Found problems: 721
2014 Ukraine Team Selection Test, 9
Let $m, n$ be odd prime numbers.
Find all pairs of integers numbers $a, b$ for which the system of equations:
$x^m+y^m+z^m=a$,
$x^n+y^n+z^n=b$
has many solutions in integers $x, y, z$.
1975 Bundeswettbewerb Mathematik, 2
Prove that no term of the sequence $10001$, $100010001$, $1000100010001$ , $...$ is prime.
2024 Israel TST, P3
Let $n$ be a positive integer and $p$ be a prime number of the form $8k+5$. A polynomial $Q$ of degree at most $2023$ and nonnegative integer coefficients less than or equal to $n$ will be called "cool" if
\[p\mid Q(2)\cdot Q(3) \cdot \ldots \cdot Q(p-2)-1.\]
Prove that the number of cool polynomials is even.
2015 IFYM, Sozopol, 5
Does there exist a natural number $n$ with exactly 3 different prime divisors $p$, $q$, and $r$, so that $p-1\mid n$, $qr-1\mid n$, $q-1\nmid n$, $r-1\nmid n$, and $3\nmid q+r$?
1992 AIME Problems, 1
Find the sum of all positive rational numbers that are less than $10$ and that have denominator $30$ when written in lowest terms.
2006 Greece JBMO TST, 2
Let $a,b,c$ be positive integers such that the numbers $k=b^c+a, l=a^b+c, m=c^a+b$ to be prime numbers. Prove that at least two of the numbers $k,l,m$ are equal.
2020 JBMO Shortlist, 8
Find all prime numbers $p$ and $q$ such that
$$1 + \frac{p^q - q^p}{p + q}$$
is a prime number.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2019 Pan-African, 2
Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?
1973 IMO Shortlist, 4
Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided into $7$ disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of $C$ are there ?
1976 Chisinau City MO, 122
The diagonals of some convex quadrilateral are mutually perpendicular and divide the quadrangle into $4$ triangles, the areas of which are expressed by prime numbers. Prove that a circle can be inscribed in this quadrilateral.
2013 All-Russian Olympiad, 3
Find all positive integers $k$ such that for the first $k$ prime numbers $2, 3, \ldots, p_k$ there exist positive integers $a$ and $n>1$, such that $2\cdot 3\cdot\ldots\cdot p_k - 1=a^n$.
[i]V. Senderov[/i]
2017 China Team Selection Test, 6
For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.
2018 Israel Olympic Revenge, 1
Let $n$ be a positive integer.
Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$
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$
2018 Malaysia National Olympiad, A6
Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$.
1999 Putnam, 6
Let $S$ be a finite set of integers, each greater than $1$. Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$. Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.
Bangladesh Mathematical Olympiad 2020 Final, #11
A prime number$ q $is called[b][i] 'Kowai' [/i][/b]number if $ q = p^2 + 10$ where $q$, $p$, $p^2-2$, $p^2-8$, $p^3+6$ are prime numbers. WE know that, at least one [b][i]'Kowai'[/i][/b] number can be found. Find the summation of all [b][i]'Kowai'[/i][/b] numbers.
2015 Iran MO (3rd round), 5
$p>30$ is a prime number. Prove that one of the following numbers is in form of $x^2+y^2$.
$$ p+1 , 2p+1 , 3p+1 , .... , (p-3)p+1$$
2021 Stars of Mathematics, 1
For every integer $n\geq 3$, let $s_n$ be the sum of all primes (strictly) less than $n$. Are there infinitely many integers $n\geq 3$ such that $s_n$ is coprime to $n$?
[i]Russian Competition[/i]
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 $a_{2013}$.
2020 Regional Olympiad of Mexico Southeast, 6
Prove that for all $a, b$ and $x_0$ positive integers, in the sequence $x_1, x_2, x_3, \cdots$ defined by
$$x_{n+1}=ax_n+b, n\geq 0$$
Exist an $x_i$ that is not prime for some $i\geq 1$
1985 AMC 12/AHSME, 12
Let's write p,q, and r as three distinct prime numbers, where 1 is not a prime. Which of the following is the smallest positive perfect cube leaving $ n \equal{} pq^2r^4$ as a divisor?
$ \textbf{(A)}\ p^8q^8r^8\qquad
\textbf{(B)}\ (pq^2r^2)^3\qquad
\textbf{(C)}\ (p^2q^2r^2)^3\qquad
\textbf{(D)}\ (pqr^2)^3\qquad
\textbf{(E)}\ 4p^3q^3r^3$
2023 OlimphÃada, 4
We say that a prime $p$ is $n$-$\textit{rephinado}$ if $n | p - 1$ and all $1, 2, \ldots , \lfloor \sqrt[\delta]{p}\rfloor$ are $n$-th residuals modulo $p$, where $\delta = \varphi+1$. Are there infinitely many $n$ for which there are infinitely many $n$-$\textit{rephinado}$ primes?
Notes: $\varphi =\frac{1+\sqrt{5}}{2}$. We say that an integer $a$ is a $n$-th residue modulo $p$ if there is an integer $x$ such that $$x^n \equiv a \text{ (mod } p\text{)}.$$
2018 VJIMC, 2
Find all prime numbers $p$ such that $p^3$ divides the determinant
\[\begin{vmatrix} 2^2 & 1 & 1 & \dots & 1\\1 & 3^2 & 1 & \dots & 1\\ 1 & 1 & 4^2 & & 1\\ \vdots & \vdots & & \ddots & \\1 & 1 & 1 & & (p+7)^2 \end{vmatrix}.\]
2011 Czech-Polish-Slovak Match, 3
Let $a$ be any integer. Prove that there are infinitely many primes $p$ such that \[ p\,|\,n^2+3\qquad\text{and}\qquad p\,|\,m^3-a \] for some integers $n$ and $m$.