Olympiad problems

2008 Czech-Polish-Slovak Match, 1

Prove that there exists a positive integer $n$, such that the number $k^2+k+n$ does not have a prime divisor less than $2008$ for any integer $k$.

2009 Singapore MO Open, 3

Tags: number theory

for $k\in\mathbb{N}$ , define $A_n$ for $n=1,2,...$ by $A_{n+1} = \frac{ nA_n+2(n+1)^{2k} }{n+2} , A_1=1$ Prove $A_n$ is integer for all $n\geq 1$, and $A_n$ is odd if and only if $n\equiv$1 or 2(mod 4)

2011 India IMO Training Camp, 1

Tags: 3d geometry , diophantine equation , number theory , geometry

Find all positive integer $n$ satisfying the conditions $a)n^2=(a+1)^3-a^3$ $b)2n+119$ is a perfect square.

2023-24 IOQM India, 29

Tags: number theory

A positive integer $n>1$ is called beautiful if $n$ can be written in one and only one way as $n=a_1+a_2+\cdots+a_k=a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k>1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. (For example 6 is beautiful since $6=3 \cdot 2 \cdot 1=3+2+1$, and this is unique. But 8 is not beautiful since $8=4+2+1+1=4 \cdot 2 \cdot 1 \cdot 1$ as well as $8=2+2+2+1+1=2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$, so uniqueness is lost.) Find the largest beautiful number less than 100.

1969 Yugoslav Team Selection Test, Problem 4

Tags: number theory

Let $a$ and $b$ be two natural numbers such that $a<b$. Prove that in each set of $b$ consecutive positive integers there are two numbers whose product is divisible by $ab$.

2018 BMT Spring, Tie 3

Tags: number theory

Let $f : Z^2 \to C$ be a function such that $f(x+11, y) = f(x, y+11) = f(x, y)$, and $f(x, y)f(z,w) = f(xz - yw,xw + yz)$. How many possible values can $f(1, 1)$ have?

1993 APMO, 2

Tags: function , number theory

Find the total number of different integer values the function \[ f(x) = [x] + [2x] + [\frac{5x}{3}] + [3x] + [4x] \] takes for real numbers $x$ with $0 \leq x \leq 100$.

2000 Romania National Olympiad, 1

Tags: number theory , irrational number

Let be two natural primes $ 1\le q \le p. $ Prove that $ \left( \sqrt{p^2+q} +p\right)^2 $ is irrational and its fractional part surpasses $ 3/4. $

2024 Bulgaria MO Regional Round, 10.3

Tags: number theory

Find all positive integers $1 \leq k \leq 6$ such that for any prime $p$, satisfying $p^2=a^2+kb^2$ for some positive integers $a, b$, there exist positive integers $x, y$, satisfying $p=x^2+ky^2$. [hide=Remark on 10.4] It also appears as ARO 2010 10.4 with the grid changed to $10 \times 10$ and $17$ changed to $5$, so it will not be posted.

2010 Peru MO (ONEM), 2

Tags: algebra , arithmetic progression , number theory

An arithmetic progression is formed by $9$ positive integers such that the product of these $9$ terms is a multiple of $3$. Prove that said product is also multiple of $81$.

2011 Purple Comet Problems, 1

Tags: number theory , relatively prime

There are relatively prime positive integers $m$ and $n$ so that \[\dfrac{\dfrac{1}{2}}{\dfrac{\dfrac{1}{3}}{\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}}+\dfrac{\dfrac{1}{3}}{\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}}}=\dfrac{m}{n}.\]Find $m+2n$.

2021 IMO Shortlist, N5

Tags: number theory

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2009 Purple Comet Problems, 4

Tags: ratio , number theory , relatively prime

There are three bags of marbles. Bag two has twice as many marbles as bag one. Bag three has three times as many marbles as bag one. Half the marbles in bag one, one third the marbles in bag two, and one fourth the marbles in bag three are green. If all three bags of marbles are dumped into a single pile, $\frac{m}{n}$ of the marbles in the pile would be green where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

2018 District Olympiad, 2

Tags: number theory , divide

Find the pairs of integers $(a, b)$ such that $a^2 + 2b^2 + 2a +1$ is a divisor of $2ab$.

2020 India National Olympiad, 5

Tags: number theory , tiling , descent , trigonometry , niven theorem , rational , galois theory

Infinitely many equidistant parallel lines are drawn in the plane. A positive integer $n \geqslant 3$ is called frameable if it is possible to draw a regular polygon with $n$ sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon. (a) Show that $3, 4, 6$ are frameable. (b) Show that any integer $n \geqslant 7$ is not frameable. (c) Determine whether $5$ is frameable. [i]Proposed by Muralidharan[/i]

2005 IMO Shortlist, 5

Tags: number theory , prime number , prime factorization

Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

2001 China National Olympiad, 3

Tags: modular arithmetic , limit , number theory

Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\neq0$ such that (i) $m<2a$; (ii) $2n|(2am-m^2+n^2)$; (iii) $n^2-m^2+2mn\leq2a(n-m)$. For $(m, n)\in A$, let \[f(m,n)=\frac{2am-m^2-mn}{n}.\] Determine the maximum and minimum values of $f$.

2021 APMO, 5

Tags: function , functional equation , number theory

Determine all Functions $f:\mathbb{Z} \to \mathbb{Z}$ such that $f(f(a)-b)+bf(2a)$ is a perfect square for all integers $a$ and $b$.

2004 District Olympiad, 1

Tags: number theory , sum of digits

Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $2,0,0,4.$ [hide= original wording] before finding a typo .. Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $1,0,0,4.$ Posts 2 and 3 reply to this wording [/hide]

2022 JBMO TST - Turkey, 1

Tags: number theory , divisibility , integer

For positive integers $a$ and $b$, if the expression $\frac{a^2+b^2}{(a-b)^2}$ is an integer, prove that the expression $\frac{a^3+b^3}{(a-b)^3}$ is an integer as well.

MathLinks Contest 3rd, 3

Tags: number theory

An integer $z$ is said to be a [i]friendly [/i] integer if $|z|$ is not the square of an integer. Determine all integers $n$ such that there exists an infinite number of triplets of distinct friendly integers $(a, b, c)$ such that $n = a+b+c$ and $abc$ is the square of an odd integer.

1977 IMO Longlists, 10

Tags: number theory , additive number theory , remainder , divisibility

Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$

2014 AMC 12/AHSME, 23

Tags: modular arithmetic , induction , function , number theory , divisibility tests

The fraction \[\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\] where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$? $\textbf{(A) }874\qquad \textbf{(B) }883\qquad \textbf{(C) }887\qquad \textbf{(D) }891\qquad \textbf{(E) }892\qquad$

1976 Spain Mathematical Olympiad, 4

Tags: number theory , divide

Show that the expression $$\frac{n^5 -5n^3 + 4n}{n + 2}$$ where n is any integer, it is always divisible by $24$.

1999 Balkan MO, 2

Tags: number theory

Let $p$ be an odd prime congruent to 2 modulo 3. Prove that at most $p-1$ members of the set $\{m^2 - n^3 - 1 \mid 0 < m,\ n < p\}$ are divisible by $p$.