Found problems: 721
2018 All-Russian Olympiad, 1
Suppose $a_1,a_2, \dots$ is an infinite strictly increasing sequence of positive integers and $p_1, p_2, \dots$ is a sequence of distinct primes such that $p_n \mid a_n$ for all $n \ge 1$. It turned out that $a_n-a_k=p_n-p_k$ for all $n,k \ge 1$. Prove that the sequence $(a_n)_n$ consists only of prime numbers.
PEN D Problems, 3
Show that \[(-1)^{\frac{p-1}{2}}{p-1 \choose{\frac{p-1}{2}}}\equiv 4^{p-1}\pmod{p^{3}}\] for all prime numbers $p$ with $p \ge 5$.
2007 Korea National Olympiad, 4
For all positive integer $ n\geq 2$, prove that product of all prime numbers less or equal than $ n$ is smaller than $ 4^{n}$.
2007 Bulgarian Autumn Math Competition, Problem 12.4
Let $p$ and $q$ be prime numbers and $\{a_{n}\}_{n=1}^{\infty}$ be a sequence of integers defined by:
\[a_{0}=0, a_{1}=1, a_{n+2}=pa_{n+1}-qa_{n}\quad\forall n\geq 0\]
Find $p$ and $q$ if there exists an integer $k$ such that $a_{3k}=-3$.
2020 AMC 10, 4
The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }11$
2012 Czech And Slovak Olympiad IIIA, 1
Find all integers for which $n$ is $n^4 -3n^2 + 9$ prime
2024 AMC 10, 3
What is the sum of the digits of the smallest prime that can be written as a sum of $5$ distinct primes?
$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$
1973 Putnam, B3
Consider an integer $p>1$ with the property that the polynomial $x^2 - x + p$ takes prime values for all integers $x$ such that $0\leq x <p$. Show that there is exactly one triple of integers $a, b, c$ satisfying the conditions:
$$b^2 -4ac = 1-4p,\;\; 0<a \leq c,\;\; -a\leq b<a.$$
2003 IMO Shortlist, 8
Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions:
(i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements;
(ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power.
What is the largest possible number of elements in $A$ ?
2016 Lusophon Mathematical Olympiad, 5
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?
2016 Thailand Mathematical Olympiad, 5
given $p_1,p_2,...$ be a sequence of integer and $p_1=2$,
for positive integer $n$, $p_{n+1}$ is the least prime factor of $np_1^{1!}p_2^{2!}...p_n^{n!}+1 $
prove that all primes appear in the sequence
(Proposed by Beatmania)
2008 All-Russian Olympiad, 3
Given a finite set $ P$ of prime numbers, prove that there exists a positive integer $ x$ such that it can be written in the form $ a^p \plus{} b^p$ ($ a,b$ are positive integers), for each $ p\in P$, and cannot be written in that form for each $ p$ not in $ P$.
2019 Dutch IMO TST, 4
Find all functions $f : Z \to Z$ satisfying
$\bullet$ $ f(p) > 0$ for all prime numbers $p$,
$\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.
1996 Iran MO (3rd Round), 4
Let $n$ be a positive integer and suppose that $\phi(n)=\frac{n}{k}$, where $k$ is the greatest perfect square such that $k \mid n$. Let $a_1,a_2,\ldots,a_n$ be $n$ positive integers such that $a_i=p_1^{a_1i} \cdot p_2^{a_2i} \cdots p_n^{a_ni}$, where $p_i$ are prime numbers and $a_{ji}$ are non-negative integers, $1 \leq i \leq n, 1 \leq j \leq n$. We know that $p_i\mid \phi(a_i)$, and if $p_i\mid \phi(a_j)$, then $p_j\mid \phi(a_i)$. Prove that there exist integers $k_1,k_2,\ldots,k_m$ with $1 \leq k_1 \leq k_2 \leq \cdots \leq k_m \leq n$ such that
\[\phi(a_{k_{1}} \cdot a_{k_{2}} \cdots a_{k_{m}})=p_1 \cdot p_2 \cdots p_n.\]
2020 New Zealand MO, 4
Determine all prime numbers $p$ such that $p^2 - 6$ and $p^2 + 6$ are both prime numbers.
2006 AMC 8, 25
Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers?
[asy]path card=((0,0)--(0,3)--(2,3)--(2,0)--cycle);
draw(card, linewidth(1));
draw(shift(2.5,0)*card, linewidth(1));
draw(shift(5,0)*card, linewidth(1));
label("$44$", (1,1.5));
label("$59$", shift(2.5,0)*(1,1.5));
label("$38$", shift(5,0)*(1,1.5));[/asy]
$ \textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 14 \qquad
\textbf{(C)}\ 15 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 17$
2014 AMC 8, 4
The sum of two prime numbers is $85$. What is the product of these two prime numbers?
$\textbf{(A) }85\qquad\textbf{(B) }91\qquad\textbf{(C) }115\qquad\textbf{(D) }133\qquad \textbf{(E) }166$
2020 Kosovo National Mathematical Olympiad, 3
Find all prime numbers $p$ such that $3^p + 5^p -1$ is a prime number.
2003 Tournament Of Towns, 1
An increasing arithmetic progression consists of one hundred positive integers. Is it possible that every two of them are relatively prime?
1994 IMO Shortlist, 3
Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.
2013 Dutch IMO TST, 3
Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.
2007 Bulgarian Autumn Math Competition, Problem 8.3
Determine all triplets of prime numbers $p<q<r$, such that $p+q=r$ and $(r-p)(q-p)-27p$ is a square.
1988 IMO Longlists, 26
The circle $x^2+ y^2 = r^2$ meets the coordinate axis at $A = (r,0), B = (-r,0), C = (0,r)$ and $D = (0,-r).$ Let $P = (u,v)$ and $Q = (-u,v)$ be two points on the circumference of the circle. Let $N$ be the point of intersection of $PQ$ and the $y$-axis, and $M$ be the foot of the perpendicular drawn from $P$ to the $x$-axis. If $r^2$ is odd, $u = p^m > q^n = v,$ where $p$ and $q$ are prime numbers and $m$ and $n$ are natural numbers, show that
\[ |AM| = 1, |BM| = 9, |DN| = 8, |PQ| = 8. \]
2024 Girls in Mathematics Tournament, 4
Find all the positive integers $a,b,c$ such that $3ab= 2c^2$ and $a^3+b^3+c^3$ is the double of a prime number.
2024 Mexican University Math Olympiad, 1
Let \( x \), \( y \), \( p \) be positive integers that satisfy the equation \( x^4 = p + 9y^4 \), where \( p \) is a prime number. Show that \( \frac{p^2 - 1}{3} \) is a perfect square and a multiple of 16.