Olympiad problems

1997 German National Olympiad, 6a

Let us define $f$ and $g$ by $f(x) = x^5 +5x^4 +5x^3 +5x^2 +1$, $g(x) = x^5 +5x^4 +3x^3 -5x^2 -1$. Determine all prime numbers $p$ such that, for at least one integer $x, 0 \le x < p-1$, both $f(x)$ and $g(x)$ are divisible by $p$. For each such $p$, find all $x$ with this property.

2010 Hanoi Open Mathematics Competitions, 7

Tags: number theory , primes , quadratics , roots

Determine all positive integer $a$ such that the equation $2x^2 - 30x + a = 0$ has two prime roots, i.e. both roots are prime numbers.

2015 Belarus Team Selection Test, 1

Tags: number theory , primes , divisible , consecutive

Given $m,n \in N$ such that $M>n^{n-1}$ and the numbers $m+1, m+2, ..., m+n$ are composite. Prove that exist distinct primes $p_1,p_2,...,p_n$ such that $M+k$ is divisible by $p_k$ for any $k=1,2,...,n$. Tuymaada Olympiad 2004, C.A.Grimm. USA

2024 Indonesia MO, 2

Tags: number theory , Inamo , primes , factorizations , Indonesia , NT construction , Indonesia MO

The triplet of positive integers $(a,b,c)$ with $a<b<c$ is called a [i]fatal[/i] triplet if there exist three nonzero integers $p,q,r$ which satisfy the equation $a^p b^q c^r = 1$. As an example, $(2,3,12)$ is a fatal triplet since $2^2 \cdot 3^1 \cdot (12)^{-1} = 1$. The positive integer $N$ is called [i]fatal[/i] if there exists a fatal triplet $(a,b,c)$ satisfying $N=a+b+c$. (a) Prove that 16 is not [i]fatal[/i]. (b) Prove that all integers bigger than 16 which are [b]not[/b] an integer multiple of 6 are fatal.

2025 Macedonian Balkan MO TST, 4

Tags: number theory , primes , Sum of Squares , BMO TST

Let $n$ be a positive integer. Prove that for every odd prime $p$ dividing $n^2 + n + 2$, there exist integers $a, b$ such that $p = a^2 + 7b^2$.

2006 Singapore Senior Math Olympiad, 1

Tags: mutliple , divides , divisible , primes , number theory

Let $a, d$ be integers such that $a,a + d, a+ 2d$ are all prime numbers larger than $3$. Prove that $d$ is a multiple of $6$.

2016 Croatia Team Selection Test, Problem 4

Tags: number theory , Diophantine equation , prime numbers , primes

Find all pairs $(p,q)$ of prime numbers such that $$ p(p^2 - p - 1) = q(2q + 3) .$$

2013 QEDMO 13th or 12th, 9

Tags: number theory , primes

Are there infinitely many different natural numbers $a_1,a_2, a_3,...$ so that for every integer $k$ only finitely many of the numbers $a_1 + k$,$a_2 + k$,$a_3 + k$,$...$ are numbers prime?

2021 Indonesia MO, 3

Tags: number theory , Sequence , primes

A natural number is called a [i]prime power[/i] if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$. Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$.

2019 Argentina National Olympiad, 5

Tags: number theory , primes , arithmetic sequence

There is an arithmetic progression of $7$ terms in which all the terms are different prime numbers. Determine the smallest possible value of the last term of such a progression. Clarification: In an arithmetic progression of difference $d$ each term is equal to the previous one plus $d$.

2022 AMC 12/AHSME, 3

Tags: AMC , AMC 12 , AMC 10 , 2022 AMC 10B , 2022 AMC 12B , 2022 AMC , primes

How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers? $\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$

2014 Contests, 4

Tags: number theory , point , circle , primes

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

2021 Nigerian MO Round 3, Problem 1

Tags: number theory , Diophantine equation , primes

Find all triples of primes $(p, q, r)$ such that $p^q=2021+r^3$.

2000 Singapore MO Open, 2

Tags: divides , divisible , primes , number theory , GCD

Show that $240$ divides all numbers of the form $p^4 - q^4$, where p and q are prime numbers strictly greater than $5$. Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$, with $p$ and $q$ prime numbers strictly greater than $5$.

2023 Brazil Team Selection Test, 3

Tags: number theory , prime factorization , function , prime numbers , primes

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2003 France Team Selection Test, 3

Tags: algebra , number theory , primes , Divisors , IMO Shortlist

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

2019 Austrian Junior Regional Competition, 4

Tags: number theory , primes , divisible , divides

Let $p, q, r$ and $s$ be four prime numbers such that $$5 <p <q <r <s <p + 10.$$ Prove that the sum of the four prime numbers is divisible by $60$. (Walther Janous)

1993 ITAMO, 2

Tags: quadratic trinomial , trinomial , Quadratic , number theory , Rational roots , rational , primes

Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.

2007 Bulgarian Autumn Math Competition, Problem 8.3

Tags: primes , square , number theory , prime numbers

Determine all triplets of prime numbers $p<q<r$, such that $p+q=r$ and $(r-p)(q-p)-27p$ is a square.

2009 Singapore Junior Math Olympiad, 3

Tags: primes , Digits , number theory

Suppose $\overline{a_1a_2...a_{2009}}$ is a $2009$-digit integer such that for each $i = 1,2,...,2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$ (Note: $\overline{xyz...}$ denotes an integer whose digits are $x, y,z,...$.)

2025 Bulgarian Spring Mathematical Competition, 9.4

Tags: functional equation , number theory , Divisibility , primes , Sophie Germain identity , function

Determine all functions $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that $f(a) + 2ab + 2f(b)$ divides $f(a)^2 + 4f(b)^2$ for any positive integers $a$ and $b$.