Olympiad problems

2018 Spain Mathematical Olympiad, 1

Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares.

2002 Canada National Olympiad, 2

Tags: number theory

Call a positive integer $n$ [b]practical[/b] if every positive integer less than or equal to $n$ can be written as the sum of distinct divisors of $n$. For example, the divisors of 6 are 1, 2, 3, and 6. Since \[ \centerline{1={\bf 1}, ~~ 2={\bf 2}, ~~ 3={\bf 3}, ~~ 4={\bf 1}+{\bf 3}, ~~ 5={\bf 2}+ {\bf 3}, ~~ 6={\bf 6},} \] we see that 6 is practical. Prove that the product of two practical numbers is also practical.

2022 Junior Balkan Team Selection Tests - Romania, P1

Tags: number theory

Let $p$ be an odd prime number. Prove that there exist nonnegative integers $x,y,z,t$ not all of which are $0$ such that $t<p$ and \[x^2+y^2+z^2=tp.\]

2016 Postal Coaching, 1

Tags: number theory

Let $n$ be an odd positive integer such that $\varphi (n)$ and $\varphi (n+1)$ are both powers of $2$ (here $\varphi(n)$ denotes Euler’s totient function). Prove that $n+1$ is a power of $2$ or $n = 5$.

2008 AMC 12/AHSME, 15

Tags: induction , modular arithmetic , number theory

Let $ k\equal{}2008^2\plus{}2^{2008}$. What is the units digit of $ k^2\plus{}2^k$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$

1993 Greece National Olympiad, 15

Tags: number theory , relatively prime , pythagorean theorem , geometry

Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime integers. Find $m + n$

2000 Vietnam Team Selection Test, 2

Tags: search , number theory

Let $k$ be a given positive integer. Deﬁne $x_{1}= 1$ and, for each $n > 1$, set $x_{n+1}$ to be the smallest positive integer not belonging to the set $\{x_{i}, x_{i}+ik | i = 1, . . . , n\}$. Prove that there is a real number $a$ such that $x_{n}= [an]$ for all $n \in\mathbb{ N}$.

2001 Estonia Team Selection Test, 5

Tags: product , power , number theory , digit , prime

Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers

2014 Contests, 1

Tags: number theory

Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$

2023 Girls in Mathematics Tournament, 2

Tags: algebra , number theory

Given $n$ a positive integer, define $T_n$ the number of quadruples of positive integers $(a,b,x,y)$ such that $a>b$ and $n= ax+by$. Prove that $T_{2023}$ is odd.

2007 China Western Mathematical Olympiad, 2

Tags: number theory

Find all natural numbers $n$ such that there exist $ x_1,x_2,\ldots,x_n,y\in\mathbb{Z}$ where $x_1,x_2,\ldots,x_n,y\neq 0$ satisfying: \[x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n \equal{} 0\] \[ny^2 \equal{} x_1^2 \plus{} x_2^2 \plus{} \ldots \plus{} x_n^2\]

2019 ELMO Shortlist, N2

Tags: number theory , function

Let $f:\mathbb N\to \mathbb N$. Show that $f(m)+n\mid f(n)+m$ for all positive integers $m\le n$ if and only if $f(m)+n\mid f(n)+m$ for all positive integers $m\ge n$. [i]Proposed by Carl Schildkraut[/i]

2004 Argentina National Olympiad, 2

Tags: diophantine equation , number theory , system of equations

Determine all positive integers $a,b,c,d$ such that$$\begin{cases} a<b \\ a^2c =b^2d \\ ab+cd =2^{99}+2^{101} \end{cases}$$

2014 HMNT, 10

Tags: number theory

Suppose that $m$ and $n$ are integers with $1 \le m \le 49$ and $n \ge 0$ such that $m$ divides $n^{n+1} + 1$. What is the number of possible values of $m$?

1977 IMO Longlists, 19

Tags: number theory

Given any integer $m>1$ prove that there exist infinitely many positive integers $n$ such that the last $m$ digits of $5^n$ are a sequence $a_m,a_{m-1},\ldots ,a_1=5\ (0\le a_j<10)$ in which each digit except the last is of opposite parity to its successor (i.e., if $a_i$ is even, then $a_{i-1}$ is odd, and if $a_i$ is odd, then $a_{i-1}$ is even).

2016 Iran MO (3rd Round), 2

Tags: number theory , polynomial , algebra

Let $P$ be a polynomial with integer coefficients. We say $P$ is [i]good [/i] if there exist infinitely many prime numbers $q$ such that the set $$X=\left\{P(n) \mod q : \quad n\in \mathbb N\right\}$$ has at least $\frac{q+1}{2}$ members. Prove that the polynomial $x^3+x$ is good.

2014 Iran Team Selection Test, 2

Tags: polynomial , number theory , algebra

find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.

2012 ELMO Shortlist, 6

Tags: floor function , algebra , polynomial , vieta , quadratic , inequalities , number theory

Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

2014 VTRMC, Problem 5

Tags: diophantine equation , number theory

Let $n\ge1$ and $r\ge2$ be positive integers. Prove that there is no integer $m$ such that $n(n+1)(n+2)=m^r$.

2018 Saint Petersburg Mathematical Olympiad, 6

Tags: number theory

$a,b$ are odd numbers. Prove, that exists natural $k$ that $b^k-a^2$ or $a^k-b^2$ is divided by $2^{2018}$.

2000 Croatia National Olympiad, Problem 2

Tags: number theory

Find all $5$-tuples of different four-digit integers with the same initial digit such that the sum of the five numbers is divisible by four of them.

2016 Vietnam National Olympiad, 3

Tags: number theory , perfect number

a) Prove that if $n$ is an odd perfect number then $n$ has the following form \[ n=p^sm^2 \] where $p$ is prime has form $4k+1$, $s$ is positive integers has form $4h+1$, and $m\in\mathbb{Z}^+$, $m$ is not divisible by $p$. b) Find all $n\in\mathbb{Z}^+$, $n>1$ such that $n-1$ and $\frac{n(n+1)}{2}$ is perfect number

2002 AIME Problems, 14

Tags: geometry , perimeter , geometric transformation , dilation , trigonometry , number theory , relatively prime

The perimeter of triangle $APM$ is $152,$ and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}.$ Given that $OP=m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

2001 Irish Math Olympiad, 1

Tags: modular arithmetic , number theory

Find the least positive integer $ a$ such that $ 2001$ divides $ 55^n\plus{}a \cdot 32^n$ for some odd $ n$.

2007 Singapore MO Open, 5

Tags: number theory , least common multiple , floor function , induction

Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.