Found problems: 715
2015 Thailand TSTST, 2
Fix a sequence $ a_1,a_2,a_3,... $ 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} $.
2015 Thailand TSTST, 1
Find all primes $1 < p < 100$ such that the equation $x^2-6y^2=p$ has an integer solution $(x, y)$.
2022 Federal Competition For Advanced Students, P1, 4
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$
holds.
[i](Walther Janous)[/i]
2015 Germany Team Selection Test, 3
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
2019 Junior Balkan MO, 1
Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number
$x^p + y^p + z^p - x - y - z$
is a product of exactly three distinct prime numbers.
1997 Bundeswettbewerb Mathematik, 2
Find a prime number $p$ such that $\frac{p+1}{2}$ and $\frac{p^2+1}{2}$ are perfect square
2014 NIMO Problems, 3
Find the number of positive integers $n$ with exactly $1974$ factors such that no prime greater than $40$ divides $n$, and $n$ ends in one of the digits $1$, $3$, $7$, $9$. (Note that $1974 = 2 \cdot 3 \cdot 7 \cdot 47$.)
[i]Proposed by Yonah Borns-Weil[/i]
2017 China Northern MO, 8
On Qingqing Grassland, there are 7 sheep numberd $1,2,3,4,5,6,7$ and 2017 wolves numberd $1,2,\cdots,2017$. We have such strange rules:
(1) Define $P(n)$: the number of prime numbers that are smaller than $n$. Only when $P(i)\equiv j\pmod7$, wolf $i$ may eat sheep $j$ (he can also choose not to eat the sheep).
(2) If wolf $i$ eat sheep $j$, he will immediately turn into sheep $j$.
(3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep.
Assume that all wolves are very smart, then how many wolves will remain in the end?
2024 Polish MO Finals, 3
Determine all pairs $(p,q)$ of prime numbers with the following property: There are positive integers $a,b,c$ satisfying
\[\frac{p}{a}+\frac{p}{b}+\frac{p}{c}=1 \quad \text{and} \quad \frac{a}{p}+\frac{b}{p}+\frac{c}{p}=q+1.\]
2010 Iran Team Selection Test, 1
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that
\[f(a+1), f(a+2), \dots, f(a+n)\]
form an arithmetic progression.
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}$.
2011 ISI B.Stat Entrance Exam, 7
[b](i)[/b] Show that there cannot exists three peime numbers, each greater than $3$, which are in arithmetic progression with a common difference less than $5$.
[b](ii)[/b] Let $k > 3$ be an integer. Show that it is not possible for $k$ prime numbers, each greater than $k$, to be in an arithmetic progression with a common difference less than or equal to $k+1$.
2018 Latvia Baltic Way TST, P16
Call a natural number [i]simple[/i] if it is not divisible by any square of a prime number (in other words it is square-free).
Prove that there are infinitely many positive integers $n$ such that both $n$ and $n+1$ are [i]simple[/i].
2016 Hong Kong TST, 4
Mable and Nora play a game according to the following steps in order:
1. Mable writes down any 2015 distinct prime numbers in ascending order in a row. The product of these primes is Marble's score.
2. Nora writes down a positive integer
3. Mable draws a vertical line between two adjacent primes she has written in step 1, and compute the product of the prime(s) on the left of the vertical line
4. Nora must add the product obtained by Marble in step 3 to the number she has written in step 2, and the sum becomes Nora's score.
If Marble and Nora's scores have a common factor greater than 1, Marble wins, otherwise Nora wins.
Who has a winning strategy?
2021 Bolivia Ibero TST, 3
Let $p=ab+bc+ac$ be a prime number where $a,b,c$ are different two by two, show that $a^3,b^3,c^3$ gives different residues modulo $p$
2015 Canada National Olympiad, 5
Let $p$ be a prime number for which $\frac{p-1}{2}$ is also prime, and let $a,b,c$ be integers not divisible by $p$. Prove that there are at most $1+\sqrt {2p}$ positive integers $n$ such that $n<p$ and $p$ divides $a^n+b^n+c^n$.
1959 Miklós Schweitzer, 1
[b]1.[/b] Let $p_n$ be the $n$th prime number. Prove that
$\sum_{n=2}^{\infty} \frac{1}{np_n-(n-1)p_{n-1}}= \infty$
[b](N.17)[/b]
2013 National Olympiad First Round, 2
How many triples $(p,q,n)$ are there such that $1/p+2013/q = n/5$ where $p$, $q$ are prime numbers and $n$ is a positive integer?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 4
$
2019 Olympic Revenge, 5
Define $f: \mathbb{N} \rightarrow \mathbb{N}$ by $$f(n) = \sum \frac{(1+\sum_{i=1}^{n} t_i)!}{(1+t_1) \cdot \prod_{i=1}^{n} (t_i!) }$$
where the sum runs through all $n$-tuples such that $\sum_{j=1}^{n}j \cdot t_j=n$ and $t_j \ge 0$ for all $1 \le j \le n$.
Given a prime $p$ greater than $3$, prove that $$\sum_{1 \le i < j <k \le p-1 } \frac{f(i)}{i \cdot j \cdot k} \equiv \sum_{1 \le i < j <k \le p-1 } \frac{2^i}{i \cdot j \cdot k} \pmod{p}.$$
1974 Miklós Schweitzer, 4
Let $ R$ be an infinite ring such that every subring of $ R$ different from $ \{0 \}$ has a finite index in $ R$. (By the index of a subring, we mean the index of its additive group in the additive group of $ R$.) Prove that the additive group of $ R$ is cyclic.
[i]L. Lovasz, J. Pelikan[/i]
STEMS 2023 Math Cat A, 8
For how many pairs of primes $(p, q)$, is $p^2 + 2pq^2 + 1$ also a prime?
2016 Iran MO (3rd Round), 1
Let $p,q$ be prime numbers ($q$ is odd). Prove that there exists an integer $x$ such that:
$$q |(x+1)^p-x^p$$
If and only if $$q \equiv 1 \pmod p$$
2002 AMC 12/AHSME, 17
Several sets of prime numbers, such as $ \{ 7, 83, 421, 659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
$ \textbf{(A)}\ 193\qquad\textbf{(B)}\ 207\qquad\textbf{(C)}\ 225\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 447$
2001 Tournament Of Towns, 2
There exists a block of 1000 consecutive positive integers containing no prime numbers, namely, $1001!+2,1001!+3,...,1001!+1001$. Does there exist a block of 1000 consecutive positive intgers containing exactly five prime numbers?
2016 China Team Selection Test, 4
Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$.
Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.