Found problems: 715
2012 Iran MO (3rd Round), 1
$P(x)$ is a nonzero polynomial with integer coefficients. Prove that there exists infinitely many prime numbers $q$ such that for some natural number $n$, $q|2^n+P(n)$.
[i]Proposed by Mohammad Gharakhani[/i]
2006 Estonia Team Selection Test, 6
Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$.
Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$.
(a) Show that there are only finitely many pairs of consecutive highly divisible
integers of the form $(a, b)$ with $a\mid b$.
(b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.
2011 IFYM, Sozopol, 5
Does there exist a strictly increasing sequence $\{a_n\}_{n=1}^\infty$ of natural numbers with the following property: for $\forall$ $c\in \mathbb{Z}$ the sequence $c+a_1,c+a_2,...,c+a_n...$ has finite number of primes? Explain your answer.
1987 Czech and Slovak Olympiad III A, 2
Given a prime $p>3$ and an odd integer $n>0$, show that the equation $$xyz=p^n(x+y+z)$$ has at least $3(n+1)$ different solutions up to symmetry. (That is, if $(x',y',z')$ is a solution and $(x'',y'',z'')$ is a permutation of the previous, they are considered to be the same solution.)
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 $p \ge 5$.
1995 Tournament Of Towns, (474) 2
Do there exist
(a) four
(b) five
distinct positive integers such that the sum of any three of them is a prime number?
(V Senderov)
2023 USAMTS Problems, 1
In the diagram below, fill the $12$ circles with numbers from the following bank so that each number is used once. Two circles connected by a single line must contain relatively prime numbers. Two circles connected by a double line must contain numbers that are not relatively prime.
$$\text{Bank: } 20, 21, 22, 23, 24, 25, 27, 28, 30 ,32, 33 ,35$$
[asy]
real HRT3 = sqrt(3) / 2;
void drawCircle(real x, real y, real r) {
path p = circle((x,y), r);
draw(p);
fill(p, white);
}
void drawCell(int gx, int gy) {
real x = 0.5 * gx;
real y = HRT3 * gy;
drawCircle(x, y, 0.35);
}
void drawEdge(int gx1, int gy1, int gx2, int gy2, bool doubled) {
real x1 = 0.5 * gx1;
real y1 = HRT3 * gy1;
real x2 = 0.5 * gx2;
real y2 = HRT3 * gy2;
if (doubled) {
real dx = x2 - x1;
real dy = y2 - y1;
real ox = -0.035 * dy / sqrt(dx * dx + dy * dy);
real oy = 0.035 * dx / sqrt(dx * dx + dy * dy);
draw((x1+ox,y1+oy)--(x2+ox,y2+oy));
draw((x1-ox,y1-oy)--(x2-ox,y2-oy));
} else {
draw((x1,y1)--(x2,y2));
}
}
drawEdge(2, 0, 4, 0, true);
drawEdge(2, 0, 1, 1, true);
drawEdge(2, 0, 3, 1, true);
drawEdge(4, 0, 3, 1, false);
drawEdge(4, 0, 5, 1, false);
drawEdge(1, 1, 0, 2, false);
drawEdge(1, 1, 2, 2, false);
drawEdge(1, 1, 3, 1, false);
drawEdge(3, 1, 2, 2, true);
drawEdge(3, 1, 4, 2, true);
drawEdge(3, 1, 5, 1, false);
drawEdge(5, 1, 4, 2, true);
drawEdge(5, 1, 6, 2, false);
drawEdge(0, 2, 1, 3, false);
drawEdge(0, 2, 2, 2, false);
drawEdge(2, 2, 1, 3, false);
drawEdge(2, 2, 3, 3, true);
drawEdge(2, 2, 4, 2, false);
drawEdge(4, 2, 3, 3, false);
drawEdge(4, 2, 5, 3, false);
drawEdge(4, 2, 6, 2, false);
drawEdge(6, 2, 5, 3, true);
drawEdge(1, 3, 3, 3, true);
drawEdge(3, 3, 5, 3, false);
drawCell(2, 0);
drawCell(4, 0);
drawCell(1, 1);
drawCell(3, 1);
drawCell(5, 1);
drawCell(0, 2);
drawCell(2, 2);
drawCell(4, 2);
drawCell(6, 2);
drawCell(1, 3);
drawCell(3, 3);
drawCell(5, 3);
[/asy]
2014 AMC 8, 4
The sum of two prime numbers is $85$. What is the product of these two prime numbers?
