Olympiad problems

2017 Pan-African Shortlist, N2

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

2022 VIASM Summer Challenge, Problem 1

Tags: number theory , prime numbers

Find all prime number pairs $(p,q)$ such that $p(p^2-p-1)=q(2q+3).$

2020 European Mathematical Cup, 3

Tags: number theory , game , prime numbers

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 China Team Selection Test, 6

Tags: number theory , Polynomials , prime numbers , 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 $p^t\mid N$.

2021/2022 Tournament of Towns, P1

Tags: number theory , prime numbers , Tournament of Towns

Let us call a positive integer $k{}$ interesting if the product of the first $k{}$ primes is divisible by $k{}$. For example the product of the first two primes is $2\cdot3 = 6$, it is divisible by 2, hence 2 is an interesting integer. What is the maximal possible number of consecutive interesting integers? [i]Boris Frenkin[/i]

1988 Balkan MO, 4

Tags: number theory , prime numbers , number theory proposed

Let $(a_{n})_{n\geq 1}$ be a sequence defined by $a_{n}=2^{n}+49$. Find all values of $n$ such that $a_{n}=pg, a_{n+1}=rs$, where $p,q,r,s$ are prime numbers with $p<q, r<s$ and $q-p=s-r$.

2011 Akdeniz University MO, 1

Tags: number theory , prime numbers

Let $m,n$ positive integers and $p$ prime number with $p=3k+2$. If $p \mid {(m+n)^2-mn}$ , prove that $$p \mid m,n$$

1973 IMO Shortlist, 4

Tags: number theory , prime numbers , combinatorics , composite numbers , counting , IMO Shortlist

Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided into $7$ disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of $C$ are there ?

2010 China Team Selection Test, 2

Tags: number theory , prime numbers , factorial

Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.

2019 IFYM, Sozopol, 1

Tags: number theory , Sequence , prime numbers

We define the sequence $a_n=(2n)^2+1$ for each natural number $n$. We will call one number [i]bad[/i], if there don’t exist natural numbers $a>1$ and $b>1$ such that $a_n=a^2+b^2$. Prove that the natural number $n$ is [i]bad[/i], if and only if $a_n$ is prime.

2016 Azerbaijan National Mathematical Olympiad, 3

Tags: number theory , prime numbers

Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or number itself) is a prime number. For example, $23$ and $37$ are "very prime" numbers, but $237$ and $357$ are not. Find the largest "prime" number (with justification!).

1969 AMC 12/AHSME, 23

Tags: factorial , number theory , prime numbers , AMC , 1969 AHSME , Ahsme , AMC 12

For any integer $n$ greater than $1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is: $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }\dfrac n2\text{ for }n\text{ even,}\,\dfrac{n+1}2\text{ for }n\text{ odd}$ $\textbf{(D) }n-1\qquad \textbf{(E) }n$

2014 Contests, 1

Tags: number theory , prime numbers , number theory proposed

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?

2011 Philippine MO, 3

Tags: number theory , prime numbers , combinatorial geometry , number theory unsolved

The $2011$th prime number is $17483$ and the next prime is $17489$. Does there exist a sequence of $2011^{2011}$ consecutive positive integers that contain exactly $2011$ prime numbers?

2006 Germany Team Selection Test, 1

Tags: algebra , polynomial , number theory , composite number , prime numbers , IMO Shortlist , Hi

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

2005 China Team Selection Test, 1

Tags: number theory , prime divisors , prime numbers , Divisibility , IMO Shortlist , power of 2 , Zsigmondy

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2023 Brazil Team Selection Test, 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 $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

1997 Mexico National Olympiad, 1

Tags: number theory , prime numbers , prime

Determine all prime numbers $p$ for which $8p^4-3003$ is a positive prime number.

1981 Putnam, B3

Tags: prime numbers , number theory

Prove that there are infinitely many positive $n$ that for all prime divisors $p$ of $n^2 + 3, \exists 0 \leq k \leq \sqrt{n}$ and $p \mid k^2+3$

2024 Baltic Way, 18

Tags: number theory , number theory proposed , Sequence , Divisors , prime numbers , Bound

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.

2011 Iran MO (3rd Round), 5

Tags: number theory , prime numbers

Suppose that $\alpha$ is a real number and $a_1<a_2<.....$ is a strictly increasing sequence of natural numbers such that for each natural number $n$ we have $a_n\le n^{\alpha}$. We call the prime number $q$ golden if there exists a natural number $m$ such that $q|a_m$. Suppose that $q_1<q_2<q_3<.....$ are all the golden prime numbers of the sequence $\{a_n\}$. [b]a)[/b] Prove that if $\alpha=1.5$, then $q_n\le 1390^n$. Can you find a better bound for $q_n$? [b]b)[/b] Prove that if $\alpha=2.4$, then $q_n\le 1390^{2n}$. Can you find a better bound for $q_n$? [i]part [b]a[/b] proposed by mahyar sefidgaran by an idea of this question that the $n$th prime number is less than $2^{2n-2}$ part [b]b[/b] proposed by mostafa einollah zade[/i]

2001 National Olympiad First Round, 23

Tags: modular arithmetic , number theory , prime numbers

Which of the followings is false for the sequence $9,99,999,\dots$? $\textbf{(A)}$ The primes which do not divide any term of the sequence are finite. $\textbf{(B)}$ Infinitely many primes divide infinitely many terms of the sequence. $\textbf{(C)}$ For every positive integer $n$, there is a term which is divisible by at least $n$ distinct prime numbers. $\textbf{(D)}$ There is an inteter $n$ such that every prime number greater than $n$ divides infinitely many terms of the sequence. $\textbf{(E)}$ None of above

2002 AMC 12/AHSME, 12

Tags: quadratics , Vieta , algebra , number theory , prime numbers , special factorizations

Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$

2022 SG Originals, Q5

Tags: number theory , polynomial , modulo , algebra , prime numbers

Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer. [i]Proposed by fattypiggy123[/i]

2023 IMAR Test, P3

Tags: number theory , permutations , prime numbers

Let $p{}$ be an odd prime number. Determine whether there exists a permutation $a_1,\ldots,a_p$ of $1,\ldots,p$ satisfying \[(i-j)a_k+(j-k)a_i+(k-i)a_j\neq 0,\] for all pairwise distinct $i,j,k.$