Olympiad problems

2001 Tournament Of Towns, 2

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?

2012 IMO Shortlist, N3

Tags: binomial coefficient , number theory , prime number

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

2015 Iran MO (2nd Round), 3

Tags: number theory , prime number , algebra

Let $n \ge 50 $ be a natural number. Prove that $n$ is expressible as sum of two natural numbers $n=x+y$, so that for every prime number $p$ such that $ p\mid x$ or $p\mid y $ we have $ \sqrt{n} \ge p $. For example for $n=94$ we have $x=80, y=14$.

2009 AMC 12/AHSME, 19

Tags: algebra , polynomial , search , vieta , prime number , number theory

For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers? $ \textbf{(A)}\ 794\qquad \textbf{(B)}\ 796\qquad \textbf{(C)}\ 798\qquad \textbf{(D)}\ 800\qquad \textbf{(E)}\ 802$

2007 Tournament Of Towns, 5

Tags: prime number , number theory

Find all (finite) increasing arithmetic progressions, consisting only of prime numbers, such that the number of terms is larger than the common difference.

2006 Germany Team Selection Test, 1

Tags: algebra , polynomial , number theory , composite number , prime number

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 USAMO, 1

Tags: induction , number theory , prime number , relatively prime , combinatorics , arrangement , divisor

Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

2010 China Team Selection Test, 2

Tags: prime number , factorial , number theory

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 $n+a_n$.

2003 APMO, 3

Tags: number theory , prime number

Let $k\ge 14$ be an integer, and let $p_k$ be the largest prime number which is strictly less than $k$. You may assume that $p_k\ge 3k/4$. Let $n$ be a composite integer. Prove: (a) if $n=2p_k$, then $n$ does not divide $(n-k)!$; (b) if $n>2p_k$, then $n$ divides $(n-k)!$.

2010 Bundeswettbewerb Mathematik, 1

Tags: number theory , prime number , digit

Exists a positive integer $n$ such that the number $\underbrace{1...1}_{n \,ones} 2 \underbrace{1...1}_{n \, ones}$ is a prime number?

2008 Bulgarian Autumn Math Competition, Problem 8.3

Tags: prime number , sum of digits , number theory

Prove that there exists a prime number $p$, such that the sum of digits of $p$ is a composite odd integer. Find the smallest such $p$.

2000 Flanders Math Olympiad, 3

Tags: number theory , prime number

Let $p_n$ be the $n$-th prime. ($p_1=2$) Define the sequence $(f_j)$ as follows: - $f_1=1, f_2=2$ - $\forall j\ge 2$: if $f_j = kp_n$ for $k<p_n$ then $f_{j+1}=(k+1)p_n$ - $\forall j\ge 2$: if $f_j = p_n^2$ then $f_{j+1}=p_{n+1}$ (a) Show that all $f_i$ are different (b) from which index onwards are all $f_i$ at least 3 digits? (c) which integers do not appear in the sequence? (d) how many numbers with less than 3 digits appear in the sequence?

2016 Azerbaijan National Mathematical Olympiad, 3

Tags: number theory , prime number

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!).

2025 Poland - First Round, 3

Tags: prime number , number theory

Let $n$ be a product of 2024 different prime numbers. Find the number of positive integers $k$, such that $$n+gcd(n, k)=k.$$

1998 IMO Shortlist, 5

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 $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

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 China Team Selection Test, 4

Tags: number theory , sequence , prime number

Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$. Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.

2017 India PRMO, 23

Tags: algebra , number theory , prime number

Suppose an integer $x$, a natural number $n$ and a prime number $p$ satisfy the equation $7x^2-44x+12=p^n$. Find the largest value of $p$.

2010 Contests, 1

Tags: function , prime number , arithmetic sequence , number theory , logarithm

Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

2015 Miklos Schweitzer, 4

Tags: limit , algebra and number theory , prime number , number theory

Let $a_n$ be a series of positive integers with $a_1=1$ and for any arbitrary prime number $p$, the set $\{a_1,a_2,\cdots,a_p\}$ is a complete remainder system modulo $p$. Prove that $\lim_{n\rightarrow \infty} \cfrac{a_n}{n}=1$.

2022 IMO Shortlist, N7

Tags: number theory , prime number

Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.

2011 IFYM, Sozopol, 2

Tags: number theory , prime number

Let $k>1$ and $n$ be natural numbers and $p=\frac{((n+1)(n+2)…(n+k))}{k!}-1$. Prove that, if $p$ is prime, then $n|k!$.

2022 Kurschak Competition, 2

Tags: number theory , prime number , square

Let $p$ and $q$ be prime numbers of the form $4k+3$. Suppose that there exist integers $x$ and $y$ such that $x^2-pqy^2=1$. Prove that there exist positive integers $a$ and $b$ such that $|pa^2-qb^2|=1$.

2015 Germany Team Selection Test, 3

Tags: combinatorics , algebra , prime number

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

1960 AMC 12/AHSME, 33

Tags: number theory , prime number

You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times... \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $2, 3, 4, ...., 59$. let $N$ be the number of primes appearing in this sequence. Then $N$ is: $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 57\qquad\textbf{(E)}\ 58 $