Found problems: 15460
1996 IMO Shortlist, 1
Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd|$ are primes?
2016 Azerbaijan BMO TST, 3
$a,b$ are positive integers and $(a!+b!)|a!b!$.Prove that $3a\ge 2b+2$.
2018 ELMO Shortlist, 4
Say a positive integer $n>1$ is $d$-coverable if for each non-empty subset $S\subseteq \{0, 1, \ldots, n-1\}$, there exists a polynomial $P$ with integer coefficients and degree at most $d$ such that $S$ is exactly the set of residues modulo $n$ that $P$ attains as it ranges over the integers. For each $n$, find the smallest $d$ such that $n$ is $d$-coverable, or prove no such $d$ exists.
[i]Proposed by Carl Schildkraut[/i]
2018 Finnish National High School Mathematics Comp, 5
Solve the diophantine equation $x^{2018}-y^{2018}=(xy)^{2017}$ when $x$ and $y$ are non-negative integers.
2020 Romania EGMO TST, P2
Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.
2020 Princeton University Math Competition, A1/B3
Compute the last two digits of $$9^{2020} + 9^{2020^2}+ ... + 9^{2020^{2020}}$$
2017 IMO Shortlist, N7
An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$, we have:
$$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$
[i]Proposed by John Berman, United States[/i]
2003 France Team Selection Test, 2
A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.
1996 Bundeswettbewerb Mathematik, 4
Find all natural numbers $n$ for which $n2^{n-1} +1$ is a perfect square.
2020 LIMIT Category 1, 12
$q$ is the smallest rational number having the following properties:
(i) $q>\frac{31}{17}$
(ii) when $q$ is written in its reduced form $\frac{a}{b}$, then $b<17$
As in part (ii) above, find $a+b$.
2015 Romania National Olympiad, 1
Show that among the square roots of the first $ 2015 $ natural numbers, we cannot choose an arithmetic sequence composed of $ 45 $ elements.
2018 Moldova Team Selection Test, 8
Let the set $A=${$ 1,2,3, \dots ,48n+24$ } , where $ n \in \mathbb {N^*}$ . Prove that there exist a subset $B $ of $A $ with $24n+12$ elements with the property : the sum of the squares of the elements of the set $B $ is equal to the sum of the squares of the elements of the set $A$ \ $B $ .
2025 ISI Entrance UGB, 6
Let $\mathbb{N}$ denote the set of natural numbers, and let $\left( a_i, b_i \right)$, $1 \leq i \leq 9$, be nine distinct tuples in $\mathbb{N} \times \mathbb{N}$. Show that there are three distinct elements in the set $\{ 2^{a_i} 3^{b_i} \colon 1 \leq i \leq 9 \}$ whose product is a perfect cube.
2000 Junior Balkan Team Selection Tests - Moldova, 8
Show that the numbers $18^n$ and $2^n + 18^n$ are having the same number of digits (as written in base 10), for every natural number $n$.
2017 Hong Kong TST, 5
Find the first digit after the decimal point of the number $\displaystyle \frac1{1009}+\frac1{1010}+\cdots + \frac1{2016}$
2007 Indonesia MO, 2
For every positive integer $ n$, $ b(n)$ denote the number of positive divisors of $ n$ and $ p(n)$ denote the sum of all positive divisors of $ n$. For example, $ b(14)\equal{}4$ and $ p(14)\equal{}24$. Let $ k$ be a positive integer greater than $ 1$.
(a) Prove that there are infinitely many positive integers $ n$ which satisfy $ b(n)\equal{}k^2\minus{}k\plus{}1$.
(b) Prove that there are finitely many positive integers $ n$ which satisfy $ p(n)\equal{}k^2\minus{}k\plus{}1$.
2011 LMT, 19
A positive six-digit integer begins and ends in $8$, and is also the product of three consecutive even numbers. What is the sum of the three even numbers?
2023 Argentina National Olympiad, 4
Lets say that a positive integer is $good$ if its equal to the the subtraction of two positive integer cubes. For example: $7$ is a $good$ prime because $2^3-1^3=7$.
Determine how much the last digit of a $good$ prime may be worth. Give all the possibilities.
1996 All-Russian Olympiad, 3
Find all natural numbers $n$, such that there exist relatively prime integers $x$ and $y$ and an integer $k > 1$ satisfying the equation $3^n =x^k + y^k$.
[i]A. Kovaldji, V. Senderov[/i]
2003 Olympic Revenge, 2
Let $x_n$ the sequence defined by any nonnegatine integer $x_0$ and $x_{n+1}=1+\prod_{0 \leq i \leq n}{x_i}$
Show that there exists prime $p$ such that $p\not|x_n$ for any $n$.
1984 Swedish Mathematical Competition, 5
Solve in natural numbers $a,b,c$ the system \[\left\{ \begin{array}{l}a^3 -b^3 -c^3 = 3abc \\
a^2 = 2(a+b+c)\\
\end{array} \right.
\]
2009 Costa Rica - Final Round, 4
Show that the number $ 3^{{4}^{5}} \plus{} 4^{{5}^{6}}$ can be expresed as the product of two integers greater than $ 10^{2009}$
2014 IFYM, Sozopol, 1
Find all pairs of natural numbers $(m,n)$, for which $m\mid 2^{\varphi(n)} +1$ and $n\mid 2^{\varphi (m)} +1$.
2013 Dutch IMO TST, 4
Determine all positive integers $n\ge 2$ satisfying $i+j\equiv\binom ni +\binom nj \pmod{2}$ for all $i$ and $j$ with $0\le i\le j\le n$.
2019 Junior Balkan Team Selection Tests - Romania, 1
Let $n$ be a nonnegative integer and $M =\{n^3, n^3+1, n^3+2, ..., n^3+n\}$.
Consider $A$ and $B$ two nonempty, disjoint subsets of $M$ such that the sum of elements of the set $A$ divides the sum of elements of the set $B$. Prove that the number of elements of the set $A$ divides the number of elements of the set $B$.