympiad, 1

Tags: number theory unsolved , number theory

For a positive integer $n$, $S(n)$ denotes the sum of its digits and $U(n)$ its unit digit. Determine all positive integers $n$ with the property that \[n = S(n) + U(n)^2.\]

1997 Taiwan National Olympiad, 6

Tags: number theory unsolved , number theory

Show that every number of the form $2^{p}3^{q}$ , where $p,q$ are nonnegative integers, divides some number of the form $a_{2k}10^{2k}+a_{2k-2}10^{2k-2}+...+a_{2}10^{2}+a_{0}$, where $a_{2i}\in\{1,2,...,9\}$

2009 Germany Team Selection Test, 2

Tags: induction , number theory , prime numbers , number theory unsolved

Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$

1994 Moldova Team Selection Test, 7

Tags: geometry , geometric transformation , number theory unsolved , number theory

Given an odd $n$, prove that there exist $2n$ integers $a_1,a_2,\cdots ,a_n$; $b_1,b_2,\cdots ,b_n$, such that for any integer $k$ ($0<k<n$), the following $3n$ integers: $a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}$ ($i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n$) are of different remainders on division by $3n$.

2011 Mongolia Team Selection Test, 1

Tags: quadratics , modular arithmetic , Ring Theory , Diophantine equation , number theory unsolved , number theory

Let $A=\{a^2+13b^2 \mid a,b \in\mathbb{Z}, b\neq0\}$. Prove that there a) exist b) exist infinitely many $x,y$ integer pairs such that $x^{13}+y^{13} \in A$ and $x+y \notin A$. (proposed by B. Bayarjargal)

2007 Croatia Team Selection Test, 1

Tags: calculus , integration , number theory unsolved , number theory

Find integral solutions to the equation \[(m^{2}-n^{2})^{2}=16n+1.\]

2010 China Team Selection Test, 3

Tags: pigeonhole principle , ceiling function , number theory unsolved , number theory

Let $k>1$ be an integer, set $n=2^{k+1}$. Prove that for any positive integers $a_1<a_2<\cdots<a_n$, the number $\prod_{1\leq i<j\leq n}(a_i+a_j)$ has at least $k+1$ different prime divisors.

2003 Iran MO (3rd Round), 10

Tags: number theory unsolved , number theory

let p be a prime and a and n be natural numbers such that (p^a -1 )/ (p-1) = 2 ^n find the number of natural divisors of na. :)

1972 IMO Longlists, 46

Tags: linear algebra , matrix , number theory unsolved , number theory

Numbers $1, 2,\cdots, 16$ are written in a $4\times 4$ square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers $1$ and $16$ lie in opposite corners. Prove that the sum of any two numbers symmetric with respect to the center of the square equals $17$.

1996 Taiwan National Olympiad, 5

Tags: number theory unsolved , number theory , Integer sequence

Dertemine integers $a_{1},a_{2},...,a_{99}=a_{0}$ satisfying $|a_{k}-a_{k-1}|\geq 1996$ for all $k=1,2,...,99$, such that $m=\max_{1\leq k\leq 99} |a_{k}-a_{k-1}|$ is minimum possible, and find the minimum value $m^{*}$ of $m$.

2016 Serbia National Math Olympiad, 6

Tags: number theory , number theory unsolved

Let $a_1, a_2, \dots, a_{2^{2016}}$ be positive integers not bigger than $2016$. We know that for each $n \leq 2^{2016}$, $a_1a_2 \dots a_{n} +1 $ is a perfect square. Prove that for some $i $ , $a_i=1$.

2006 MOP Homework, 2

Tags: number theory unsolved , number theory

Let $c$ be a fixed positive integer, and let ${a_n}^{\inf}_{n=1}$ be a sequence of positive integers such that $a_n < a_{n+1} < a_n+c$ for every positive integer $n$. Let $s$ denote the infinite string of digits obtained by writing the terms in the sequence consecutively from left to right, starting from the first term. For every positive integer $k$, let $s_k$ denote the number whose decimal representation is identical to the $k$ most left digits of $s$. Prove that for every positive integer $m$ there exists a positive integer $k$ such that $s_k$ is divisible by $m$.

2004 Postal Coaching, 2

Tags: algorithm , Stanford , college , least common multiple , greatest common divisor , number theory unsolved , number theory

(a) Find all triples $(x,y,z)$ of positive integers such that $xy \equiv 2 (\bmod{z})$ , $yz \equiv 2 (\bmod{x})$ and $zx \equiv 2 (\bmod{y} )$ (b) Let $n \geq 1$ be an integer. Give an algoritm to determine all triples $(x,y,z)$ such that '2' in part (a) is replaced by 'n' in all three congruences.

2010 Mexico National Olympiad, 3

Tags: number theory unsolved , number theory

Let $p$, $q$, and $r$ be distinct positive prime numbers. Show that if \[pqr\mid (pq)^r+(qr)^p+(rp)^q-1,\] then \[(pqr)^3\mid 3((pq)^r+(qr)^p+(rp)^q-1).\]

2000 Junior Balkan Team Selection Tests - Romania, 1

Tags: modular arithmetic , number theory unsolved , number theory

Solve in natural the equation $9^x-3^x=y^4+2y^3+y^2+2y$ _____________________________ Azerbaijan Land of the Fire :lol:

1983 Federal Competition For Advanced Students, P2, 4

Tags: function , number theory unsolved , number theory

The sequence $ (x_n)_{n \in \mathbb{N}}$ is defined by $ x_1\equal{}2, x_2\equal{}3,$ and $ x_{2m\plus{}1}\equal{}x_{2m}\plus{}x_{2m\minus{}1}$ for $ m \ge 1;$ $ x_{2m}\equal{}x_{2m\minus{}1}\plus{}2x_{2m\minus{}2}$ for $ m \ge 2.$ Determine $ x_n$ as a function of $ n$.

2003 IberoAmerican, 3

Tags: quadratics , induction , modular arithmetic , number theory unsolved , number theory

The sequences $(a_n),(b_n)$ are defined by $a_0=1,b_0=4$ and for $n\ge 0$ \[a_{n+1}=a_n^{2001}+b_n,\ \ b_{n+1}=b_n^{2001}+a_n\] Show that $2003$ is not divisor of any of the terms in these two sequences.

1991 Federal Competition For Advanced Students, P2, 6

Tags: calculus , induction , number theory unsolved , number theory

Find the number of ten-digit natural numbers (which do not start with zero) containing no block $ 1991$.

2013 Gulf Math Olympiad, 4

Tags: number theory , greatest common divisor , number theory unsolved

Let $m,n$ be integers. It is known that there are integers $a,b$ such that $am+bn=1$ if, and only if, the greatest common divisor of $m,n$ is 1. [i]You are not required to prove this[/i]. Now suppose that $p,q$ are different odd primes. In each case determine if there are integers $a,b$ such that $ap+bq=1$ so that the given condition is satisfied: [list] a. $p$ divides $b$ and $q$ divides $a$; b. $p$ divides $a$ and $q$ divides $b$; c. $p$ does not divide $a$ and $q$ does not divide $b$. [/list]