Found problems: 85335
2016 Purple Comet Problems, 19
Find the positive integer $n$ such that the least common multiple of $n$ and $n - 30$ is $n + 1320$.
2014 PUMaC Combinatorics B, 8
There are $60$ friends who want to visit each others home during summer vacation. Everyday, they decide to either stay home or visit the home of everyone who stayed home that day. Find the minimum number of days required for everyone to have visited their friends’ homes.
2015 BMT Spring, P1
Suppose $z_0,z_1,\ldots,z_{n-1}$ are complex numbers such that $z_k=e^{2k\pi i/n}$ for $k=0,1,2,\ldots,n-1$. Prove that for any complex number $z$, $\sum_{k=0}^{n-1}|z-z_k|\ge n$.
2023 Baltic Way, 19
Show that $S(2^{2^{2 \cdot 2023}})>2023$, where $S(m)$ denotes the digit sum of $m$.
2015 Online Math Open Problems, 8
Determine the number of sequences of positive integers $1 = x_0 < x_1 < \dots < x_{10} = 10^{5}$ with the property that for each $m=0,\dots,9$ the number $\tfrac{x_{m+1}}{x_m}$ is a prime number.
[i]Proposed by Evan Chen[/i]
2008 Argentina Iberoamerican TST, 1
Find all integers $ x$ such that $ x(x\plus{}1)(x\plus{}7)(x\plus{}8)$ is a perfect square
It's a nice problem ...hope you enjoy it!
Daniel
1979 Poland - Second Round, 3
In space there is a line $ k $ and a cube with a vertex $ M $ and edges $ \overline{MA} $, $ \overline{MB} $, $ \overline{MC} $, of length$ 1$. Prove that the length of the orthogonal projection of edge $ MA $ on the line $ k $ is equal to the area of the orthogonal projection of a square with sides $ MB $ and $ MC $ onto a plane perpendicular to the line $ k $.
[hide=original wording]W przestrzeni dana jest prosta $ k $ oraz sześcian o wierzchołku $ M $ i krawędziach $ \overline{MA} $, $ \overline{MB} $, $ \overline{MC} $, długości 1. Udowodnić, że długość rzutu prostokątnego krawędzi $ MA $ na prostą $ k $ jest równa polu rzutu prostokątnego kwadratu o bokach $ MB $ i $ MC $ na płaszczyznę prostopadłą do prostej $ k $.[/hide]
2017 Istmo Centroamericano MO, 1
Let $ABC$ be a triangle with $\angle ABC = 90^o$ and $AB> BC$. Let $D$ be a point on side $AB$ such that $BD = BC$. Let $E$ be the foot of the perpendicular from $D$ on $AC$, and $F$ the reflection of $B$ wrt $CD$. Show that $EC$ is the bisector of angle $\angle BEF$.
2012 Balkan MO Shortlist, N1
A sequence $(a_n)_{n=1}^{\infty}$ of positive integers satisfies the condition $a_{n+1} = a_n +\tau (n)$ for all positive integers $n$ where $\tau (n)$ is the number of positive integer divisors of $n$. Determine whether two consecutive terms of this sequence can be perfect squares.
2002 ITAMO, 3
Let $A$ and $B$ are two points on a plane, and let $M$ be the midpoint of $AB$. Let $r$ be a line and let $R$ and $S$ be the projections of $A$ and $B$ onto $r$. Assuming that $A$, $M$, and $R$ are not collinear, prove that the circumcircle of triangle $AMR$ has the same radius as the circumcircle of $BSM$.
2009 China Team Selection Test, 3
Prove that for any odd prime number $ p,$ the number of positive integer $ n$ satisfying $ p|n! \plus{} 1$ is less than or equal to $ cp^\frac{2}{3}.$ where $ c$ is a constant independent of $ p.$
2005 Sharygin Geometry Olympiad, 1
The chords $AC$ and $BD$ of the circle intersect at point $P$. The perpendiculars to $AC$ and $BD$ at points $C$ and $D$, respectively, intersect at point $Q$. Prove that the lines $AB$ and $PQ$ are perpendicular.
2019 BMT Spring, 4
The area of right triangle $ ABC $ is 4, and the length of hypotenuse $ AB $ is 12. Compute the perimeter of $ \triangle ABC $.
2019 Saudi Arabia IMO TST, 1
Let $a_0$ be an arbitrary positive integer. Let $(a_n)$ be infinite sequence of positive integers such that for every positive integer $n$, the term $a_n$ is the smallest positive integer such that $a_0 + a_1 +... + a_n$ is divisible by $n$. Prove that there exist $N$ such that $a_{n+1} = a_n$ for all $n \ge N$
1996 Bundeswettbewerb Mathematik, 1
For a given set of points in space it is allowed to mirror a point from the set with respect to another point from the set, and to include the image in the set. Starting with a set of seven vertices of a cube, is it possible to include the eight vertex in the set after finitely many such steps?
