Olympiad problems

1971 IMO Longlists, 2

Let us denote by $s(n)= \sum_{d|n} d$ the sum of divisors of a positive integer $n$ ($1$ and $n$ included). If $n$ has at most $5$ distinct prime divisors, prove that $s(n) < \frac{77}{16} n.$ Also prove that there exists a natural number $n$ for which $s(n) < \frac{76}{16} n$ holds.

2012 All-Russian Olympiad, 2

Tags: modular arithmetic , number theory proposed , number theory

Does there exist natural numbers $a,b,c$ all greater than $10^{10}$ such that their product is divisible by each of these numbers increased by $2012$?

1978 Canada National Olympiad, 1

Tags: modular arithmetic , number theory proposed , number theory

Let $n$ be an integer. If the tens digit of $n^2$ is 7, what is the units digit of $n^2$?

2015 Serbia National Math Olympiad, 6

Tags: number theory , number theory proposed , exponential equations , congruence , Serbia

In nonnegative set of integers solve the equation: $$(2^{2015}+1)^x + 2^{2015}=2^y+1$$

2008 Bosnia and Herzegovina Junior BMO TST, 2

Tags: modular arithmetic , number theory proposed , number theory

Let $ x,y,z$ be positive integers. If $ 7$ divides $ (x\plus{}6y)(2x\plus{}5y)(3x\plus{}4y)$ than prove that $ 343$ also divides it.

2005 Croatia National Olympiad, 1

Tags: function , number theory proposed , number theory

Find all positive integer solutions of the equation $k!l! = k!+l!+m!.$

2002 Tournament Of Towns, 1

Tags: number theory proposed , number theory

John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary. John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well. What was Mary's Number?

2011 ELMO Problems, 5

Tags: modular arithmetic , quadratics , number theory proposed , number theory

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]

2007 Junior Tuymaada Olympiad, 1

Tags: number theory proposed , number theory

Positive integers $ a<b$ are given. Prove that among every $ b$ consecutive positive integers there are two numbers whose product is divisible by $ ab$.

2007 Korea - Final Round, 3

Tags: number theory proposed , number theory

Find all triples of $ (x, y, z)$ of positive intergers satisfying $ 1\plus{}{4}^{x}\plus{}{4}^{y}\equal{}z^2$.

2023 Romania EGMO TST, P2

Tags: function , pigeonhole principle , induction , modular arithmetic , number theory proposed , 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 $a \in \mathbb{N}$.

2004 Germany Team Selection Test, 1

Tags: analytic geometry , number theory proposed , number theory

Consider the real number axis (i. e. the $x$-axis of a Cartesian coordinate system). We mark the points $1$, $2$, ..., $2n$ on this axis. A flea starts at the point $1$. Now it jumps along the real number axis; it can jump only from a marked point to another marked point, and it doesn't visit any point twice. After the ($2n-1$)-th jump, it arrives at a point from where it cannot jump any more after this rule, since all other points are already visited. Hence, with its $2n$-th jump, the flea breaks this rule and gets back to the point $1$. Assume that the sum of the (non-directed) lengths of the first $2n-1$ jumps of the flea was $n\left(2n-1\right)$. Show that the length of the last ($2n$-th) jump is $n$.

2013 China Team Selection Test, 2

Tags: limit , modular arithmetic , quadratics , number theory proposed , 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.

2006 Germany Team Selection Test, 1

Tags: number theory , prime numbers , number theory proposed

Does there exist a natural number $n$ in whose decimal representation each digit occurs at least $2006$ times and which has the property that you can find two different digits in its decimal representation such that the number obtained from $n$ by interchanging these two digits is different from $n$ and has the same set of prime divisors as $n$ ?

1972 Bundeswettbewerb Mathematik, 2

Tags: number theory proposed , number theory

Prove: out of $ 79$ consecutive positive integers, one can find at least one whose sum of digits is divisible by $ 13$. Show that this isn't true for $ 78$ consecutive integers.

2006 Turkey Team Selection Test, 1

Tags: induction , quadratics , Gauss , modular arithmetic , number theory , prime numbers , number theory proposed

For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$.

2012 Cono Sur Olympiad, 4

Tags: number theory proposed , number theory

4. Find the biggest positive integer $n$, lesser thar $2012$, that has the following property: If $p$ is a prime divisor of $n$, then $p^2 - 1$ is a divisor of $n$.

1985 Dutch Mathematical Olympiad, 2

Tags: modular arithmetic , floor function , number theory proposed , number theory

Among the numbers $ 11n \plus{} 10^{10}$, where $ 1 \le n \le 10^{10}$ is an integer, how many are squares?

2019 International Zhautykov OIympiad, 2

Tags: izho , algebra , number theory , number theory proposed , inequalities

Find the biggest real number $C$, such that for every different positive real numbers $a_1,a_2...a_{2019}$ that satisfy inequality : $\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$

2022 Kazakhstan National Olympiad, 2

Tags: Kazakhstan , Kazakhstan 2022 , number theory proposed , number theory , prime numbers

Given a prime number $p$. It is known that for each integer $a$ such that $1<a<p/2$ there exist integer $b$ such that $p/2<b<p$ and $p|ab-1$. Find all such $p$.

2010 Indonesia TST, 1

Tags: algebra , polynomial , number theory , relatively prime , number theory proposed

find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \] for every natural numbers $ a $ and $ b $

2012 Korea National Olympiad, 3

Tags: modular arithmetic , quadratics , number theory proposed , number theory

Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes. \[ 2^m p^2 + 1 = q^5 \]

2004 Bundeswettbewerb Mathematik, 4

Tags: number theory proposed , number theory

Prove that there exist infinitely many pairs $\left(x;\;y\right)$ of different positive rational numbers, such that the numbers $\sqrt{x^2+y^3}$ and $\sqrt{x^3+y^2}$ are both rational.

2002 Polish MO Finals, 1

Tags: number theory proposed , number theory

Find all the natural numbers $a,b,c$ such that: 1) $a^2+1$ and $b^2+1$ are primes 2) $(a^2+1)(b^2+1)=(c^2+1)$

2014 ELMO Shortlist, 9

Tags: logarithms , number theory proposed , number theory

Let $d$ be a positive integer and let $\varepsilon$ be any positive real. Prove that for all sufficiently large primes $p$ with $\gcd(p-1,d) \neq 1$, there exists an positive integer less than $p^r$ which is not a $d$th power modulo $p$, where $r$ is defined by \[ \log r = \varepsilon - \frac{1}{\gcd(d,p-1)}. \][i]Proposed by Shashwat Kishore[/i]