Found problems: 1048
Let $n = p_1p_2... p_{2017}$ be the positive integer where $p_1, p_2, ..., p_{2017}$ are $2017$ distinct odd primes. A triangle is called [i]nice [/i] if it is a right triangle with integer side lengths and the inradius is $n$. Find the number of nice triangles (two triangles are consider different if their tuples of length of sides are different)
The numbers $1,2,\dots, 49$ are written on unit squares of a $7\times 7$ chessboard such that consequtive numbers are on unit squares sharing a common edge. At most how many prime numbers can a row have?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 3
$
Does there exist a natural number $n$ in whose decimal representation each digit occurs at least $2006$ times and which has the property that you can find two different digits in its decimal representation such that the number obtained from $n$ by interchanging these two digits is different from $n$ and has the same set of prime divisors as $n$ ?
Prove that there are at most three primes between $10$ and $10^{10}$ all of whose decimal digits are $1$.
Find the solutions in prime numbers of the following equation
$p^4 + q^4 + r^4 + 119 = s^2 .$
Prove that there exists a prime number $p$ such that the minimum positive integer $n$ such that $p|2^n -1$ is $3^{2013}$.
Find the sum of all prime numbers between $ 1$ and $ 100$ that are simultaneously $ 1$ greater than a multiple of $ 4$ and $ 1$ less than a multiple of $ 5$.
$ \textbf{(A)}\ 118\qquad \textbf{(B)}\ 137\qquad \textbf{(C)}\ 158\qquad \textbf{(D)}\ 187 \qquad \textbf{(E)}\ 245$
Find all right triangles with integer side lengths in which two congruent circles with prime radius can be inscribed such that they are externally tangent, both touch the hypotenuse, and each is tangent to another leg of the right triangle.
Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.
Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.
Let $n\ge 3$ be an integer such that for every prime factor $q$ of $n-1$ exists an integer $a > 1$ such that $a^{n-1} \equiv 1 \,(\mod n \, )$ and $a^{\frac{n-1} {q}}\not\equiv 1 \,(\mod n \, )$. Prove that $n$ is not prime.
Let $p$ be an odd prime number, and $k$ be an odd number not divisible by $p$. Consider a field $K$ be a field with $kp+1$ elements, and $A = \{x_1,x_2, \dots, x_t\}$ be the set of elements of $K^*$, whose order is not $k$ in the multiplicative group $(K^*,\cdot)$. Prove that the polynomial $P(X)=(X+x_1)(X+x_2)\dots(X+x_t)$ has at least $p$ coefficients equal to $1$.
For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$
[i]Proposed by Suhaimi Ramly, Malaysia[/i]
Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations
$ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$
$ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$
Find all possible values of the product $ p_1p_2p_3p_4$
Find all prime numbers $p,q$ for which $pq$ divides $(5^p-2^p)(5^q-2^q)$.
Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.
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]
For a positive integer $n$, $a_1, a_2, \cdots a_k$ are all positive integers without repetition that are not greater than $n$ and relatively prime to $n$. If $k>8$, prove the following. $$\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}$$
For each positive integer $k$ denote $C(k)$ to be sum of its distinct prime divisors. For example $C(1)=0,C(2)=2,C(45)=8$. Find all positive integers $n$ for which $C(2^n+1)=C(n)$.
Are there any sequences of positive integers $ 1 \leq a_{1} < a_{2} < a_{3} < \ldots$ such that for each integer $ n$, the set $ \left\{a_{k} \plus{} n\ |\ k \equal{} 1, 2, 3, \ldots\right\}$ contains finitely many prime numbers?
$15.$ Let $n$ be the largest integer that is the product of exactly $3$ distinct prime numbers, $x,y,$ and $10x+y,$ where $x$ and $y$ are digits. What is the sum of digits of $n ?$
Find all pairs of prime numbers $p$ and $q$ such that:
\[
p(p + q) = q^p+ 1.
\]
An infinite sequence $a_1, a_2,\ldots$ of positive integers is such that $a_n \geq 2$ and $a_{n+2}$ divides $a_{n+1} + a_n$ for all $n \geq 1$. Prove that there exists a prime which divides infinitely many terms of the sequence.
a) Prove that there doesn't exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: gcd(a_i+j,a_j+i)=1$
b) Let $p$ be an odd prime number. Prove that there exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: p \not | gcd(a_i+j,a_j+i)$
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.