Found problems: 85335
2015 Dutch BxMO/EGMO TST, 1
Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$.
Prove that $m$ is a divisor of $n$.
2018 Polish Junior MO First Round, 7
Square $ABCD$ with sides of length $4$ is a base of a cuboid $ABCDA'B'C'D'$. Side edges $AA'$, $BB'$, $CC'$, $DD'$ of this cuboid have length $7$. Points $K, L, M$ lie respectively on line segments $AA'$, $BB'$, $CC'$, and $AK = 3$, $BL = 2$, $CM = 5$. Plane passing through points $K, L, M$ cuts cuboid on two blocks. Calculate volumes of these blocks.
2010 JBMO Shortlist, 5
Let $ x, y, z > 0 $ with $ x \leq 2, \;y \leq 3 \;$ and $ x+y+z = 11 $
Prove that $xyz \leq 36$
2013 National Olympiad First Round, 6
What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base-$5$ representation?
$
\textbf{(A)}\ 760
\qquad\textbf{(B)}\ 756
\qquad\textbf{(C)}\ 755
\qquad\textbf{(D)}\ 752
\qquad\textbf{(E)}\ 750
$
2004 Turkey Junior National Olympiad, 3
On the evening, more than $\frac 13$ of the students of a school are going to the cinema. On the same evening, More than $\frac {3}{10}$ are going to the theatre, and more than $\frac {4}{11}$ are going to the concert. At least how many students are there in this school?
2022 VIASM Summer Challenge, Problem 1
Given prime numbers $p$ and $q.$
a) Assume that $2^xq=p^y+1,$ with $x,y$ are integers greater than $1$. Can $x$ be a composite number?
b) Assume that $2^uq=p^v-1,$ with $u$ is a prime number and $v$ is an integer greater than $1$. Find all possible values of $p.$
1968 Yugoslav Team Selection Test, Problem 1
Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.
2009 Mexico National Olympiad, 2
Consider a triangle $ABC$ and a point $M$ on side $BC$. Let $P$ be the intersection of the perpendiculars from $M$ to $AB$ and from $B$ to $BC$, and let $Q$ be the intersection of the perpendiculars from $M$ to $AC$ and from $C$ to $BC$. Show that $PQ$ is perpendicular to $AM$ if and only if $M$ is the midpoint of $BC$.
2008 Princeton University Math Competition, A2
What is the polynomial of smallest degree that passes through $(-2, 2), (-1, 1), (0, 2),(1,-1)$, and $(2, 10)$?
2013 India Regional Mathematical Olympiad, 5
In a triangle $ABC$, let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $BC$. The circumcircle of triangle $ABP$ intersects the line $BH$ again at $Q$, and the circumcircle of triangle $ACP$ intersects the line $CH$ again at $R$. Prove that $H$ is the incentre of triangle $PQR$.
Kyiv City MO Juniors Round2 2010+ geometry, 2017.7.4
On the sides $AD$ and $BC$ of a rectangle $ABCD$ select points $M, N$ and $P, Q$ respectively such that $AM = MN = ND = BP = PQ = QC$. On segment $QC$ selected point $X$, different from the ends of the segment. Prove that the perimeter of $\vartriangle ANX$ is more than the perimeter of $\vartriangle MDX$.
2007 Today's Calculation Of Integral, 208
Find the values of real numbers $a,\ b$ for which the function $f(x)=a|\cos x|+b|\sin x|$ has local minimum at $x=-\frac{\pi}{3}$ and satisfies $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\{f(x)\}^{2}dx=2$.
PEN N Problems, 15
In the sequence $00$, $01$, $02$, $03$, $\cdots$, $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$, $39$, or $28$, but not by $30$ or $20$). What is the maximal number of terms that could remain on their places?
2018 Saint Petersburg Mathematical Olympiad, 7
In $10\times 10$ square we choose $n$ cells. In every chosen cell we draw one arrow from the angle to opposite angle. It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of $n$?
