Olympiad problems

2020-21 KVS IOQM India, 12

Let $A = \{m : m$ an integer and the roots of $x^2 + mx + 2020 = 0$ are positive integers $\}$ and $B= \{n : n$ an integer and the roots of $x^2 + 2020x + n = 0$ are negative integers $\}$. Suppose $a$ is the largest element of $A$ and $b$ is the smallest element of $B$. Find the sum of digits of $a + b$.

TNO 2008 Junior, 12

Tags: number theory

(a) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $11n$ is twice the sum of the digits of $n$. (b) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $5n + 1$ is six times the sum of the digits of $n$.

1964 Poland - Second Round, 3

Tags: arithmetic sequence , number theory , arithmetic progression , divisible

Prove that if three prime numbers form an arithmetic progression whose difference is not divisible by 6, then the smallest of these numbers is $3 $.

2012 South East Mathematical Olympiad, 3

Tags: number theory

For composite number $n$, let $f(n)$ denote the sum of the least three divisors of $n$, and $g(n)$ the sum of the greatest two divisors of $n$. Find all composite numbers $n$, such that $g(n)=(f(n))^m$ ($m\in N^*$).

2009 JBMO Shortlist, 3

Tags: number theory

Find all pairs $(x,y)$ of integers which satisfy the equation $(x + y)^2(x^2 + y^2) = 2009^2$

2025 Canada National Olympiad, 2

Tags: number theory

Determine all positive integers $a$, $b$, $c$, $p$, where $p$ and $p+2$ are odd primes and \[2^ap^b=(p+2)^c-1.\]

2017 Turkey MO (2nd round), 4

Tags: number theory , prime number

Let $d(n)$ be number of prime divisors of $n$. Prove that one can find $k,m$ positive integers for any positive integer $n$ such that $k-m=n$ and $d(k)-d(m)=1$

2012 Brazil Team Selection Test, 3

Tags: modular arithmetic , number theory , prime , divisibility

Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.

PEN L Problems, 2

Tags: linear recurrence , number theory , greatest common divisor

The Fibonacci sequence $\{F_{n}\}$ is defined by \[F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.\] Show that $\gcd (F_{m}, F_{n})=F_{\gcd (m, n)}$ for all $m, n \in \mathbb{N}$.

2004 China Girls Math Olympiad, 7

Tags: number theory

Let $ p$ and $ q$ be two coprime positive integers, and $ n$ be a non-negative integer. Determine the number of integers that can be written in the form $ ip \plus{} jq$, where $ i$ and $ j$ are non-negative integers with $ i \plus{} j \leq n$.

2015 Indonesia MO Shortlist, N4

Tags: number theory , exponential equation , exponential , greatest common divisor , diophantine equation

Suppose that the natural number $a, b, c, d$ satisfy the equation $a^ab^{a + b} = c^cd^{c + d}$. (a) If gcd $(a, b) = $ gcd $(c, d) = 1$, prove that $a = c$ and $b = d$. (b) Does the conclusion $a = c$ and $b = d$ apply, without the condition gcd $(a, b) = $ gcd $(c, d) = 1$?

2012 ELMO Shortlist, 4

Tags: factorial , 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]

1974 IMO, 6

Tags: coefficient , algebra , polynomial , number theory , degree

Let $P(x)$ be a polynomial with integer coefficients. We denote $\deg(P)$ its degree which is $\geq 1.$ Let $n(P)$ be the number of all the integers $k$ for which we have $(P(k))^{2}=1.$ Prove that $n(P)- \deg(P) \leq 2.$

1993 IMO Shortlist, 1

Tags: modular arithmetic , number theory , relatively prime , partition , algebra

a) Show that the set $ \mathbb{Q}^{ + }$ of all positive rationals can be partitioned into three disjoint subsets. $ A,B,C$ satisfying the following conditions: \[ BA = B; \& B^2 = C; \& BC = A; \] where $ HK$ stands for the set $ \{hk: h \in H, k \in K\}$ for any two subsets $ H, K$ of $ \mathbb{Q}^{ + }$ and $ H^2$ stands for $ HH.$ b) Show that all positive rational cubes are in $ A$ for such a partition of $ \mathbb{Q}^{ + }.$ c) Find such a partition $ \mathbb{Q}^{ + } = A \cup B \cup C$ with the property that for no positive integer $ n \leq 34,$ both $ n$ and $ n + 1$ are in $ A,$ that is, \[ \text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34. \]