$\textbf{(A) }85\qquad\textbf{(B) }91\qquad\textbf{(C) }115\qquad\textbf{(D) }133\qquad \textbf{(E) }166$
PEN E Problems, 24
Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.
2010 Contests, 4
With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?
2024 Kyiv City MO Round 1, Problem 2
Write the numbers from $1$ to $16$ in the cells of a of a $4 \times 4$ square so that:
1. Each cell contains exactly one number;
2. Each number is written exactly once;
3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the
original $4 \times 4$ square, the sum of numbers in them is a prime number
The figure below shows examples of such pairs of cells, sums of numbers in which have to be prime.
[img]https://i.ibb.co/fqX05dY/Kyiv-MO-2024-Round-1-8-2.png[/img]
[i]Proposed by Mykhailo Shtandenko[/i]
2021 Argentina National Olympiad, 2
Let $m$ be a positive integer for which there exists a positive integer $n$ such that the multiplication $mn$ is a perfect square and $m- n$ is prime. Find all $m$ for $1000\leq m \leq 2021.$
2011 All-Russian Olympiad Regional Round, 9.7
Find all prime numbers $p$, $q$ and $r$ such that the fourth power of any of them minus one is divisible by the product of the other two. (Author: V. Senderov)
2017 India PRMO, 23
Suppose an integer $x$, a natural number $n$ and a prime number $p$ satisfy the equation $7x^2-44x+12=p^n$. Find the largest value of $p$.
2019 Singapore MO Open, 4
Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$.
[i]Proposed by fattypiggy123[/i]
2025 China National Olympiad, 5
Let $p$ be a prime number and $f$ be a bijection from $\left\{0,1,\ldots,p-1\right\}$ to itself. Suppose that for integers $a,b \in \left\{0,1,\ldots,p-1\right\}$, $|f(a) - f(b)|\leqslant 2024$ if $p \mid a^2 - b$. Prove that there exists infinite many $p$ such that there exists such an $f$ and there also exists infinite many $p$ such that there doesn't exist such an $f$.
2019 All-Russian Olympiad, 5
In a kindergarten, a nurse took $n$ congruent cardboard rectangles and gave them to $n$ kids, one per each. Each kid has cut its rectangle into congruent squares(the squares of different kids could be of different sizes). It turned out that the total number of the obtained squares is a prime number. Prove that all the initial squares were in fact squares.
2001 May Olympiad, 4
Using only prime numbers, a set is formed with the following conditions:
Any one-digit prime number can be in the set.
For a prime number with more than one digit to be in the set, the number that results from deleting only the first digit and also the number that results from deleting only the last digit must be in the set.
Write, of the sets that meet these conditions, the one with the greatest number of elements.
Justify why there cannot be one with more elements.
Remember that the number $1$ is not prime.
2022 HMNT, 5
Alice is once again very bored in class. On a whim, she chooses three primes $p$, $q$, $r$ independently and uniformly at random from the set of primes at most 30. She then calculates the roots of $px^2+qx+r$. What is the probability that at least one of her roots is an integer?
2017 Turkey MO (2nd round), 4
Let $d(n)$ be number of prime divisors of $n$. Prove that one can find $k,m$ positive integers for any positive integer $n$ such that $k-m=n$ and $d(k)-d(m)=1$
2018 IFYM, Sozopol, 1
Find all prime numbers $p$ and all positive integers $n$, such that
$n^8 - n^2 = p^5 + p^2$
2010 Contests, 1
$a)$ Let $p$ and $q$ be distinct prime numbers such that $p+q^2$ divides $p^2+q$. Prove that $p+q^2$ divides $pq-1$.
$b)$ Find all prime numbers $p$ such that $p+121$ divides $p^2+11$.
1955 Miklós Schweitzer, 4
[b]4.[/b] Find all positive integers $\alpha , \beta (\alpha >1)$ and all prime numbers $p, q, r$ which satisfy the equation $p^{\alpha}= q^{\beta}+r^{\alpha}$ ($\alpha , \beta , p, q, r$ need not necessarily be different). [b](N. 12)[/b]
2008 Tuymaada Olympiad, 8
250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer $ t$ there are four chosen numbers $ a_1$, $ a_2$, $ a_3$, $ a_4$, such that $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \minus{} t$ is divisible by 23.
[i]Author: K. Kokhas[/i]
2018 Brazil Team Selection Test, 2
Prove that there is an integer $n>10^{2018}$ such that the sum of all primes less than $n$ is relatively prime to $n$.
[i](R. Salimov)[/i]