Found problems: 364
2006 Korea National Olympiad, 2
Alice and Bob are playing "factoring game." On the paper, $270000(=2^43^35^4)$ is written and each person picks one number from the paper(call it $N$) and erase $N$ and writes integer $X,Y$ such that $N=XY$ and $\text{gcd}(X,Y)\ne1.$ Alice goes first and the person who can no longer make this factoring loses. If two people use optimal strategy, prove that Alice always win.
2014 Ukraine Team Selection Test, 12
Prove that for an arbitrary prime $p \ge 3$ the number of positive integers $n$, for which $p | n! +1$ does not exceed $cp^{2/3}$, where c is a constant that does not depend on $p$.
2016 JBMO Shortlist, 2
The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?
2014 IMO Shortlist, N5
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.
[i]Proposed by Belgium[/i]
2022 Mexico National Olympiad, 1
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.
2021 Baltic Way, 17
Distinct positive integers $a, b, c, d$ satisfy
$$\begin{cases} a \mid b^2 + c^2 + d^2,\\
b\mid a^2 + c^2 + d^2,\\
c \mid a^2 + b^2 + d^2,\\
d \mid a^2 + b^2 + c^2,\end{cases}$$
and none of them is larger than the product of the three others. What is the largest possible number of primes among them?
2019 Durer Math Competition Finals, 2
Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.
Mathley 2014-15, 7
Find all primes $p,q, r$ such that $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$.
Titu Andreescu, Mathematics Department, College of Texas, USA
2017 JBMO Shortlist, NT5
Find all positive integers $n$ such that there exists a prime number $p$, such that $p^n-(p-1)^n$ is a power of $3$.
Note. A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer.
2012 Tournament of Towns, 5
Let $p$ be a prime number. A set of $p + 2$ positive integers, not necessarily distinct, is called [i]interesting [/i] if the sum of any $p$ of them is divisible by each of the other two. Determine all interesting sets.
2012 EGMO, 5
The numbers $p$ and $q$ are prime and satisfy
\[\frac{p}{{p + 1}} + \frac{{q + 1}}{q} = \frac{{2n}}{{n + 2}}\]
for some positive integer $n$. Find all possible values of $q-p$.
[i]Luxembourg (Pierre Haas)[/i]
2021 South Africa National Olympiad, 3
Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.
2015 India IMO Training Camp, 2
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.
[i]Proposed by Belgium[/i]
2013 Danube Mathematical Competition, 2
Let $a, b, c, n$ be four integers, where n$\ge 2$, and let $p$ be a prime dividing both $a^2+ab+b^2$ and $a^n+b^n+c^n$, but not $a+b+c$. for instance, $a \equiv b \equiv -1 (mod \,\, 3), c \equiv 1 (mod \,\, 3), n$ a positive even integer, and $p = 3$ or $a = 4, b = 7, c = -13, n = 5$, and $p = 31$ satisfy these conditions. Show that $n$ and $p - 1$ are not coprime.