Olympiad problems

2004 Mediterranean Mathematics Olympiad, 1

Find all natural numbers $m$ such that \[1! \cdot 3! \cdot 5! \cdots (2m-1)! = \biggl( \frac{m(m+1)}{2}\biggr) !.\]

2010 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: inequalities , number theory , number theory unsolved

Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$. [list](a) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the smallest? (b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?[/list]

1997 Greece National Olympiad, 3

Tags: number theory unsolved , number theory

Find all integer solutions to \[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997}.\]

1997 Greece National Olympiad, 4

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

A polynomial $P$ with integer coefficients has at least $13$ distinct integer roots. Prove that if an integer $n$ is not a root of $P$, then $|P(n)| \geq 7 \cdot 6!^2$, and give an example for equality.

2012 Korea - Final Round, 2

Tags: number theory , relatively prime , number theory unsolved

Let $n$ be a given positive integer. Prove that there exist infinitely many integer triples $(x,y,z)$ such that \[nx^2+y^3=z^4,\ \gcd (x,y)=\gcd (y,z)=\gcd (z,x)=1.\]

2013 Baltic Way, 17

Tags: modular arithmetic , number theory unsolved , number theory

Let $c$ and $n > c$ be positive integers. Mary's teacher writes $n$ positive integers on a blackboard. Is it true that for all $n$ and $c$ Mary can always label the numbers written by the teacher by $a_1,\ldots, a_n$ in such an order that the cyclic product $(a_1-a_2)\cdot(a_2-a_3)\cdots(a_{n-1}-a_n)\cdot(a_n-a_1)$ would be congruent to either $0$ or $c$ modulo $n$?

2020 Thailand TST, 4

Tags: induction , LaTeX , number theory unsolved , number theory

For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.

2014 India IMO Training Camp, 3

Tags: function , number theory unsolved , number theory

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that $f^{k}((n,m))=(m,n)$,where $f^{k}(x)=f(f(f(...f(x))))$,$f$ being composed with itself $k$ times.

2004 Federal Competition For Advanced Students, P2, 2

Tags: number theory , prime numbers , number theory unsolved

Show that every set $ \{p_1,p_2,\dots,p_k\}$ of prime numbers fulfils the following: The sum of all unit fractions (that are fractions of the type $ \frac{1}{n}$), whose denominators are exactly the $ k$ given prime factors (but in arbitrary powers with exponents unequal zero), is an unit fraction again. How big is this sum if $ \frac{1}{2004}$ is among this summands? Show that for every set $ \{p_1,p_2,\dots,p_k\}$ containing $ k$ prime numbers ($ k>2$) is the sum smaller than $ \frac{1}{N}$ with $ N=2\cdot 3^{k-2}(k-2)!$

2005 Korea National Olympiad, 1

Tags: algorithm , number theory , greatest common divisor , relatively prime , Euclidean algorithm , number theory unsolved

For two positive integers a and b, which are relatively prime, find all integer that can be the great common divisor of $a+b$ and $\frac{a^{2005}+b^{2005}}{a+b}$.

2007 Croatia Team Selection Test, 1

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

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

2012 Kazakhstan National Olympiad, 3

Tags: induction , number theory unsolved , number theory

The sequence $a_{n}$ defined as follows: $a_{1}=4, a_{2}=17$ and for any $k\geq1$ true equalities $a_{2k+1}=a_{2}+a_{4}+...+a_{2k}+(k+1)(2^{2k+3}-1)$ $a_{2k+2}=(2^{2k+2}+1)a_{1}+(2^{2k+3}+1)a_{3}+...+(2^{3k+1}+1)a_{2k-1}+k$ Find the smallest $m$ such that $(a_{1}+...a_{m})^{2012^{2012}}-1$ divided $2^{2012^{2012}}$

2017 Bundeswettbewerb Mathematik, 1

Tags: blackboard , combinatorics , combinatorics unsolved , number theory , number theory unsolved , divisible , Divisibility

The numbers $1,2,3,\dots,2017$ are on the blackboard. Amelie and Boris take turns removing one of those until only two numbers remain on the board. Amelie starts. If the sum of the last two numbers is divisible by $8$, then Amelie wins. Else Boris wins. Who can force a victory?

