Olympiad problems

2011 Belarus Team Selection Test, 1

Given natural number $a>1$ and different odd prime numbers $p_1,p_2,...,p_n$ with $a^{p_1}\equiv 1$ (mod $p_2$), $a^{p_2}\equiv 1$ (mod $p_3$), ..., $a^{p_n}\equiv 1$(mod $p_1$). Prove that a) $(a-1)\vdots p_i$ for some $i=1,..,n$ b) Can $(a-1)$ be divisible by $p_i $for exactly one $i$ of $i=1,...,n$? I. Bliznets

2001 Estonia National Olympiad, 4

Tags: composite , sum of powers , prime , number theory

Prove that for any integer $a > 1$ there is a prime $p$ for which $1+a+a^2+...+ a^{p-1}$ is composite.

2021 Nigerian MO Round 3, Problem 1

Tags: diophantine equation , prime , number theory

Find all triples of primes $(p, q, r)$ such that $p^q=2021+r^3$.

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 $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{p}$

1978 Chisinau City MO, 159

Tags: number theory , divide , prime , factorial , divisible

Prove that the product of numbers $1, 2, ..., n$ ($n \ge 2$) is divisible by their sum if and only if the number $n + 1$ is not prime.

2012 Tournament of Towns, 2

Tags: number theory , divisor , prime

Let $C(n)$ be the number of prime divisors of a positive integer n. (For example, $C(10) = 2,C(11) = 1, C(12) = 2$). Consider set S of all pairs of positive integers $(a, b)$ such that $a\ne b$ and $C(a + b) = C(a) + C(b)$. Is set $S$ finite or infinite?

1979 IMO Longlists, 25

Tags: series summation , divisibility , prime , number theory , relatively prime

If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

2014 Czech-Polish-Slovak Junior Match, 3

Tags: prime , number theory

Find with all integers $n$ when $|n^3 - 4n^2 + 3n - 35|$ and $|n^2 + 4n + 8|$ are prime numbers.

2007 Bulgarian Autumn Math Competition, Problem 8.3

Tags: prime , number theory , square , prime number

Determine all triplets of prime numbers $p<q<r$, such that $p+q=r$ and $(r-p)(q-p)-27p$ is a square.

2022 Cyprus JBMO TST, 2

Tags: prime , prime number , number theory

Determine all pairs of prime numbers $(p, q)$ which satisfy the equation \[ p^3+q^3+1=p^2q^2 \]

2015 Caucasus Mathematical Olympiad, 1

Tags: prime , natural number , product , prime number , number theory

Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.

2016 Mediterranean Mathematics Olympiad, 4

Tags: prime , number theory

Determine all integers $n\ge1$ for which the number $n^8+n^6+n^4+4$ is prime. (Proposed by Gerhard Woeginger, Austria)

2012 Czech-Polish-Slovak Junior Match, 2

Tags: prime , diophantine , number theory , diophantine equation

Determine all three primes $(a, b, c)$ that satisfied the equality $a^2+ab+b^2=c^2+3$.

2007 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: modulo , prime , number theory

Let $p > 3$ be a prime number and $ x$ an integer, denote by $r ( x )\in \{ 0 , 1 , ... , p - 1 \}$ to the rest of $x$ modulo $p$ . Let $x_1, x_2, ... , x_k$ ( $2 < k < p$) different integers modulo $p$ and not divisible by $p$. We say that a number $a \in \{ 1 , 2 ,..., p -1 \}$ is [i]good [/i] if $r ( a x_1) < r ( a x_2) <...< r ( a x_k)$. Show that there are at most $\frac{2 p}{k + 1}-{ 1}$ [i]good [/i] numbers.

2002 VJIMC, Problem 2

Tags: number theory , prime , divisibility

Let $p>3$ be a prime number and $n=\frac{2^{2p}-1}3$. Show that $n$ divides $2^n-2$.

1998 Estonia National Olympiad, 2

Tags: prime , number theory , digit

Find all prime numbers of the form $10101...01$.

1989 Austrian-Polish Competition, 3

Tags: prime , digit , number theory

Find all natural numbers $N$ (in decimal system) with the following properties: (i) $N =\overline{aabb}$, where $\overline{aab}$ and $\overline{abb}$ are primes, (ii) $N = P_1P_2P_3$, where $P_k (k = 1,2,3)$ is a prime consisting of $k$ (decimal) digits.

2023 Germany Team Selection Test, 1

Tags: number theory , prime , power of 2

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

2023 Belarus Team Selection Test, 1.3

Tags: number theory , prime factorization , prime number , prime , function

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

2016 Lusophon Mathematical Olympiad, 5

Tags: number theory , prime , prime number , sequence

A numerical sequence is called lusophone if it satisfies the following three conditions: i) The first term of the sequence is number $1$. ii) To obtain the next term of the sequence we can multiply the previous term by a positive prime number ($2,3,5,7,11, ...$) or add $1$. (iii) The last term of the sequence is the number $2016$. For example: $1\overset{{\times 11}}{\to}11 \overset{{\times 61}}{\to} 671 \overset{{+1}}{\to}672 \overset{{\times 3}}{\to}2016$ How many Lusophone sequences exist in which (as in the example above) the add $1$ operation was used exactly once and not multiplied twice by the same prime number?

2021 Azerbaijan EGMO TST, 1

Tags: number theory , prime , prime number

p is a prime number, k is a positive integer Find all (p, k): $k!=(p^3-1)(p^3-p)(p^3-p^2)$

2013 India Regional Mathematical Olympiad, 2

Tags: divisor , prime

Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.

2012 APMO, 3

Tags: modular arithmetic , number theory , prime , divisibility

Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.

2008 Singapore Junior Math Olympiad, 4

Tags: prime , sum , number theory

Six distinct positive integers $a,b,c.d,e, f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $10$ prime numbers while Jill claims that she has $9$ prime numbers among the sums. Who has the correct claim?

2010 Estonia Team Selection Test, 1

Tags: prime , gcd , power of prime , coprime , number theory

For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$ Let $n$ be a positive integer. Prove that the following conditions are equivalent: (i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$, (ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.