2012 Singapore Junior Math Olympiad, 2
Does there exist an integer $A$ such that each of the ten digits $0, 1, . . . , 9$ appears exactly once as a digit in exactly one of the numbers $A, A^2, A^ 3$ ?
2012 Hanoi Open Mathematics Competitions, 2
Compare the numbers $P = 2^a,Q = 3, T = 2^b$, where $a=\sqrt2 , b=1+\frac{1}{\sqrt2}$
(A) $P < Q < T$, (B) $T < P < Q$, (C) $P < T < Q$, (D) $T < Q < P$, (E) $ Q < P < T$
1992 Baltic Way, 19
Let $ C$ be a circle in plane. Let $ C_1$ and $ C_2$ be nonintersecting circles touching $ C$ internally at points $ A$ and $ B$ respectively. Let $ t$ be a common tangent of $ C_1$ and $ C_2$ touching them at points $ D$ and $ E$ respectively, such that both $ C_1$ and $ C_2$ are on the same side of $ t$. Let $ F$ be the point of intersection of $ AD$ and $ BE$. Show that $ F$ lies on $ C$.
2023 APMO, 1
Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1, 2, \dots , n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.
2011 National Olympiad First Round, 19
For which inequality, there exists a line such that the region defined by the inequality and the line intersect in exactly two distinct points?
$\textbf{(A)}\ x^2+y^2\leq 1 \qquad\textbf{(B)}\ |x+y|+|x-y| \leq 1 \qquad\textbf{(C)}\ |x|^3+|y|^3 \leq 1 \\ \textbf{(D)}\ |x|+|y| \leq 1 \qquad\textbf{(E)}\ |x|^{1/2} + |y|^{1/2} \leq 1$
2017 Estonia Team Selection Test, 10
Let $ABC$ be a triangle with $AB = \frac{AC}{2 }+ BC$. Consider the two semicircles outside the triangle with diameters $AB$ and $BC$. Let $X$ be the orthogonal projection of $A$ onto the common tangent line of those semicircles. Find $\angle CAX$.
2009 India IMO Training Camp, 2
Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) \equal{} 1$, are elements of V.
$ (a,b)$ is connected to $ (a,b \plus{} kab)$ by an edge and to $ (a \plus{} kab,b)$ by another edge for all integer k.
Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.
PEN A Problems, 65
Clara computed the product of the first $n$ positive integers and Valerid computed the product of the first $m$ even positive integers, where $m \ge 2$. They got the same answer. Prove that one of them had made a mistake.
1989 Irish Math Olympiad, 2
A 3x3 magic square, with magic number $m$, is a $3\times 3$ matrix such that the entries on each row, each column and each diagonal sum to $m$. Show that if the square has positive integer entries, then $m$ is divisible by $3$, and each entry of the square is at most $2n-1$, where $m=3n$. An example of a magic square with $m=6$ is
\[\left( \begin{array}{ccccc}
2 & 1 & 3\\
3 & 2 & 1\\
1 & 3 & 2
\end{array} \right)\]
2023 IFYM, Sozopol, 7
In an acute scalene triangle $ABC$, the incircle $\omega$ touches the sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $P$ be the foot of the perpendicular from $F$ to $DE$. The line $BP$ intersects segment $AC$ at $K$, and the line $AP$ intersects segment $BC$ at $L$. The altitude through vertex $C$ in $\triangle ABC$ intersects the circumcircle of $\triangle CKL$ at a point $Q$. Prove that line $PQ$ passes through the center of $\omega$.
2020 Macedonia Additional BMO TST, 1
Let $P$ and $Q$ be interior points in $\Delta ABC$ such that $PQ$ doesn't contain any vertices of $\Delta ABC$.
Let $A_1$, $B_1$, and $C_1$ be the points of intersection of $BC$, $CA$, and $AB$ with $AQ$, $BQ$, and $CQ$, respectively.
Let $K$, $L$, and $M$ be the intersections of $AP$, $BP$, and $CP$ with $B_1C_1$, $C_1A_1$, and $A_1B_1$, respectively.
Prove that $A_1K$, $B_1L$, and $C_1M$ are concurrent.