Found problems: 15460
2014 Korea Junior Math Olympiad, 4
Positive integers $p, q, r$ satisfy $gcd(a,b,c) = 1$.
Prove that there exists an integer $a$ such that $gcd(p,q+ar) = 1$.
2024 IFYM, Sozopol, 2
Let \( p \neq 3 \) be a prime number. Prove that there exist natural numbers \( a \), \( b \), \( c \), \( d \), none of which are divisible by \( p \), such that \( a^2 + 3b^5 + 5c^6 + 7d^7 \) is divisible by \( p^{1000} \).
2002 All-Russian Olympiad, 4
From the interval $(2^{2n},2^{3n})$ are selected $2^{2n-1}+1$ odd numbers. Prove that there are two among the selected numbers, none of which divides the square of the other.
2019 MMATHS, 1
$S$ is a set of positive integers with the following properties:
(a) There are exactly $3$ positive integers missing from $S$.
(b) If $a$ and $b$ are elements of $S$, then $a + b$ is an element of $S$. (We allow a and b to be the same.)
Find all possibilities for the set $S$ (with proof).
2019 Estonia Team Selection Test, 10
Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.
DMM Individual Rounds, 2007 Tie
[b]p1.[/b] Let $p_b(m)$ be the sum of digits of $m$ when $m$ is written in base $b$. (So, for example, $p_2(5) = 2$). Let $f(0) = 2007^{2007}$, and for $n \ge 0$ let $f(n + 1) = p_7(f(n))$. What is $f(10^{10000})$?
[b]p2.[/b] Compute:
$$\sum^{\infty}_{n=1}\frac{(-1)^{n+1}4n}{n^4 - 8n^2 + 4}.$$
[b]p3.[/b] $ABCDEFGH$ is an octagon whose eight interior angles all have the same measure. The lengths of the eight sides of this octagon are, in some order, $$2, 2\sqrt2, 4, 4\sqrt2, 6, 7, 7, \,\,\, and \,\,\, 8.$$
Find the area of $ABCDEFGH$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2004 Switzerland - Final Round, 6
Determine all $k$ for which there exists a natural number n such that $1^n + 2^n + 3^n + 4^n$ with exactly $k$ zeros at the end.
1995 All-Russian Olympiad, 1
Can the numbers $1,2,3,\ldots,100$ be covered with $12$ geometric progressions?
[i]A. Golovanov[/i]
2018 Iran MO (3rd Round), 2
Prove that for every prime number $p$ there exist infinity many natural numbers $n$ so that they satisfy:
$2^{2^{2^{ \dots ^{2^n}}}} \equiv n^{2^{2^{\dots ^{2}}}} (mod p)$
Where in both sides $2$ appeared $1397$ times
2025 CMIMC Algebra/NT, 4
Consider the system of equations $$\log_x y +\log_y z + \log_z x =8$$ $$\log_{\log_y x}z = -3$$ $$\log_z y + \log_x z = 16$$
Find $z.$
2012 Saint Petersburg Mathematical Olympiad, 1
Find all integer $b$ such that $[x^2]-2012x+b=0$ has odd number of roots.
2021 Science ON all problems, 1
Consider the prime numbers $p_1,p_2,\dots ,p_{2021}$ such that the sum
$$p_1^4+p_2^4+\dots +p_{2021}^4$$
is divisible by $6060$. Prove that at least $4$ of these prime numbers are less than $2021$.
$\textit{Stefan Bălăucă}$
2007 ITest, 39
Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the sum of the real solutions to the equation \[\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}.\] Find $a+b$.
OMMC POTM, 2024 5
Every integer $> 2024$ is given a color, white or black. The product of any two white integers is a black integer. Prove that there are two black integers that have a difference of one.
2008 All-Russian Olympiad, 1
Do there exist $ 14$ positive integers, upon increasing each of them by $ 1$,their product increases exactly $ 2008$ times?
1979 IMO Shortlist, 5
Let $n \geq 2$ be an integer. Find the maximal cardinality of a set $M$ of pairs $(j, k)$ of integers, $1 \leq j < k \leq n$, with the following property: If $(j, k) \in M$, then $(k,m) \not \in M$ for any $m.$
2017 International Zhautykov Olympiad, 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 $C(2^n+1)=C(n)$.
1979 Bulgaria National Olympiad, Problem 6
The set $M=\{1,2,\ldots,2n\}~(n\ge2)$ is partitioned into $k$ nonintersecting subsets $M_1,M_2,\ldots,M_k$, where $k^3+1\le n$. Prove that there exist $k+1$ even numbers $2j_1,2j_2,\ldots,2j_{k+1}$ in $M$ that are in one and the same subset $M_j$ $(1\le j\le k)$ such that the numbers $2j_1-1,2j_2-1,\ldots,2j_{k+1}-1$ are also in one and the same subset $M_r$ $(1\le r\le k)$.
1991 IMO Shortlist, 17
Find all positive integer solutions $ x, y, z$ of the equation $ 3^x \plus{} 4^y \equal{} 5^z.$
2024/2025 TOURNAMENT OF TOWNS, P3
There are five positive integers written in a row. Each one except for the first one is the minimal positive integer that is not a divisor of the previous one. Can all these five numbers be distinct?
Boris Frenkin
2020 European Mathematical Cup, 3
Let $p$ be a prime number. Troy and Abed are playing a game. Troy writes a positive integer $X$ on the board, and gives a sequence $(a_n)_{n\in\mathbb{N}}$ of positive integers to Abed. Abed now makes a sequence of moves. The $n$-th move is the following:
$$\text{ Replace } Y \text{ currently written on the board with either } Y + a_n \text{ or } Y \cdot a_n.$$
Abed wins if at some point the number on the board is a multiple of $p$. Determine whether Abed can win, regardless of Troy’s choices, if
$a) p = 10^9 + 7$;
$b) p = 10^9 + 9$.
[i]Remark[/i]: Both $10^9 + 7$ and $10^9 + 9$ are prime.
[i]Proposed by Ivan Novak[/i]
2010 CHKMO, 4
Find all non-negative integers $ m$ and $ n$ that satisfy the equation:
\[ 107^{56}(m^2\minus{}1)\minus{}2m\plus{}5\equal{}3\binom{113^{114}}{n}\]
(If $ n$ and $ r$ are non-negative integers satisfying $ r\le n$, then $ \binom{n}{r}\equal{}\frac{n}{r!(n\minus{}r)!}$ and $ \binom{n}{r}\equal{}0$ if $ r>n$.)
2013 IFYM, Sozopol, 5
Determine all increasing sequences $\{a_n\}_{n=1}^\infty$ of natural numbers with the following property: for each two natural numbers $i$ and $j$ (not necessarily different), the numbers $i+j$ and $a_i+a_j$ have an equal number of distinct natural divisors.
2020 Simon Marais Mathematics Competition, B4
[i]The following problem is open in the sense that no solution is currently known to part (b).[/i]
Let $n\geq 2$ be an integer, and $P_n$ be a regular polygon with $n^2-n+1$ vertices.
We say that $n$ is $\emph{taut}$ if it is possible to choose $n$ of the vertices of $P_n$ such that the pairwise distances between the chosen vertices are all distinct.
(a) show that if $n-1$ is prime then $n$ is taut.
(b) Which integers $n\geq 2$ are taut?
2002 Rioplatense Mathematical Olympiad, Level 3, 1
Determine all pairs $(a, b)$ of positive integers for which $\frac{a^2b+b}{ab^2+9}$ is an integer number.