2020 Durer Math Competition Finals, 1

Tags: induction , modular arithmetic , strong induction , number theory , relatively prime , prime numbers , number theory unsolved

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant. [The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]

1996 USAMO, 2

Tags: ratio , inequalities , number theory unsolved , number theory

For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of the elements of $S$. Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\sigma(S)$ as $S$ ranges over the nonempty subsets of $A$. Prove that this collection of sums can be partitioned into $n$ classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.

2011 India IMO Training Camp, 3

Tags: function , number theory unsolved , number theory

Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that \[(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0\] for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that \[a_{k+2011}=a_{k+2011^{2011}}.\]

2007 India National Olympiad, 3

Tags: quadratics , inequalities , number theory unsolved , number theory

Let $ m$ and $ n$ be positive integers such that $ x^2 \minus{} mx \plus{}n \equal{} 0$ has real roots $ \alpha$ and $ \beta$. Prove that $ \alpha$ and $ \beta$ are integers [b]if and only if[/b] $ [m\alpha] \plus{} [m\beta]$ is the square of an integer. (Here $ [x]$ denotes the largest integer not exceeding $ x$)

2006 German National Olympiad, 1

Tags: number theory unsolved , number theory

Find all $n\in \mathbb Z^+$, so that \[ z_n = \underbrace{ 101\dots101}_{2n+1 \text{ digits} } \] is prime.

1979 IMO Longlists, 49

Tags: inequalities , number theory unsolved , number theory

Let there be given two sequences of integers $f_i(1), f_i(2), \cdots (i = 1, 2)$ satisfying: $(i) f_i(nm) = f_i(n)f_i(m)$ if $\gcd(n,m) = 1$; $(ii)$ for every prime $P$ and all $k = 2, 3, 4, \cdots$, $f_i(P^k) = f_i(P)f_i(P^{k-1}) - P^2f(P^{k-2}).$ Moreover, for every prime $P$: $(iii) f_1(P) = 2P,$ $(iv) f_2(P) < 2P.$ Prove that $|f_2(n)| < f_1(n)$ for all $n$.

2010 South East Mathematical Olympiad, 4

Tags: number theory unsolved , number theory

Let $a$ and $b$ be positive integers such that $1\leq a<b\leq 100$. If there exists a positive integer $k$ such that $ab|a^k+b^k$, we say that the pair $(a, b)$ is good. Determine the number of good pairs.

2010 Tournament Of Towns, 1

Tags: number theory unsolved , number theory

Each of six fruit baskets contains pears, plums and apples. The number of plums in each basket equals the total number of apples in all other baskets combined while the number of apples in each basket equals the total number of pears in all other baskets combined. Prove that the total number of fruits is a multiple of $31$.

2006 China Girls Math Olympiad, 3

Tags: number theory unsolved , number theory

Show that for any $i=1,2,3$, there exist infinity many positive integer $n$, such that among $n$, $n+2$ and $n+28$, there are exactly $i$ terms that can be expressed as the sum of the cubes of three positive integers.

2006 Estonia National Olympiad, 3

Tags: modular arithmetic , number theory unsolved , number theory

Prove or disprove the following statements. a) For every integer $ n \ge 3$, there exist $ n$ pairwise distinct positive integers such that the product of any two of them is divisible by the sum of the remaining $ n \minus{} 2$ numbers. b) For some integer $ n \ge 3$, there exist $ n$ pairwise distinct positive integers, such that the sum of any $ n \minus{} 2$ of them is divisible by the product of the remaining two numbers.

2014 South East Mathematical Olympiad, 6

Tags: irrational number , number theory unsolved , number theory

Let $a,b$ and $c$ be integers and $r$ a real number such that $ar^2+br+c=0$ with $ac\not =0$.Prove that $\sqrt{r^2+c^2}$ is an irrational number

2007 Germany Team Selection Test, 2

Tags: LaTeX , number theory unsolved , number theory

Find all quadruple $ (m,n,p,q) \in \mathbb{Z}^4$ such that \[ p^m q^n \equal{} (p\plus{}q)^2 \plus{} 1.\]