Found problems: 1581
2005 Polish MO Finals, 2
The points $A, B, C, D$ lie in this order on a circle $o$. The point $S$ lies inside $o$ and has properties $\angle SAD=\angle SCB$ and $\angle SDA= \angle SBC$. Line which in which angle bisector of $\angle ASB$ in included cut the circle in points $P$ and $Q$. Prove $PS =QS$.
2013 National Olympiad First Round, 9
Let $ABC$ be a triangle with $|AB|=18$, $|AC|=24$, and $m(\widehat{BAC}) = 150^\circ$. Let $D$, $E$, $F$ be points on sides $[AB]$, $[AC]$, $[BC]$, respectively, such that $|BD|=6$, $|CE|=8$, and $|CF|=2|BF|$. Let $H_1$, $H_2$, $H_3$ be the reflections of the orthocenter of triangle $ABC$ over the points $D$, $E$, $F$, respectively. What is the area of triangle $H_1H_2H_3$?
$
\textbf{(A)}\ 70
\qquad\textbf{(B)}\ 72
\qquad\textbf{(C)}\ 84
\qquad\textbf{(D)}\ 96
\qquad\textbf{(E)}\ 108
$
2002 AMC 10, 24
Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $ 20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $ 10$ vertical feet above the bottom?
$ \textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 7.5 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 15$
2006 MOP Homework, 1
Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can not use a needle to peer through the tiling? Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can use a needle to through the tiling? What if it is a $6 \times 6$ board?
2008 Sharygin Geometry Olympiad, 9
(A.Zaslavsky, 9--10) The reflections of diagonal $ BD$ of a quadrilateral $ ABCD$ in the bisectors of angles $ B$ and $ D$ pass
through the midpoint of diagonal $ AC$. Prove that the reflections of diagonal $ AC$ in the bisectors of angles $ A$ and $ C$ pass
through the midpoint of diagonal $ BD$ (There was an error in published condition of this problem).
2012 Sharygin Geometry Olympiad, 21
Two perpendicular lines pass through the orthocenter of an acute-angled triangle. The sidelines of the triangle cut on each of these lines two segments: one lying inside the triangle and another one lying outside it. Prove that the product of two internal segments is equal to the product of two external segments.
[i]Nikolai Beluhov and Emil Kolev[/i]
2012 Iran MO (3rd Round), 3
Cosider ellipse $\epsilon$ with two foci $A$ and $B$ such that the lengths of it's major axis and minor axis are $2a$ and $2b$ respectively. From a point $T$ outside of the ellipse, we draw two tangent lines $TP$ and $TQ$ to the ellipse $\epsilon$. Prove that
\[\frac{TP}{TQ}\ge \frac{b}{a}.\]
[i]Proposed by Morteza Saghafian[/i]
2014 Contests, 2
Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.
1988 China Team Selection Test, 3
A polygon $\prod$ is given in the $OXY$ plane and its area exceeds $n.$ Prove that there exist $n+1$ points $P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})$ in $\prod$ such that $\forall i,j \in \{1, 2, \ldots, n+1\}$, $x_j - x_i$ and $y_j - y_i$ are all integers.
2004 Tuymaada Olympiad, 1
50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine.
[i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]
2011 Philippine MO, 2
In triangle $ABC$, let $X$ and $Y$ be the midpoints of $AB$ and $AC$, respectively. On segment $BC$, there is a point $D$, different from its midpoint, such that $\angle{XDY}=\angle{BAC}$. Prove that $AD\perp BC$.
2011 Switzerland - Final Round, 10
On each square of an $n\times n$-chessboard, there are two bugs. In a move, each bug moves to a (vertically of horizontally) adjacent square. Bugs from the same square always move to different squares. Determine the maximal number of free squares that can occur after one move.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 10)[/i]
2000 Vietnam National Olympiad, 2
Two circles $ (O_1)$ and $ (O_2)$ with respective centers $ O_1$, $ O_2$ are given on a plane. Let $ M_1$, $ M_2$ be points on $ (O_1)$, $ (O_2)$ respectively, and let the lines $ O_1M_1$ and $ O_2M_2$ meet at $ Q$. Starting simultaneously from these positions, the points $ M_1$ and $ M_2$ move clockwise on their own circles with the same angular velocity.
(a) Determine the locus of the midpoint of $ M_1M_2$.
(b) Prove that the circumcircle of $ \triangle M_1QM_2$ passes through a fixed point.
2012 Germany Team Selection Test, 2
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.
[i]Proposed by Härmel Nestra, Estonia[/i]
2020 MBMT, 20
Sam colors each tile in a 4 by 4 grid white or black. A coloring is called [i]rotationally symmetric[/i] if the grid can be rotated 90, 180, or 270 degrees to achieve the same pattern. Two colorings are called [i]rotationally distinct[/i] if neither can be rotated to match the other. How many rotationally distinct ways are there for Sam to color the grid such that the colorings are [i]not[/i] rotationally symmetric?
[i]Proposed by Gabriel Wu[/i]
2016 Indonesia TST, 3
Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$
1996 Turkey Junior National Olympiad, 2
Write out the positive integers consisting of only $1$s, $6$s, and $9$s in ascending order as in: $1,6,9,11,16,\dots$.
a. Find the order of $1996$ in the sequence.
b. Find the $1996$th term in the sequence.
2008 Balkan MO, 1
Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.
2012 Sharygin Geometry Olympiad, 15
Given triangle $ABC$. Consider lines $l$ with the next property: the reflections of $l$ in the sidelines of the triangle concur. Prove that all these lines have a common point.
2002 Balkan MO, 3
Two circles with different radii intersect in two points $A$ and $B$. Let the common tangents of the two circles be $MN$ and $ST$ such that $M,S$ lie on the first circle, and $N,T$ on the second. Prove that the orthocenters of the triangles $AMN$, $AST$, $BMN$ and $BST$ are the four vertices of a rectangle.
Today's calculation of integrals, 857
Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$
(1) Find $f(x).$
(2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.
2009 Ukraine National Mathematical Olympiad, 3
In triangle $ABC$ points $M, N$ are midpoints of $BC, CA$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PCA = \angle MAC .$ Prove that $\angle PNA = \angle AMB .$
1976 Bundeswettbewerb Mathematik, 4
Each vertex of the 3-dimensional Euclidean space either is coloured red or blue. Prove that within those squares being possible in this space with edge length 1 there is at least one square either with three red vertices or four blue vertices !
2009 Spain Mathematical Olympiad, 6
Inside a circle of center $ O$ and radius $ r$, take two points $ A$ and $ B$ symmetrical about $ O$. We consider a variable point $ P$ on the circle and draw the chord $ \overline{PP'}\perp \overline{AP}$. Let $ C$ is the symmetric of $ B$ about $ \overline{PP'}$ ($ \overline{PP}'$ is the axis of symmetry) . Find the locus of point $ Q \equal{} \overline{PP'}\cap\overline{AC}$ when we change $ P$ in the circle.
Today's calculation of integrals, 865
Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.