Olympiad problems

2021 JBMO TST - Turkey, 5

$d(n)$ shows the number of positive integer divisors of positive integer $n$. For which positive integers $n$ one cannot find a positive integer $k$ such that $\underbrace{d(\dots d(d}_{k\ \text{times}} (n) \dots )$ is a perfect square.

2004 Bulgaria Team Selection Test, 1

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

Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.

2014 Turkey MO (2nd round), 2

Tags: modular arithmetic , number theory proposed , number theory

Find all all positive integers x,y,and z satisfying the equation $x^3=3^y7^z+8$

1992 Baltic Way, 2

Tags: number theory proposed , number theory

Denote by $ d(n)$ the number of all positive divisors of a natural number $ n$ (including $ 1$ and $ n$). Prove that there are infinitely many $ n$, such that $ n/d(n)$ is an integer.

2007 China Northern MO, 3

Tags: number theory proposed , number theory

Let $ n$ be a positive integer and $ [ \ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied: $ (1)$ $ n$ is not a perfect square; $ (2)$ $ a^{3}$ divides $ n^{2}$.

1984 Canada National Olympiad, 1

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

Prove that the sum of the squares of $1984$ consecutive positive integers cannot be the square of an integer.

2013 China Team Selection Test, 2

Tags: pigeonhole principle , number theory proposed , number theory

For the positive integer $n$, define $f(n)=\min\limits_{m\in\Bbb Z}\left|\sqrt2-\frac mn\right|$. Let $\{n_i\}$ be a strictly increasing sequence of positive integers. $C$ is a constant such that $f(n_i)<\dfrac C{n_i^2}$ for all $i\in\{1,2,\ldots\}$. Show that there exists a real number $q>1$ such that $n_i\geqslant q^{i-1}$ for all $i\in\{1,2,\ldots \}$.

2005 All-Russian Olympiad Regional Round, 10.7

Tags: number theory proposed , number theory

10.7 Find all pairs $(a,b)$ of natural numbers s.t. $a^n+b^n$ is a perfect $n+1$th power for all $n\in\mathbb{N}$. ([i]V. Senderov[/i])

2014 Lithuania Team Selection Test, 2

Tags: number theory proposed , number theory

Finite set $A$ has such property: every six its distinct elements’ sum isn’t divisible by $6$. Does there exist such set $A$ consisting of $11$ distinct natural numbers?

2012 Iran MO (3rd Round), 3

Tags: modular arithmetic , number theory proposed , number theory

$p$ is an odd prime number. Prove that there exists a natural number $x$ such that $x$ and $4x$ are both primitive roots modulo $p$. [i]Proposed by Mohammad Gharakhani[/i]

2021 China Team Selection Test, 3

Tags: number theory , sum of digits , decimal representation , China TST , China , number theory proposed

Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following: There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements, $$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$ where $S(n)$ denotes sum of digits of decimal representation of $n$.

2009 Pan African, 2

Tags: function , induction , number theory proposed , number theory

Find all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ for which $f(0)=0$ and \[f(x^2-y^2)=f(x)f(y) \] for all $x,y\in\mathbb{N}_0$ with $x>y$.

1951 Miklós Schweitzer, 9

Tags: modular arithmetic , number theory proposed , number theory

Let $ \{m_1,m_2,\dots\}$ be a (finite or infinite) set of positive integers. Consider the system of congruences (1) $ x\equiv 2m_i^2 \pmod{2m_i\minus{}1}$ ($ i\equal{}1,2,...$ ). Give a necessary and sufficient condition for the system (1) to be solvable.

2012 Bulgaria National Olympiad, 1

Tags: number theory proposed , number theory

The sequence $a_1,a_2,a_3\ldots $, consisting of natural numbers, is defined by the rule: \[a_{n+1}=a_{n}+2t(n)\] for every natural number $n$, where $t(n)$ is the number of the different divisors of $n$ (including $1$ and $n$). Is it possible that two consecutive members of the sequence are squares of natural numbers?

1990 IMO Longlists, 87

Tags: number theory proposed , number theory

Let $m$ be an positive odd integer not divisible by $3$. Prove that $\left[4^m -(2+\sqrt 2)^m\right]$ is divisible by $112.$

2010 Contests, 1

Tags: number theory proposed , number theory

Find all triplets of natural numbers $(a,b,c)$ that satisfy the equation $abc=a+b+c+1$.

2004 Tuymaada Olympiad, 4

Tags: number theory , least common multiple , number theory proposed

It is known that $m$ and $n$ are positive integers, $m > n^{n-1}$, and all the numbers $m+1$, $m+2$, \dots, $m+n$ are composite. Prove that there exist such different primes $p_1$, $p_2$, \dots, $p_n$ that $p_k$ divides $m+k$ for $k = 1$, 2, \dots, $n$. [i]Proposed by C. A. Grimm [/i]

1985 IMO Longlists, 23

Tags: floor function , inequalities , number theory , relatively prime , number theory proposed

Let $\mathbb N = {1, 2, 3, . . .}$. For real $x, y$, set $S(x, y) = \{s | s = [nx+y], n \in \mathbb N\}$. Prove that if $r > 1$ is a rational number, there exist real numbers $u$ and $v$ such that \[S(r, 0) \cap S(u, v) = \emptyset, S(r, 0) \cup S(u, v) = \mathbb N.\]

2005 Indonesia MO, 3

Tags: number theory proposed , number theory

Let $ k$ and $ m$ be positive integers such that $ \displaystyle\frac12\left(\sqrt{k\plus{}4\sqrt{m}}\minus{}\sqrt{k}\right)$ is an integer. (a) Prove that $ \sqrt{k}$ is rational. (b) Prove that $ \sqrt{k}$ is a positive integer.

2002 Turkey Team Selection Test, 1

Tags: number theory proposed , number theory

If $ab(a+b)$ divides $a^2 + ab+ b^2$ for different integers $a$ and $b$, prove that \[|a-b|>\sqrt[3]{ab}.\]

1993 Iran MO (3rd Round), 4

Tags: number theory , number theory proposed

Prove that there exists a subset $S$ of positive integers such that we can represent each positive integer as difference of two elements of $S$ in exactly one way.

2013 Iran MO (3rd Round), 2

Tags: algebra , polynomial , Vieta , number theory proposed , number theory , Vieta Jumping

Suppose that $a,b$ are two odd positive integers such that $2ab+1 \mid a^2 + b^2 + 1$. Prove that $a=b$. (15 points)

1985 IMO Longlists, 29

Tags: number theory proposed , number theory

[i]a)[/i] Call a four-digit number $(xyzt)_B$ in the number system with base $B$ stable if $(xyzt)_B = (dcba)_B - (abcd)_B$, where $a \leq b \leq c \leq d$ are the digits of $(xyzt)_B$ in ascending order. Determine all stable numbers in the number system with base $B.$ [i]b)[/i] With assumptions as in [i]a[/i], determine the number of bases $B \leq 1985$ such that there exists a stable number with base $B.$

2009 Baltic Way, 6

Tags: number theory proposed , number theory

Let $ a$ and $ b$ be integers such that the equation $ x^3\minus{}ax^2\minus{}b\equal{}0$ has three integer roots. Prove that $ b\equal{}dk^2$, where $ d$ and $ k$ are integers and $ d$ divides $ a$.

2014 ELMO Shortlist, 3

Tags: algebra , polynomial , induction , binomial theorem , number theory proposed , number theory

Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$. [i]Proposed by Michael Kural[/i]