Olympiad problems

2010 Balkan MO, 4

For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.

2009 Olympic Revenge, 4

Tags: floor function , number theory proposed , number theory

Let $d_i(k)$ the number of divisors of $k$ greater than $i$. Let $f(n)=\sum_{i=1}^{\lfloor \frac{n^2}{2} \rfloor}d_i(n^2-i)-2\sum_{i=1}^{\lfloor \frac{n}{2} \rfloor}d_i(n-i)$. Find all $n \in N$ such that $f(n)$ is a perfect square.

1993 Baltic Way, 1

Tags: number theory proposed , number theory

$a_1a_2a_3$ and $a_3a_2a_1$ are two three-digit decimal numbers, with $a_1$ and $a_3$ different non-zero digits. Squares of these numbers are five-digit numbers $b_1b_2b_3b_4b_5$ and $b_5b_4b_3b_2b_1$ respectively. Find all such three-digit numbers.

2008 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory proposed , number theory

In a sequence of natural numbers $ a_1,a_2,...,a_n$ every number $ a_k$ represents sum of the multiples of the $ k$ from sequence. Find all possible values for $ n$.

2008 CHKMO, 2

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

is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?

1992 Balkan MO, 1

Tags: number theory proposed , number theory

For all positive integers $m,n$ define $f(m,n) = m^{3^{4n}+6} - m^{3^{4n}+4} - m^5 + m^3$. Find all numbers $n$ with the property that $f(m, n)$ is divisible by 1992 for every $m$. [i]Bulgaria[/i]

2014 Contests, 3

Tags: Diophantine equation , number theory proposed , number theory

Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.

1979 IMO Longlists, 19

Tags: number theory proposed , number theory

For $k = 1, 2, \ldots$ consider the $k$-tuples $(a_1, a_2, \ldots, a_k)$ of positive integers such that \[a_1 + 2a_2 + \cdots + ka_k = 1979.\] Show that there are as many such $k$-tuples with odd $k$ as there are with even $k$.

2014 District Olympiad, 1

Tags: number theory proposed , number theory

Find with proof all positive $3$ digit integers $\overline{abc}$ satisfying \[ b\cdot \overline{ac}=c \cdot \overline{ab} +10 \]

2009 Baltic Way, 8

Tags: modular arithmetic , number theory proposed , number theory

Determine all positive integers $n$ for which there exists a partition of the set \[\{n,n+1,n+2,\ldots ,n+8\}\] into two subsets such that the product of all elements of the first subset is equal to the product of all elements of the second subset.

IMSC 2024, 6

Tags: algebra , polynomial , number theory , number theory proposed , algebra proposed , Lifting the Exponent , Divisibility

Let $a\equiv 1\pmod{4}$ be a positive integer. Show that any polynomial $Q\in\mathbb{Z}[X]$ with all positive coefficients such that $$Q(n+1)((a+1)^{Q(n)}-a^{Q(n)})$$ is a perfect square for any $n\in\mathbb{N}^{\ast}$ must be a constant polynomial. [i]Proposed by Vlad Matei, Romania[/i]

2010 Middle European Mathematical Olympiad, 12

Tags: number theory proposed , number theory

We are given a positive integer $n$ which is not a power of two. Show that ther exists a positive integer $m$ with the following two properties: (a) $m$ is the product of two consecutive positive integers; (b) the decimal representation of $m$ consists of two identical blocks with $n$ digits. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 8)[/i]

2005 Romania Team Selection Test, 3

Tags: modular arithmetic , floor function , quadratics , induction , algebra , functional equation , number theory proposed

Let $n\geq 0$ be an integer and let $p \equiv 7 \pmod 8$ be a prime number. Prove that \[ \sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 . \] [i]Călin Popescu[/i]

1997 South africa National Olympiad, 2

Tags: calculus , number theory proposed , number theory

Find all natural numbers with the property that, when the first digit is moved to the end, the resulting number is $\dfrac{7}{2}$ times the original one.

2011 Mexico National Olympiad, 4

Tags: number theory proposed , number theory

Find the smallest positive integer that uses exactly two different digits when written in decimal notation and is divisible by all the numbers from $1$ to $9$.

2003 India IMO Training Camp, 7

Tags: algebra , polynomial , limit , number theory proposed , number theory

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

2011 IberoAmerican, 2

Tags: number theory proposed , number theory

Find all positive integers $n$ for which exist three nonzero integers $x, y, z$ such that $x+y+z=0$ and: \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{n}\]

2014 Contests, 3

Tags: number theory , greatest common divisor , least common multiple , inequalities , number theory proposed

Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$

2014 Dutch IMO TST, 1

Tags: number theory proposed , number theory

Determine all pairs $(a,b)$ of positive integers satisfying \[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]

1993 Iran MO (3rd Round), 1

Tags: number theory proposed , number theory

Prove that there exist infinitely many positive integers which can't be represented as sum of less than $10$ odd positive integers' perfect squares.

2001 Brazil National Olympiad, 2

Tags: ceiling function , logarithms , number theory , greatest common divisor , relatively prime , number theory proposed

Given $a_0 > 1$, the sequence $a_0, a_1, a_2, ...$ is such that for all $k > 0$, $a_k$ is the smallest integer greater than $a_{k-1}$ which is relatively prime to all the earlier terms in the sequence. Find all $a_0$ for which all terms of the sequence are primes or prime powers.

2007 China Second Round Olympiad, 3

Tags: function , number theory proposed , number theory , number theory unsolved

For positive integers $k,m$, where $1\le k\le 5$, define the function $f(m,k)$ as \[f(m,k)=\sum_{i=1}^{5}\left[m\sqrt{\frac{k+1}{i+1}}\right]\] where $[x]$ denotes the greatest integer not exceeding $x$. Prove that for any positive integer $n$, there exist positive integers $k,m$, where $1\le k\le 5$, such that $f(m,k)=n$.

1999 Irish Math Olympiad, 5

Tags: induction , number theory proposed , number theory

The sequence $ u_n$, $ n\equal{}0,1,2,...$ is defined by $ u_0\equal{}0, u_1\equal{}1$ and for each $ n \ge 1$, $ u_{n\plus{}1}$ is the smallest positive integer greater than $ u_n$ such that $ \{ u_0,u_1,...,u_{n\plus{}1} \}$ contains no three elements in arithmetic progression. Find $ u_{100}$.

2012 ELMO Shortlist, 4

Tags: factorial , number theory proposed , number theory

Do there exist positive integers $b,n>1$ such that when $n$ is expressed in base $b$, there are more than $n$ distinct permutations of its digits? For example, when $b=4$ and $n=18$, $18 = 102_4$, but $102$ only has $6$ digit arrangements. (Leading zeros are allowed in the permutations.) [i]Lewis Chen.[/i]

2010 Iran MO (3rd Round), 3

Tags: abstract algebra , modular arithmetic , number theory proposed , number theory

If $p$ is a prime number, what is the product of elements like $g$ such that $1\le g\le p^2$ and $g$ is a primitive root modulo $p$ but it's not a primitive root modulo $p^2$, modulo $p^2$?($\frac{100}{6}$ points)