Olympiad problems

PEN E Problems, 35

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

1994 China Team Selection Test, 2

Tags: algebra , polynomial , calculus , integration , gauss , prime number , number theory

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

2009 Princeton University Math Competition, 5

Tags: number theory , prime number

Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$, $b$ and $c$.

2017 China Team Selection Test, 6

Tags: number theory , prime number , algebra , polynomial

For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.

2021 Bangladeshi National Mathematical Olympiad, 1

Tags: number theory , prime number , counting

How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?

2011 Putnam, B2

Tags: number theory , prime number , algebra , polynomial , rational root theorem

Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0.$ Which primes appear in seven or more elements of $S?$

2017 Dutch IMO TST, 2

Tags: number theory , prime number

Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum. Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)

2016 IberoAmerican, 1

Tags: number theory , prime number

Find all prime numbers $p,q,r,k$ such that $pq+qr+rp = 12k+1$

2010 Contests, 1

Tags: prime number , number theory

a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers. b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?

PEN C Problems, 4

Tags: floor function , quadratic , inequalities , number theory , prime number , quadratic residue

Let $M$ be an integer, and let $p$ be a prime with $p>25$. Show that the set $\{M, M+1, \cdots, M+ 3\lfloor \sqrt{p} \rfloor -1\}$ contains a quadratic non-residue to modulus $p$.

1995 AIME Problems, 10

Tags: calculus , integration , number theory , prime number

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

1993 IMO Shortlist, 2

Tags: modular arithmetic , number theory , divisibility , prime number , composite number

A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$ a.) Show that every prime number $n$ has property $P.$ b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$

2021 Stars of Mathematics, 1

Tags: number theory , prime number

For every integer $n\geq 3$, let $s_n$ be the sum of all primes (strictly) less than $n$. Are there infinitely many integers $n\geq 3$ such that $s_n$ is coprime to $n$? [i]Russian Competition[/i]

2023 India IMO Training Camp, 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 $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2021 Argentina National Olympiad, 5

Tags: number theory , prime number

The sequence $a_n (n\geq 1)$ of natural numbers is defined as $a_{n+1}=a_n+b_n,$ where $b_n$ is the number that has the same digits as $a_n$ but in the opposite order ($b_n$ can start with $0$). For example, if $a_1=180,$ then $a_2=261, a_3=423.$ a) Decide if $a_1$ can be chosen so that $a_7$ is prime. b) Decide if $a_1$ can be chosen so that $a_5$ is prime.

Russian TST 2015, P2

Tags: number theory , prime number

Let $p\geqslant 5$ be a prime number. Prove that the set $\{1,2,\ldots,p - 1\}$ can be divided into two nonempty subsets so that the sum of all the numbers in one subset and the product of all the numbers in the other subset give the same remainder modulo $p{}$.

2014 Iran Team Selection Test, 2

Tags: function , prime number , number theory

is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $i) \exists n\in \mathbb{N}:f(n)\neq n$ $ii)$ the number of divisors of $m$ is $f(n)$ if and only if the number of divisors of $f(m)$ is $n$

2022 IMO Shortlist, N6

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

2005 IMO Shortlist, 5

Tags: number theory , prime number , prime factorization

Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

2025 Polish MO Finals, 2

Tags: number theory , prime number

Positive integers $k, m, n ,p $ integers are such that $p=2^{2^n}+1$ is prime and $p\mid 2^k-m$. Prove that there exists a positive integer $l$ such that $p^2\mid 2^l-m$.

2007 Bulgarian Autumn Math Competition, Problem 12.4

Tags: prime number , number theory , linear recurrence

Let $p$ and $q$ be prime numbers and $\{a_{n}\}_{n=1}^{\infty}$ be a sequence of integers defined by: \[a_{0}=0, a_{1}=1, a_{n+2}=pa_{n+1}-qa_{n}\quad\forall n\geq 0\] Find $p$ and $q$ if there exists an integer $k$ such that $a_{3k}=-3$.

2024 Baltic Way, 17

Tags: prime number , divisibility , bound , number theory , fermat's little theorem

Do there exist infinitely many quadruples $(a,b,c,d)$ of positive integers such that the number $a^{a!} + b^{b!} - c^{c!} - d^{d!}$ is prime and $2 \leq d \leq c \leq b \leq a \leq d^{2024}$?

PEN E Problems, 24

Tags: inequalities , floor function , limit , number theory , prime number , logarithm

Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.

2016 Tuymaada Olympiad, 5

Tags: prime number , number theory

The ratio of prime numbers $p$ and $q$ does not exceed 2 ($p\ne q$). Prove that there are two consecutive positive integers such that the largest prime divisor of one of them is $p$ and that of the other is $q$.

2008 Irish Math Olympiad, 1

Tags: prime number , number theory

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$