Found problems: 85335
1996 Estonia National Olympiad, 1
Prove that for any positive numbers $x,y$ it holds that $x^xy^y \ge x^yy^x$.
2017 Bosnia And Herzegovina - Regional Olympiad, 3
Find prime numbers $p$, $q$, $r$ and $s$, pairwise distinct, such that their sum is prime number and numbers $p^2+qr$ and $p^2+qs$ are perfect squares
2023 MIG, 1
What is $1-2+3-4$?
$\textbf{(A) } {-}2\qquad\textbf{(B) } {-}1\qquad\textbf{(C) } 1\qquad\textbf{(D) } 4\qquad\textbf{(E) } 9$
1986 IMO Longlists, 53
For given positive integers $r, v, n$ let $S(r, v, n)$ denote the number of $n$-tuples of non-negative integers $(x_1, \cdots, x_n)$ satisfying the equation $x_1 +\cdots+ x_n = r$ and such that $x_i \leq v$ for $i = 1, \cdots , n$. Prove that
\[S(r, v, n)=\sum_{k=0}^{m} (-1)^k \binom nk \binom{r - (v + 1)k + n - 1}{n-1}\]
Where $m=\left\{n,\left[\frac{r}{v+1}\right]\right\}.$
2016 Israel Team Selection Test, 1
A square $ABCD$ is given. A point $P$ is chosen inside the triangle $ABC$ such that $\angle CAP = 15^\circ = \angle BCP$. A point $Q$ is chosen such that $APCQ$ is an isosceles trapezoid: $PC \parallel AQ$, and $AP=CQ, AP\nparallel CQ$. Denote by $N$ the midpoint of $PQ$. Find the angles of the triangle $CAN$.
2012 AMC 10, 15
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?
[asy]
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dotfactor=4;
draw((0,0)--(0,2));
draw((0,0)--(1,0));
draw((1,0)--(1,2));
draw((0,1)--(2,1));
draw((0,0)--(1,2));
draw((0,2)--(2,1));
draw((0,2)--(2,2));
draw((2,1)--(2,2));
label("$A$",(0,2),NW);
label("$B$",(1,2),N);
label("$C$",(4/5,1.55),W);
dot((0,2));
dot((1,2));
dot((4/5,1.6));
dot((2,1));
dot((0,0));
[/asy]
$ \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{2}{9}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{\sqrt2}{4} $
2016 Romania Team Selection Tests, 2
Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$.
[i]Proposed by El Salvador[/i]
1966 IMO Shortlist, 10
How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?
2011 Israel National Olympiad, 4
Let $\alpha_1,\alpha_2,\alpha_3$ be three congruent circles that are tangent to each other. A third circle $\beta$ is tangent to them at points $A_1,A_2,A_3$ respectively. Let $P$ be a point on $\beta$ which is different from $A_1,A_2,A_3$. For $i=1,2,3$, let $B_i$ be the second intersection point of the line $PA_i$ with circle $\alpha_i$. Prove that $\Delta B_1B_2B_3$ is equilateral.
2023-IMOC, A1
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for all positive integers $n$, there exists an unique positive integer $k$, satisfying $f^k(n)\leq n+k+1$.
2012 Belarus Team Selection Test, 2
$A, B, C, D, E$ are five points on the same circle, so that $ABCDE$ is convex and we have $AB = BC$ and $CD = DE$. Suppose that the lines $(AD)$ and $(BE)$ intersect at $P$, and that the line $(BD)$ meets line $(CA)$ at $Q$ and line $(CE)$ at $T$. Prove that the triangle $PQT$ is isosceles.
(I. Voronovich)
2015 HMMT Geometry, 2
Let $ABC$ be a triangle with orthocenter $H$; suppose $AB=13$, $BC=14$, $CA=15$. Let $G_A$ be the centroid of triangle $HBC$, and define $G_B$, $G_C$ similarly. Determine the area of triangle $G_AG_BG_C$.