2012 Canadian Mathematical Olympiad Qualification Repechage, 3

Tags: geometric sequence , number theory

We say that $(a,b,c)$ form a [i]fantastic triplet[/i] if $a,b,c$ are positive integers, $a,b,c$ form a geometric sequence, and $a,b+1,c$ form an arithmetic sequence. For example, $(2,4,8)$ and $(8,12,18)$ are fantastic triplets. Prove that there exist infinitely many fantastic triplets.

2009 JBMO Shortlist, 4

Tags: number theory , sum of powers

Determine all prime numbers $p_1, p_2,..., p_{12}, p_{13}, p_1 \le p_2 \le ... \le p_{12} \le p_{13}$, such that $p_1^2+ p_2^2+ ... + p_{12}^2 = p_{13}^2$ and one of them is equal to $2p_1 + p_9$.

2023 Kazakhstan National Olympiad, 5

Tags: number theory

Given are positive integers $a, b, m, k$ with $k \geq 2$. Prove that there exist infinitely many $n$, such that $\gcd (\varphi_m(n), \lfloor \sqrt[k] {an+b} \rfloor)=1$, where $\varphi_m(n)$ is the $m$-th iteration of $\varphi(n)$.

2001 Argentina National Olympiad, 1

Tags: combinatorics , number theory

Sergio thinks of a positive integer $S$, less than or equal to $100$. Iván must guess the number that Sergio thought of, using the following procedure: in each step, he chooses two positive integers $A$ and $B$ less than $100$, and asks Sergio what is the greatest common factor between $A+ S$ and $B$. Give a sequence of seven steps that ensures Iván guesses the number $S$ that Sergio thought of. Clarification:In each step, Sergio correctly answers Iván's question.

2022 IFYM, Sozopol, 4

Tags: number theory

a) Prove that for each positive integer $n$ the number or ordered pairs of integers $(x,y)$ for which $x^2-xy+y^2=n$ is finite and is multiple of 6. b) Find all ordered pairs of integers $(x,y)$ for which $x^2-xy+y^2=727$.

2021 Malaysia IMONST 1, 7

Tags: number theory , digit

Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$. How many passcodes satisfy these conditions?

2024 EGMO, 5

Tags: function , number theory

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that the following conditions are true for every pair of positive integers $(x, y)$: $(i)$: $x$ and $f(x)$ have the same number of positive divisors. $(ii)$: If $x \nmid y$ and $y \nmid x$, then: $$\gcd(f(x), f(y)) > f(\gcd(x, y))$$

2019 APMO, 1

Tags: function , number theory , algebra , functional equation

Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$ for all positive integers $a,b$.

2008 Postal Coaching, 2

Tags: number theory , prime , prime number , divide

Prove that an integer $n \ge 2$ is a prime if and only if $\phi (n)$ divides $(n - 1)$ and $(n + 1)$ divides $\sigma (n)$. [Here $\phi$ is the Totient function and $\sigma $ is the divisor - sum function.] [hide=Hint]$n$ is squarefree[/hide]

2002 Spain Mathematical Olympiad, Problem 3

Tags: function , number theory

The function $g$ is defined about the natural numbers and satisfies the following conditions: $g(2) = 1$ $g(2n) = g(n)$ $g(2n+1) = g(2n) +1.$ Where $n$ is a natural number such that $1 \leq n \leq 2002$. Find the maximum value $M$ of $g(n).$ Also, calculate how many values of $n$ satisfy the condition of $g(n) = M.$

2019 JBMO Shortlist, N4

Tags: number theory

Find all integers $x,y$ such that $x^3(y+1)+y^3(x+1)=19$. [i]Proposed by Bulgaria[/i]