Olympiad problems

2014 CentroAmerican, 1

A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?

2003 Iran MO (3rd Round), 3

Tags: prime number , number theory

assume that A is a finite subset of prime numbers, and a is an positive integer. prove that there are only finitely many positive integers m s.t: prime divisors of a^m-1 are contained in A.

2024 Mexican University Math Olympiad, 1

Tags: number theory , prime number , perfect square

Let $ x $, $ y $, $ p $ be positive integers that satisfy the equation $ x^4 = p + 9y^4 $, where $ p $ is a prime number. Show that $ \frac{p^2 - 1}{3} $ is a perfect square and a multiple of 16.

2017 Pan-African Shortlist, N2

Tags: number theory , prime number , sum of cubes , algebra , factorization

For which prime numbers $p$ can we find three positive integers $n$, $x$ and $y$ such that $p^n = x^3 + y^3$?

2001 May Olympiad, 4

Tags: number theory , prime number

Using only prime numbers, a set is formed with the following conditions: Any one-digit prime number can be in the set. For a prime number with more than one digit to be in the set, the number that results from deleting only the first digit and also the number that results from deleting only the last digit must be in the set. Write, of the sets that meet these conditions, the one with the greatest number of elements. Justify why there cannot be one with more elements. Remember that the number $1$ is not prime.

1955 Miklós Schweitzer, 4

Tags: number theory , prime number

[b]4.[/b] Find all positive integers $\alpha , \beta (\alpha >1)$ and all prime numbers $p, q, r$ which satisfy the equation $p^{\alpha}= q^{\beta}+r^{\alpha}$ ($\alpha , \beta , p, q, r$ need not necessarily be different). [b](N. 12)[/b]

2025 Francophone Mathematical Olympiad, 4

Tags: number theory , prime number

Determine all sequences of strictly positive integers $a_1, a_2, a_3, \ldots$ satisfying the following two conditions: [list] [*]There exists an integer $M > 0$ such that, for all indices $n \geqslant 1$, $0 < a_n \leqslant M$. [*]For any prime number $p$ and for any index $n \geqslant 1$, the number \[ a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p} \] is a multiple of $p$. [/list]

2013 Dutch IMO TST, 3

Tags: prime number , number theory

Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.

2018 PUMaC Live Round, 4.1

Tags: number theory , prime number

The number $400000001$ can be written as $p\cdot q$, where $p$ and $q$ are prime numbers. Find the sum of the prime factors of $p+q-1$.

1991 AIME Problems, 5

Tags: number theory , prime number , prime factorization

Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $ 20!$ be the resulting product?

2016 JBMO TST - Turkey, 7

Tags: number theory , prime number , dumb bounding

Find all pairs $(p, q)$ of prime numbers satisfying \[ p^3+7q=q^9+5p^2+18p. \]

2018 IFYM, Sozopol, 1

Tags: number theory , prime number

Find all prime numbers $p$ and all positive integers $n$, such that $n^8 - n^2 = p^5 + p^2$

2023 India Regional Mathematical Olympiad, 2

Tags: number theory , prime number

Given a prime number $p$ such that $2p$ is equal to the sum of the squares of some four consecutive positive integers. Prove that $p-7$ is divisible by 36.

2013 Singapore Junior Math Olympiad, 3

Tags: number theory , prime number

Find all prime numbers which can be presented as a sum of two primes and difference of two primes at the same time.

2020 Junior Balkan Team Selection Tests - Moldova, 10

Tags: number theory , prime number , perfect square

Find all pairs of prime numbers $(p, q)$ for which the numbers $p+q$ and $p+4q$ are simultaneously perfect squares.

2024 CAPS Match, 6

Tags: combinatorics , number theory , combinatorial number theory , prime number , triplets , positive integer

Determine whether there exist infinitely many triples $(a, b, c)$ of positive integers such that every prime $p$ divides \[\left\lfloor\left(a+b\sqrt{2024}\right)^p\right\rfloor-c.\]

2023 Brazil Team Selection Test, 3

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

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

2017 Iran MO (3rd round), 1

Tags: number theory , divisibility , prime number , prime

Let $n$ be a positive integer. Consider prime numbers $p_1,\dots ,p_k$. Let $a_1,\dots,a_m$ be all positive integers less than $n$ such that are not divisible by $p_i$ for all $1 \le i \le n$. Prove that if $m\ge 2$ then $$\frac{1}{a_1}+\dots+\frac{1}{a_m}$$ is not an integer.

2003 AMC 10, 14

Tags: number theory , prime number

Let $ n$ be the largest integer that is the product of exactly $ 3$ distinct prime numbers, $ d$, $ e$, and $ 10d\plus{}e$, where $ d$ and $ e$ are single digits. What is the sum of the digits of $ n$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$

2024 Baltic Way, 18

Tags: sequence , prime number , bound , number theory , divisor

An infinite sequence $a_1, a_2,\ldots$ of positive integers is such that $a_n \geq 2$ and $a_{n+2}$ divides $a_{n+1} + a_n$ for all $n \geq 1$. Prove that there exists a prime which divides infinitely many terms of the sequence.

2012 IMO Shortlist, N3

Tags: number theory , prime number , binomial coefficient

Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.

2020 European Mathematical Cup, 3

Tags: number theory , game , prime number

Let $p$ be a prime number. Troy and Abed are playing a game. Troy writes a positive integer $X$ on the board, and gives a sequence $(a_n)_{n\in\mathbb{N}}$ of positive integers to Abed. Abed now makes a sequence of moves. The $n$-th move is the following: $$\text{ Replace } Y \text{ currently written on the board with either } Y + a_n \text{ or } Y \cdot a_n.$$ Abed wins if at some point the number on the board is a multiple of $p$. Determine whether Abed can win, regardless of Troy’s choices, if $a) p = 10^9 + 7$; $b) p = 10^9 + 9$. [i]Remark[/i]: Both $10^9 + 7$ and $10^9 + 9$ are prime. [i]Proposed by Ivan Novak[/i]

2017 Iran Team Selection Test, 4

Tags: number theory , prime number , modular arithmetic

We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$. Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$. Is there always a number $x$ that satisfies all the equations? [i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]

2016 IMC, 5

Tags: prime number , permutation

Let $S_n$ denote the set of permutations of the sequence $(1,2,\dots, n)$. For every permutation $\pi=(\pi_1, \dots, \pi_n)\in S_n$, let $\mathrm{inv}(\pi)$ be the number of pairs $1\le i < j \le n$ with $\pi_i>\pi_j$; i. e. the number of inversions in $\pi$. Denote by $f(n)$ the number of permutations $\pi\in S_n$ for which $\mathrm{inv}(\pi)$ is divisible by $n+1$. Prove that there exist infinitely many primes $p$ such that $f(p-1)>\frac{(p-1)!}{p}$, and infinitely many primes $p$ such that $f(p-1)<\frac{(p-1)!}{p}$. (Proposed by Fedor Petrov, St. Petersburg State University)

2015 Thailand TSTST, 2

Tags: prime number , number theory

Fix a sequence $ a_1,a_2,a_3,... $ of integers satisfying the following condition:for all prime numbers $ p $ and all positive integers $ k $, we have $ a_{pk+1}=pa_k-3a_p+13 $.Determine all possible values of $ a_{2013} $.