2017 Taiwan TST Round 2, 2
Given a $ \triangle ABC $ and three points $ D, E, F $ such that $ DB = DC, $ $ EC = EA, $ $ FA = FB, $ $ \measuredangle BDC = \measuredangle CEA = \measuredangle AFB. $ Let $ \Omega_D $ be the circle with center $ D $ passing through $ B, C $ and similarly for $ \Omega_E, \Omega_F. $ Prove that the radical center of $ \Omega_D, \Omega_E, \Omega_F $ lies on the Euler line of $ \triangle DEF. $
[i]Proposed by Telv Cohl[/i]
2016 AMC 8, 20
The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$?
$\textbf{(A) }20\qquad\textbf{(B) }30\qquad\textbf{(C) }60\qquad\textbf{(D) }120\qquad \textbf{(E) }180$
2024 BMT, 5
Let $U$ and $C$ be two circles, and kite $BERK$ have vertices that lie on $U$ and sides that are tangent to $C.$ Given that the diagonals of the kite measure $5$ and $6,$ find the ratio of the area of $U$ to the area of $C.$
2002 India National Olympiad, 4
Is it true that there exist 100 lines in the plane, no three concurrent, such that they intersect in exactly 2002 points?
2023 Romania National Olympiad, 4
Let $ABC$ be a triangle with $\angle BAC = 90^{\circ}$ and $\angle ACB = 54^{\circ}.$ We construct bisector $BD (D \in AC)$ of angle $ABC$ and consider point $E \in (BD)$ such that $DE = DC.$ Show that $BE = 2 \cdot AD.$
2014 Middle European Mathematical Olympiad, 5
Let $ABC$ be a triangle with $AB < AC$. Its incircle with centre $I$ touches the sides $BC, CA,$ and $AB$ in the points $D, E,$ and $F$ respectively. The angle bisector $AI$ intersects the lines $DE$ and $DF$ in the points $X$ and $Y$ respectively. Let $Z$ be the foot of the altitude through $A$ with respect to $BC$.
Prove that $D$ is the incentre of the triangle $XYZ$.
2010 CHMMC Fall, 14
A $4$-dimensional hypercube of edge length $1$ is constructed in $4$-space with its edges parallel to
the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given
by $(a, b, c, d)$, where each of $a, b, c,$ and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given
by $x + y + z + w = 2$ intersects the hypercube at $6$ of its vertices. Compute the $3$-dimensional
volume of the solid formed by the intersection.
2001 National Olympiad First Round, 11
For how many integers $n$, does the equation system \[\begin{array}{rcl}
2x+3y &=& 7\\
5x + ny &=& n^2
\end{array}\] have a solution over integers?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2013 Cuba MO, 6
$2013$ people run a marathon on a straight road $4m$ wide broad. At any given moment, no two runners are closer
$2$ m from each other. Prove that there are two runners that at that moment are more than $1052$ m from each other.
Note: Consider runners as points.
1998 Taiwan National Olympiad, 3
Let $ m,n$ be positive integers, and let $ F$ be a family of $ m$-element subsets of $ \{1,2,...,n\}$ satisfying $ A\cap B \not \equal{} \emptyset$ for all $ A,B\in F$. Determine the maximum possible number of elements in $ F$.
2004 Bulgaria Team Selection Test, 2
Find all primes $p \ge 3$ such that $p- \lfloor p/q \rfloor q$ is a square-free integer for any prime $q<p$.
2019 Switzerland Team Selection Test, 1
Let $ABC$ be a triangle and $D, E, F$ be the foots of altitudes drawn from $A,B,C$ respectively. Let $H$ be the orthocenter of $ABC$. Lines $EF$ and $AD$ intersect at $G$. Let $K$ the point on circumcircle of $ABC$ such that $AK$ is a diameter of this circle. $AK$ cuts $BC$ in $M$. Prove that $GM$ and $HK$ are parallel.
2017 AMC 12/AHSME, 8
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?
$\textbf{(A)} \text{ } \frac{\sqrt{3}-1}{2} \qquad \textbf{(B)} \text{ } \frac{1}{2} \qquad \textbf{(C)} \text{ } \frac{\sqrt{5}-1}{2} \qquad \textbf{(D)} \text{ } \frac{\sqrt{2}}{2} \qquad \textbf{(E)} \text{ } \frac{\sqrt{6}-1}{2}$