Olympiad problems

Mathematical Minds 2024, P6

Consider the sequence $a_1, a_2, \dots$ of positive integers such that $a_1=2$ and $a_{n+1}=a_n^4+a_n^3-3a_n^2-a_n+2$, for all $n\geqslant 1$. Prove that there exist infinitely many prime numbers that don't divide any term of the sequence. [i]Proposed by Pavel Ciurea[/i]

2013 Thailand Mathematical Olympiad, 1

Tags: divide , prime , number theory

Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$

2024 Indonesia MO, 2

Tags: factorization , number theory , nt construction , prime

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.

2013 Taiwan TST Round 1, 1

Tags: number theory , prime , prime number

Starting from 37, adding 5 before each previous term, forms the following sequence: \[37,537,5537,55537,555537,...\] How many prime numbers are there in this sequence?

1941 Moscow Mathematical Olympiad, 085

Tags: prime , remainder , number theory

Prove that the remainder after division of the square of any prime $p > 3$ by $12$ is equal to $1$.

2014 Estonia Team Selection Test, 1

Tags: combinatorics , prime

In Wonderland, the government of each country consists of exactly $a$ men and $b$ women, where $a$ and $b$ are fixed natural numbers and $b > 1$. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least $n$ among whom are women, are formed (where $n$ is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.

2005 iTest, 17

Tags: number theory , sum of digits , prime

On the $2004$ iTest, we defined an [i]optimus [/i] prime to be any prime number whose digits sum to a prime number. (For example, $83$ is an optimus prime, because it is a prime number and its digits sum to $11$, which is also a prime number.) Given that you select a prime number under $100$, find the probability that is it not an optimus prime.

2010 Saudi Arabia Pre-TST, 2.2

Tags: prime , consecutive , number theory , sum of squares

Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.

2003 Cuba MO, 1

Tags: number theory , digit , prime

Given the following list of numbers: $$1990, 1991, 1992, ..., 2002, 2003, 2003, 2003, ..., 2003$$ where the number $2003$ appears $12$ times. Is it possible to write these numbers in some order so that the $100$-digit number that we get is prime?

1994 Poland - Second Round, 6

Tags: number theory , divide , prime

Let $p$ be a prime number. Prove that there exists $n \in Z$ such that $p | n^2 -n+3$ if and only if there exists $m \in Z$ such that $p | m^2 -m+25$.

2019 Tournament Of Towns, 1

Tags: factor , inequalities , number theory , prime , number of divisors , prime factorization , divisor

Let us call the number of factors in the prime decomposition of an integer $n > 1$ the complexity of $n$. For example, [i]complexity [/i] of numbers $4$ and $6$ is equal to $2$. Find all $n$ such that all integers between $n$ and $2n$ have complexity a) not greater than the complexity of $n$. b) less than the complexity of $n$. (Boris Frenkin)

2022 AMC 12/AHSME, 3

Tags: prime

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

2021 Azerbaijan EGMO TST, 1

Tags: prime number , prime , number theory

p is a prime number, k is a positive integer Find all (p, k): $k!=(p^3-1)(p^3-p)(p^3-p^2)$

2016 Latvia Baltic Way TST, 17

Tags: number theory , diophantine equation , diophantine , prime

Can you find five prime numbers $p, q, r, s, t$ such that $p^3+q^3+r^3+s^3 =t^3$?

2019 Saudi Arabia Pre-TST + Training Tests, 5.1

Tags: number theory , prime , divide , divisible

Let $n$ be a positive integer and $p > n+1$ a prime. Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$

1978 IMO Longlists, 17

Tags: quadratic , number theory , prime , sum of squares

Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$. Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.

2022 Olimphíada, 1

Tags: divisibility , prime

Let $p,q$ prime numbers such that $$p+q \mid p^3-q^3$$ Show that $p=q$.

2021 Indonesia TST, C

Tags: combinatorics , subset , counting , prime , summation

Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.

2021 Dutch Mathematical Olympiad, 5

Tags: number theory , prime , divisor

We consider an integer $n > 1$ with the following property: for every positive divisor $d$ of $n$ we have that $d + 1$ is a divisor of$ n + 1$. Prove that $n$ is a prime number.

2013 Hanoi Open Mathematics Competitions, 1

Tags: minimum , number theory , prime

Write $2013$ as a sum of $m$ prime numbers. The smallest value of $m$ is: (A): $2$, (B): $3$, (C): $4$, (D): $1$, (E): None of the above.

2022 Serbia National Math Olympiad, P6

Tags: prime , algebra

Let $p$ and $q$ be different primes, and $\alpha\in (0, 3)$ a real number. Prove that in sequence $$\left[ \alpha \right] , \left[ 2\alpha \right] , \left[ 3\alpha \right] \dots$$ exists number less than $2pq$, divisible by $p$ or $q$.

2018 Peru IMO TST, 10

Tags: recurrence relation , number theory , number theory with sequences , product , prime

For each positive integer $m> 1$, let $P (m)$ be the product of all prime numbers that divide $m$. Define the sequence $a_1, a_2, a_3,...$ as followed: $a_1> 1$ is an arbitrary positive integer, $a_{n + 1} = a_n + P (a_n)$ for each positive integer $n$. Prove that there exist positive integers $j$ and $k$ such that $a_j$ is the product of the first $k$ prime numbers.

1965 Poland - Second Round, 4

Tags: number theory , prime

Find all prime numbers $ p $ such that $ 4p^2 + 1 $ and $ 6p^2 + 1 $ are also prime numbers.

2022 Bulgarian Spring Math Competition, Problem 10.4

Tags: algebra , system of equations , prime

Find the smallest odd prime $p$, such that there exist coprime positive integers $k$ and $\ell$ which satisfy \[4k-3\ell=12\quad \text{ and }\quad \ell^2+\ell k +k^2\equiv 3\text{ }(\text{mod }p)\]

2014 Indonesia MO Shortlist, N1

Tags: number theory , diophantine equation , prime

(a) Let $k$ be an natural number so that the equation $ab + (a + 1) (b + 1) = 2^k$ does not have a positive integer solution $(a, b)$. Show that $k + 1$ is a prime number. (b) Show that there are natural numbers $k$ so that $k + 1$ is prime numbers and equation $ab + (a + 1) (b + 1) = 2^k$ has a positive integer solution $(a, b)$.