2013 North Korea Team Selection Test, 2
Let $ a_1 , a_2 , \cdots , a_k $ be numbers such that $ a_i \in \{ 0,1,2,3 \} ( i= 1, 2, \cdots ,k) $. Let $ z = ( x_k , x_{k-1} , \cdots , x_1 )_4 $ be a base 4 expansion of $ z \in \{ 0, 1, 2, \cdots , 4^k -1 \} $. Define $ A $ as follows:
\[ A = \{ z | p(z)=z, z=0, 1, \cdots ,4^k-1 \}\]
where
\[ p(z) = \sum_{i=1}^{k} a_i x_i 4^{i-1} . \]
Prove that the number of elements in $ X $ is a power of 2.
2022 Dutch Mathematical Olympiad, 2
A set consisting of at least two distinct positive integers is called [i]centenary [/i] if its greatest element is $100$. We will consider the average of all numbers in a centenary set, which we will call the average of the set. For example, the average of the centenary set $\{1, 2, 20, 100\}$ is $\frac{123}{4}$ and the average of the centenary set $\{74, 90, 100\}$ is $88$. Determine all integers that can occur as the average of a centenary set.
XMO (China) 2-15 - geometry, 10.2
Given acute triangle $\vartriangle ABC$ with orthocenter $H$ and circumcenter $O$ ($O \ne H$) . Let $\Gamma$ be the circumcircle of $\vartriangle BOC$ . Segment $OH$ untersects $\Gamma$ at point $P$. Extension of $AO$ intersects $\Gamma$ at point $K$. If $AP \perp OH$, prove that $PK$ bisects $BC$.
[img]https://cdn.artofproblemsolving.com/attachments/a/b/267053569c41692f47d8f4faf2a31ebb4f4efd.png[/img]
1998 IMO Shortlist, 3
Let $x,y$ and $z$ be positive real numbers such that $xyz=1$. Prove that
\[
\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)}
\geq \frac{3}{4}.
\]
2021 Czech-Austrian-Polish-Slovak Match, 2
In an acute triangle $ABC$, the incircle $\omega$ touches $BC$ at $D$. Let $I_a$ be the excenter of $ABC$ opposite to $A$, and let $M$ be the midpoint of $DI_a$. Prove that the circumcircle of triangle $BMC$ is tangent to $\omega$.
[i]Patrik Bak (Slovakia)[/i]
Swiss NMO - geometry, 2010.2
Let $ \triangle{ABC}$ be a triangle with $ AB\not\equal{}AC$. The incircle with centre $ I$ touches $ BC$, $ CA$, $ AB$ at $ D$, $ E$, $ F$, respectively. Furthermore let $ M$ the midpoint of $ EF$ and $ AD$ intersect the incircle at $ P\not\equal{}D$.
Show that $ PMID$ ist cyclic.
2010 Moldova Team Selection Test, 3
Let $ ABCD$ be a convex quadrilateral. We have that $ \angle BAC\equal{}3\angle CAD$, $ AB\equal{}CD$, $ \angle ACD\equal{}\angle CBD$. Find angle $ \angle ACD$
2012 239 Open Mathematical Olympiad, 6
In an $n$-element set $S$, several subsets $A_1, A_2, \ldots , A_k$ are distinguished, each consists of at least two, but not all elements of $S$. What is the largest $k$ that it’s possible to write down the elements of $S$ in a row in the order such that we don’t find all of the element of an $A_i$ set in the consecutive elements of the row?
2013 Tournament of Towns, 4
Each of $100$ stones has a sticker showing its true weight. No two stones weight the same. Mischievous Greg wants to rearrange stickers so that the sum of the numbers on the stickers for any group containing from $1$ to $99$ stones is different from the true weight of this group. Is it always possible?
2018 Lusophon Mathematical Olympiad, 6
In a $3 \times 25$ board, $1 \times 3$ pieces are placed (vertically or horizontally) so that they occupy entirely $3$ boxes on the board and do not have a common point.
What is the maximum number of pieces that can be placed, and for that number, how many configurations are there?
[hide=original formulation]
Num tabuleiro 3 × 25 s˜ao colocadas pe¸cas 1 × 3 (na vertical ou na horizontal) de modo que ocupem inteiramente 3 casas do tabuleiro e n˜ao se toquem em nenhum ponto.
Qual ´e o n´umero m´aximo de pe¸cas que podem ser colocadas, e para esse n´umero,
quantas configura¸c˜oes existem?
[url=https://www.obm.org.br/content/uploads/2018/09/Provas_OMCPLP_2018.pdf]source[/url][/hide]