Olympiad problems

2012 Morocco TST, 2

Find all positive integer $n$ and prime number $p$ such that $p^2+7^n$ is a perfect square

2012 JBMO ShortLists, 2

Tags: number theory , prime numbers

Do there exist prime numbers $p$ and $q$ such that $p^2(p^3-1)=q(q+1)$ ?

2024 Czech and Slovak Olympiad III A, 6

Tags: geometry , congruent triangles , side bash , prime numbers

Find all right triangles with integer side lengths in which two congruent circles with prime radius can be inscribed such that they are externally tangent, both touch the hypotenuse, and each is tangent to another leg of the right triangle.

2007 Tuymaada Olympiad, 4

Tags: probability , logarithms , limit , ratio , greatest common divisor , number theory , prime numbers

Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.

2023 Belarus Team Selection Test, 1.3

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 $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2019 BMT Spring, 6

Tags: number theory , prime numbers

Define $ f(n) = \dfrac{n^2 + n}{2} $. Compute the number of positive integers $ n $ such that $ f(n) \leq 1000 $ and $ f(n) $ is the product of two prime numbers.

1995 AIME Problems, 10

Tags: calculus , integration , number theory , prime numbers

What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?

2010 Contests, 3

Tags: ratio , number theory , prime numbers , arithmetic sequence , number theory proposed

Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$

2021 Pan-African, 3

Tags: number theory , prime numbers , geometric sequence , PAMO

Let $(a_i)_{i\in \mathbb{N}}$ and $(p_i)_{i\in \mathbb{N}}$ be two sequences of positive integers such that the following conditions hold: $\bullet ~~a_1\ge 2$. $\bullet~~ p_n$ is the smallest prime divisor of $a_n$ for every integer $n\ge 1$ $\bullet~~ a_{n+1}=a_n+\frac{a_n}{p_n}$ for every integer $n\ge 1$ Prove that there is a positive integer $N$ such that $a_{n+3}=3a_n$ for every integer $n>N$

2010 Albania Team Selection Test, 4

Tags: algebra , polynomial , function , inequalities , number theory , prime numbers , number theory unsolved

With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?

2008 Tuymaada Olympiad, 2

Tags: AMC , USA(J)MO , USAMO , number theory , prime numbers , number theory unsolved

Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]

2021 Turkey Team Selection Test, 1

Tags: legendre symbol , prime numbers , Divisibility , number theory

Let $n$ be a positive integer. Prove that \[\frac{20 \cdot 5^n-2}{3^n+47}\] is not an integer.

2019 Korea Junior Math Olympiad., 3

Tags: number theory , prime numbers , KJMO

Find all pairs of prime numbers $p,\,q(p\le q)$ satisfying the following condition: There exists a natural number $n$ such that $2^{n}+3^{n}+\cdots+(2pq-1)^{n}$ is a multiple of $2pq$.

2025 6th Memorial "Aleksandar Blazhevski-Cane", P1

Tags: number theory , prime numbers , Diophantine equation

Determine all triples of prime numbers $(p, q, r)$ that satisfy \[p2^q + r^2 = 2025.\] Proposed by [i]Ilija Jovcevski[/i]

2015 Caucasus Mathematical Olympiad, 1

Tags: number theory , prime , Natural Numbers , Product , prime numbers

Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.

2007 Macedonia National Olympiad, 3

Tags: number theory , prime numbers , number theory proposed

Natural numbers $a, b$ and $c$ are pairwise distinct and satisfy \[a | b+c+bc, b | c+a+ca, c | a+b+ab.\] Prove that at least one of the numbers $a, b, c$ is not prime.

2023 Brazil Team Selection Test, 4

Tags: number theory , prime numbers , multiple

Find all positive integers $n$ with the following property: There are only a finite number of positive multiples of $n$ that have exactly $n$ positive divisors.

1999 China Team Selection Test, 2

Tags: number theory , prime numbers , number theory unsolved

Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.

2021 Junior Balkan Team Selection Tests - Romania, P2

Tags: number theory , prime numbers , Romanian TST , TST

Find all the pairs of positive integers $(x,y)$ such that $x\leq y$ and \[\frac{(x+y)(xy-1)}{xy+1}=p,\]where $p$ is a prime number.

2008 Irish Math Olympiad, 1

Tags: number theory , prime numbers , number theory proposed

Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations $ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$ $ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$ Find all possible values of the product $ p_1p_2p_3p_4$

2020 JBMO Shortlist, 4

Tags: Junior , Balkan , shortlist , 2020 , number theory , prime numbers

Find all prime numbers $p$ such that $(x + y)^{19} - x^{19} - y^{19}$ is a multiple of $p$ for any positive integers $x$, $y$.

2015 Korea National Olympiad, 4

Tags: inequalities , number theory , relatively prime , prime numbers , Coprime integers

For a positive integer $n$, $a_1, a_2, \cdots a_k$ are all positive integers without repetition that are not greater than $n$ and relatively prime to $n$. If $k>8$, prove the following. $$\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}$$

2014 Contests, 4

Tags: modular arithmetic , number theory , prime numbers , AMC

The sum of two prime numbers is $85$. What is the product of these two prime numbers? $\textbf{(A) }85\qquad\textbf{(B) }91\qquad\textbf{(C) }115\qquad\textbf{(D) }133\qquad \textbf{(E) }166$

2022 China Team Selection Test, 2

Tags: combinatorics , number theory , prime numbers

Let $p$ be a prime, $A$ is an infinite set of integers. Prove that there is a subset $B$ of $A$ with $2p-2$ elements, such that the arithmetic mean of any pairwise distinct $p$ elements in $B$ does not belong to $A$.

2024 SG Originals, Q5

Tags: combinatorics , primes , grid , number theory , prime numbers

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]