Olympiad problems

2013 BMT Spring, 7

Tags: number theory

Consider the infinite polynomial $G(x) = F_1x+F_2x^2 +F_3x^3 +...$ defined for $0 < x <\frac{\sqrt5 -1}{2}$ where Fk is the $k$th term of the Fibonacci sequence defined to be $F_k = F_{k-1} + F_{k-2}$ with $F_1 = 1$, $F_2 = 1$. Determine the value a such that $G(a) = 2$.

2016 Peru IMO TST, 4

Tags: positive integer , number theory , divide , min , max

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

1997 Iran MO (2nd round), 3

Tags: number theory

Let $a,b$ be positive integers and $p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}$ be a prime number. Find the maximum value of $p$ and justify your answer.

2013 Junior Balkan MO, 1

Tags: number theory

Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.

2000 JBMO ShortLists, 5

Tags: number theory

Find all pairs of integers $(m,n)$ such that the numbers $A=n^2+2mn+3m^2+2$, $B=2n^2+3mn+m^2+2$, $C=3n^2+mn+2m^2+1$ have a common divisor greater than $1$.

1978 All Soviet Union Mathematical Olympiad, 258

Tags: number theory , relatively prime

Let $f(x) = x^2 - x + 1$. Prove that for every natural $m>1$ the numbers $$m, f(m), f(f(m)), ...$$ are relatively prime.

2024-IMOC, N7

Tags: number theory , functional equation

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$|xf(y)-yf(x)|$$ is a perfect square for every $x,y \in \mathbb{N}$

2014 Peru IMO TST, 4

Tags: number theory

A positive integer is called lonely if the sum of the reciprocals of its positive divisors (including 1 and itself) is different from the sum of the reciprocals of the positive divisors of any positive integer. a) Prove that every prime number is lonely. b) Prove that there are infinitely many positive integers that are not lonely.

2018 Greece JBMO TST, 4

Tags: number theory

Find all positive integers $x,y,z$ with $z$ odd, which satisfy the equation: $$2018^x=100^y + 1918^z$$

2013 Albania Team Selection Test, 1

Tags: ratio , function , number theory

Find the 3-digit number whose ratio with the sum of its digits it's minimal.

2004 Romania Team Selection Test, 18

Tags: modular arithmetic , quadratic , number theory

Let $p$ be a prime number and $f\in \mathbb{Z}[X]$ given by \[ f(x) = a_{p-1}x^{p-2} + a_{p-2}x^{p-3} + \cdots + a_2x+ a_1 , \] where $a_i = \left( \tfrac ip\right)$ is the Legendre symbol of $i$ with respect to $p$ (i.e. $a_i=1$ if $ i^{\frac {p-1}2} \equiv 1 \pmod p$ and $a_i=-1$ otherwise, for all $i=1,2,\ldots,p-1$). a) Prove that $f(x)$ is divisible with $(x-1)$, but not with $(x-1)^2$ iff $p \equiv 3 \pmod 4$; b) Prove that if $p\equiv 5 \pmod 8$ then $f(x)$ is divisible with $(x-1)^2$ but not with $(x-1)^3$. [i]Sugested by Calin Popescu.[/i]

2022 Chile National Olympiad, 6

Tags: number theory

Determine if there is a power of 5 that begins with 2022.

2023 Grosman Mathematical Olympiad, 4

Tags: quadratic , number theory , prime number

Let $q$ be an odd prime number. Prove that it is impossible for all $(q-1)$ numbers \[1^2+1+q, 2^2+2+q, \dots, (q-1)^2+(q-1)+q\] to be products of two primes (not necessarily distinct).

2017 IFYM, Sozopol, 2

Tags: number theory

With $\sigma (n)$ we denote the sum of the positive divisors of the natural number $n$. Prove that there exist infinitely many natural numbers $n$, for which $n$ divides $2^{\sigma (n)} -1$.

2010 May Olympiad, 4

Tags: number theory , multiple

Find all natural numbers of $90$ digits that are multiples of $13$ and have the first $43$ digits equal to each other and nonzero, the last $43$ digits equal to each other, and the middle $4$ digits are $2, 0, 1, 0$, in that order.

2012 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: number theory , board , combinatorics

In each square of a $100 \times 100$ board there is written an integer. The allowed operation is to choose four squares that form the figure or any of its reflections or rotations, and add $1$ to each of the four numbers. The aim is, through operations allowed, achieving a board with the smallest possible number of different residues modulo $33$. What is the minimum number that can be achieved with certainty?

2013 HMNT, 9

Tags: number theory

For an integer $n \ge 0$, let $f(n)$ be the smallest possible value of $ |x + y|$, where $x$ and $y$ are integers such that $3x - 2y = n$. Evaluate $f(0) + f(1) + f(2) +...+ f(2013)$.

2022 APMO, 3

Tags: algebra , number theory

Find all positive integers $k<202$ for which there exist a positive integers $n$ such that $$\bigg {\{}\frac{n}{202}\bigg {\}}+\bigg {\{}\frac{2n}{202}\bigg {\}}+\cdots +\bigg {\{}\frac{kn}{202}\bigg {\}}=\frac{k}{2}$$

2016 Taiwan TST Round 2, 2

Tags: number theory , divisibility

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Russian TST 2019, P1

Tags: number theory , divisor

A positive integer $n{}$ is called [i]discontinuous[/i] if for all its natural divisors $1 = d_0 < d_1 <\cdots<d_k$, written out in ascending order, there exists $1 \leqslant i \leqslant k$ such that $d_i > d_{i-1}+\cdots+d_1+d_0+1$. Prove that there are infinitely many positive integers $n{}$ such that $n,n+1,\ldots,n+2019$ are all discontinuous.

2000 Polish MO Finals, 3

Tags: number theory

The sequence $p_1, p_2, p_3, ...$ is defined as follows. $p_1$ and $p_2$ are primes. $p_n$ is the greatest prime divisor of $p_{n-1} + p_{n-2} + 2000$. Show that the sequence is bounded.

2023 Czech-Polish-Slovak Junior Match, 1

Tags: number theory , perfect square , sum of digits

Let $S(n)$ denote the sum of all digits of natural number $n$. Determine all natural numbers $n$ for which both numbers $n + S(n)$ and $n - S(n)$ are square powers of non-zero integers.

1976 Canada National Olympiad, 5

Tags: search , number theory

Prove that a positive integer is a sum of at least two consecutive positive integers if and only if it is not a power of two.

1999 Taiwan National Olympiad, 4

Tags: number theory

Let $P^{*}$ be the set of primes less than $10000$. Find all possible primes $p\in P^{*}$ such that for each subset $S=\{p_{1},p_{2},...,p_{k}\}$ of $P^{*}$ with $k\geq 2$ and each $p\not\in S$, there is a $q\in P^{*}-S$ such that $q+1$ divides $(p_{1}+1)(p_{2}+1)...(p_{k}+1)$.

MBMT Team Rounds, 2020.41

Tags: number theory

What are the last two digits of $$2^{3^{4^{...^{2019}}}} ?$$