Olympiad problems

2003 Poland - Second Round, 4

Prove that for any prime number $p > 3$ exist integers $x, y, k$ that meet conditions: $0 < 2k < p$ and $kp + 3 = x^2 + y^2$.

2019 Switzerland Team Selection Test, 2

Tags: number theory , prime number

Find the largest prime $p$ such that there exist positive integers $a,b$ satisfying $$p=\frac{b}{2}\sqrt{\frac{a-b}{a+b}}.$$

2024 Assara - South Russian Girl's MO, 2

Tags: number theory , prime number

Let $p$ be a prime number. Positive integers numbers $a$ and $b$ are such $\frac{p}{a}+\frac{p}{b}=1$ and $a+b$ is divisible by $p$. What values can an expression $\frac{a+b}{p}$ take? [i]Yu.A.Karpenko[/i]

2012 National Olympiad First Round, 18

Tags: number theory , prime number

If the representation of a positive number as a product of powers of distinct prime numbers contains no even powers other than $0$s, we will call the number singular. At most how many consequtive singular numbers are there? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \text{None}$

2001 Tournament Of Towns, 2

Tags: prime number , number theory

There exists a block of 1000 consecutive positive integers containing no prime numbers, namely, $1001!+2,1001!+3,...,1001!+1001$. Does there exist a block of 1000 consecutive positive intgers containing exactly five prime numbers?

1997 Korea National Olympiad, 4

Tags: number theory , prime number

For any prime number $p>2,$ and an integer $a$ and $b,$ if $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{(p-1)^3}=\frac{a}{b},$ prove that $a$ is divisible by $p.$

2019 Singapore MO Open, 4

Tags: number theory , prime number , algebra , polynomial

Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$. [i]Proposed by fattypiggy123[/i]

1987 IMO Longlists, 69

Tags: number theory , polynomial , prime number , quadratic

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i] [b][i]Original Formulation[/i][/b] Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes. [i]Proposed by Soviet Union. [/i]

2019 Olympic Revenge, 5

Tags: function , number theory , prime number , modular arithmetic

Define $f: \mathbb{N} \rightarrow \mathbb{N}$ by $$f(n) = \sum \frac{(1+\sum_{i=1}^{n} t_i)!}{(1+t_1) \cdot \prod_{i=1}^{n} (t_i!) }$$ where the sum runs through all $n$-tuples such that $\sum_{j=1}^{n}j \cdot t_j=n$ and $t_j \ge 0$ for all $1 \le j \le n$. Given a prime $p$ greater than $3$, prove that $$\sum_{1 \le i < j <k \le p-1 } \frac{f(i)}{i \cdot j \cdot k} \equiv \sum_{1 \le i < j <k \le p-1 } \frac{2^i}{i \cdot j \cdot k} \pmod{p}.$$

2022 Austrian MO National Competition, 4

Tags: prime number , number theory , diophantine equation , diophantine

Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$ holds. [i](Walther Janous)[/i]

2018 Harvard-MIT Mathematics Tournament, 4

Tags: number theory , prime number

Distinct prime numbers $p,q,r$ satisfy the equation $$2pqr+50pq=7pqr+55pr=8pqr+12qr=A$$ for some positive integer $A.$ What is $A$?

2006 ISI B.Stat Entrance Exam, 3

Tags: induction , modular arithmetic , number theory , prime number

Prove that $n^4 + 4^{n}$ is composite for all values of $n$ greater than $1$.

2015 Thailand TSTST, 1

Tags: number theory , prime number

Find all primes $1 < p < 100$ such that the equation $x^2-6y^2=p$ has an integer solution $(x, y)$.

Russian TST 2015, P1

Tags: prime number , number theory

Prove that there exist two natural numbers $a,b$ such that $|a-m|+|b-n|>1000$ for any relatively prime natural numbers $m,n$.

1989 IMO Longlists, 93

Tags: modular arithmetic , number theory , prime number , sequence , power of number

Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

2019 Turkey Team SeIection Test, 2

Tags: number theory , sequence , prime number

$(a_{n})_{n=1}^{\infty}$ is an integer sequence, $a_{1}=1$, $a_{2}=2$ and for $n\geq{1}$, $a_{n+2}=a_{n+1}^{2}+(n+2)a_{n+1}-a_{n}^{2}-na_{n}$. $a)$ Prove that the set of primes that divides at least one term of the sequence can not be finite. $b)$ Find 3 different prime numbers that do not divide any terms of this sequence.

2015 JBMO TST - Turkey, 1

Tags: number theory , prime number

Let $p,q$ be prime numbers such that their sum isn't divisible by $3$. Find the all $(p,q,r,n)$ positive integer quadruples satisfy: $$p+q=r(p-q)^n$$ [i]Proposed by Şahin Emrah[/i]

2001 Bundeswettbewerb Mathematik, 4

Tags: prime factorization , prime number , number theory

Prove: For each positive integer is the number of divisors whose decimal representations ends with a 1 or 9 not less than the number of divisors whose decimal representations ends with 3 or 7.

2021 Kyiv City MO Round 1, 8.5

Tags: number theory , prime number

For a prime number $p > 3$, define the following irreducible fraction: $$\frac{m}{n} = \frac{p-1}{2} + \frac{p-2}{3} + \ldots + \frac{2}{p-1} - 1$$ Prove that $m$ is divisible by $p$. [i]Proposed by Oleksii Masalitin[/i]

2004 Baltic Way, 8

Tags: algebra , polynomial , inequalities , function , number theory , prime number , logarithm

Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.

1974 Putnam, A3

Tags: number theory , perfect square , prime number

A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squares ($p = m^2 +n^2$ , with $m$ and $n$ integers) if and only if $p \equiv 1$ (mod $4$). Assuming this result, find which primes $p > 2$ can be written in each of the following forms, using integers $x$ and $y$: a) $x^2 +16y^2, $ b) $4x^2 +4xy+ 5y^2.$

2006 National Olympiad First Round, 2

Tags: number theory , prime number

If $p$ and $p^2+2$ are prime numbers, at most how many prime divisors can $p^3+3$ have? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

2019 Switzerland Team Selection Test, 4

Tags: number theory , polynomial , prime number , algebra

Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties: $(a)$ $P(x)>x$ for all positive integers $x$. $(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some $l\geq 0$ such that $m\mid p_l$.

1990 IMO Longlists, 98

Tags: number theory , decimal representation , prime number , digit

Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

1998 IMO, 6

Tags: number theory , prime number , algebra , functional equation

Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]