Olympiad problems

2017 Romania National Olympiad, 4

Find all prime numbers with $n \ge 3$ digits, having the property: for every $k \in \{1, 2, . . . , n -2\}$, deleting any $k$ of its digits leaves a prime number.

1969 Czech and Slovak Olympiad III A, 3

Tags: number theory , prime , fraction , sequence

Let $p$ be a prime. How many different (infinite) sequences $\left(a_k\right)_{k\ge0}$ exist such that for every positive integer $n$ \[\frac{a_0}{a_1}+\frac{a_0}{a_2}+\cdots+\frac{a_0}{a_n}+\frac{p}{a_{n+1}}=1?\]

2013 Balkan MO Shortlist, N1

Tags: number theory , prime , reciprocal sum

Let $p$ be a prime number. Determine all triples $(a,b,c)$ of positive integers such that $a + b + c < 2p\sqrt{p}$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{p}$

2007 Postal Coaching, 3

Tags: power of 3 , number theory , prime

Suppose $n$ is a natural number such that $4^n + 2^n + 1$ is a prime. Prove that $n = 3^k$ for some nonnegative integer $k$.

2000 Estonia National Olympiad, 1

Tags: number theory , power of prime , prime , remainder

Find all prime numbers whose sixth power does not give remainder $1$ when dividing by $504$

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): $k!=(p^3-1)(p^3-p)(p^3-p^2)$

2022 Mexico National Olympiad, 1

Tags: reciprocal , prime

A number $x$ is "Tlahuica" if there exist prime numbers $p_1,\ p_2,\ \dots,\ p_k$ such that \[x=\frac{1}{p_1}+\frac{1}{p_2}+\dots+\frac{1}{p_k}.\] Find the largest Tlahuica number $x$ such that $0<x<1$ and there exists a positive integer $m\leq 2022$ such that $mx$ is an integer.

2022 AMC 12/AHSME, 3

Tags: prime

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

1997 Israel National Olympiad, 3

Tags: inequalities , product , number theory , prime

Let $n?$ denote the product of all primes smaller than $n$. Prove that $n? > n$ holds for any natural number $n > 3$.

1999 Estonia National Olympiad, 1

Tags: divisible , odd , number theory , prime

Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$.

1997 Estonia National Olympiad, 1

Tags: number theory , composite , prime

Prove that a positive integer $n$ is composite if and only if there exist positive integers $a,b,x,y$ such that $a+b = n$ and $\frac{x}{a}+\frac{y}{b}= 1$.

2013 IMAR Test, 1

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

Given a prime $p \geq 5$ , show that there exist at least two distinct primes $q$ and $r$ in the range $2, 3, \ldots p-2$ such that $q^{p-1} \not\equiv 1 \pmod{p^2}$ and $r^{p-1} \not\equiv 1 \pmod{p^2}$.

1995 IMO, 6

Tags: modular arithmetic , number theory , counting , prime

Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $ p$?

1955 Moscow Mathematical Olympiad, 299

Tags: number theory , arithmetic progression , sequence , divisible , prime

Suppose that primes $a_1, a_2, . . . , a_p$ form an increasing arithmetic progression and $a_1 > p$. Prove that if $p$ is a prime, then the difference of the progression is divisible by $p$.

2013 Abels Math Contest (Norwegian MO) Final, 3

Tags: number theory , prime , divide

A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$. Show that $p$ divides $a$.

2001 Estonia Team Selection Test, 5

Tags: product , power , number theory , digit , prime

Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers

2009 Cuba MO, 1

Tags: number theory , remainder , prime

Show that when a prime number is divided by $30$, the remainder is $1$ or a prime number. Shows that if it is divided by $60$ or $90$ the same thing does not happen.

2018 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: sequence , prime , composite , infinitely many solutions , number theory

We observe that number $10001=73\cdot137$ is not prime. Show that every member of infinite sequence $10001, 100010001, 1000100010001,...$ is not prime

1995 India National Olympiad, 6

Tags: number theory , prime , greatest common divisor

Find all primes $p$ for which the quotient \[ \dfrac{2^{p-1} - 1 }{p} \] is a square.

2023 Germany Team Selection Test, 1

Tags: power of 2 , prime , number theory

Does there exist a positive odd integer $n$ so that there are primes $p_1$, $p_2$ dividing $2^n-1$ with $p_1-p_2=2$?

1997 Mexico National Olympiad, 1

Tags: number theory , prime number , prime

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

2016 Costa Rica - Final Round, N2

Tags: prime , number theory

Determine all positive integers $a$ and $b$ for which $a^4 + 4b^4$ be a prime number.

2021 Indonesia TST, C

Tags: subset , counting , prime , summation , combinatorics

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 $p$.

2017 Puerto Rico Team Selection Test, 5

Tags: number theory , odd , prime

Find a pair prime numbers $(p, q)$, $p> q$ of , if any, such that $\frac{p^2 - q^2}{4}$ is an odd integer.

2018 India PRMO, 1

Tags: prime , divisor , number theory

A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is $1709$. If $n$ is the last page number, what is the largest prime factor of $n$?