Found problems: 1581
A rectangular piece of paper has top edge $AD$. A line $L$ from $A$ to the bottom edge makes an angle $x$ with the line $AD$. We want to trisect $x$. We take $B$ and $C$ on the vertical ege through $A$ such that $AB = BC$. We then fold the paper so that $C$ goes to a point $C'$ on the line $L$ and $A$ goes to a point $A'$ on the horizontal line through $B$. The fold takes $B$ to $B'$. Show that $AA'$ and $AB'$ are the required trisectors.
A point is chosen on each side of a regular $2009$-gon. Let $S$ be the area of the $2009$-gon with vertices at these points. For each of the chosen points, reflect it across the midpoint of its side. Prove that the $2009$-gon with vertices at the images of these reflections also has area $S.$
[i](4 points)[/i]
a) Two legs of an angle $\alpha$ on a plane are mirrors. Prove that after several reflections in the mirrors any ray leaves in the direction opposite the one from which it came if and only if $\alpha = \frac{90^o}{n}$ for an integer $n$. Find the number of reflections.
b) Given three planar mirrors in space forming an octant (trihedral angle with right planar angles), prove that any ray of light coming into this mirrored octant leaves it, after several reflections in the mirrors, in the direction opposite to the one from which it came. Find the number of reflections.
Let $A_{1}A_{2}A_{3}$ be an acute triangle, and let $O$ and $H$ be its circumcenter and orthocenter, respectively. For $1\leq i \leq 3$, points $P_{i}$ and $Q_{i}$ lie on lines $OA_{i}$ and $A_{i+1}A_{i+2}$ (where $A_{i+3}=A_{i}$), respectively, such that $OP_{i}HQ_{i}$ is a parallelogram. Prove that
\[\frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.\]
Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, and circumcircle $\Gamma$. It is known that $AB = 7$, $BC = 8$, $CA = 9$. Let $M$ denote the midpoint of major arc $\widehat{BAC}$ of $\Gamma$, and let $D$ denote the intersection of $\Gamma$ with the circumcircle of $\triangle IMO$ (other than $M$). Let $E$ denote the reflection of $D$ over line $IO$. Find the integer closest to $1000 \cdot \frac{BE}{CE}$.
[i]Proposed by Evan Chen[/i]
Let $ Ax,By$ be two perpendicular semi-straight lines, being not complanar, (non-coplanar rays) such that $ AB$ is the their common perpendicular, and let $ M$ and $ N$ be the two variable points on $ Ax$ and $ Bx,$ respectively, such that $ AM \plus{} BN \equal{} MN.$
[b](a)[/b] Prove that there exist infinitely many lines being co-planar with each of the straight lines $ MN.$
[b](b)[/b] Prove that there exist infinitely many rotations around a fixed axis $ \delta$ mapping the line $ Ax$ onto a line coplanar with each of the lines $ MN.$
For a set $A$ of integers, define $f(A)=\{x^2+xy+y^2: x,y\in A\}$. Is there a constant $c$ such that for all positive integers $n$, there exists a set $A$ of size $n$ such that $|f(A)|\le cn$?
[i]David Yang.[/i]
In triangle $ABC$, $D$, $E$ and $F$ are the points of tangency of incircle with the center of $I$ to $BC$, $CA$ and $AB$ respectively. Let $M$ be the foot of the perpendicular from $D$ to $EF$. $P$ is on $DM$ such that $DP = MP$. If $H$ is the orthocenter of $BIC$, prove that $PH$ bisects $ EF$.
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.
A regular dodecahedron is a convex polyhedra that its faces are regular pentagons. The regular dodecahedron has twenty vertices and there are three edges connected to each vertex. Suppose that we have marked ten vertices of the regular dodecahedron.
[b]a)[/b] prove that we can rotate the dodecahedron in such a way that at most four marked vertices go to a place that there was a marked vertex before.
[b]b)[/b] prove that the number four in previous part can't be replaced with three.
[i]proposed by Kasra Alishahi[/i]
Let $ABCD$ be a cyclic quadrilaterla and let $E$ be the point of intersection of its diagonals $AC$ and $BD$. Suppose $AD$ and $BC$ meet in $F$. Let the midpoints of $AB$ and $CD$ be $G$ and $H$ respectively. If $\Gamma $ is the circumcircle of triangle $EGH$, prove that $FE$ is tangent to $\Gamma $.
Let $ k$ be the inscribed circle of non-isosceles triangle $ \triangle ABC$, which center is $ S$. Circle $ k$ touches sides $ BC,CA,AB$ in points $ P,Q,R$ respectively. Line $ QR$ intersects $ BC$ in point $ M$. Let a circle which contains points $ B$ and $ C$ touch $ k$ in point $ N$. Circumscribed circle of $ \triangle MNP$ intersects line $ AP$ in point $ L$, different from $ P$. Prove that points $ S,L$ and $ M$ are collinear.
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]
In the acute-angled triangle $ ABC$, $ D$ is the intersection of the altitude passing through $ A$ with $ BC$ and $ I_a$ is the excenter of the triangle with respect to $ A$. $ K$ is a point on the extension of $ AB$ from $ B$, for which $ \angle AKI_a\equal{}90^\circ\plus{}\frac 34\angle C$. $ I_aK$ intersects the extension of $ AD$ at $ L$. Prove that $ DI_a$ bisects the angle $ \angle AI_aB$ iff $ AL\equal{}2R$. ($ R$ is the circumradius of $ ABC$)
Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. Circle $\omega_A$ is externally tangent to $\omega$ and tangent to sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Let $r_A$ be the line $A_1A_2$. Define $r_B$ and $r_C$ in a similar fashion. Lines $r_A$, $r_B$ and $r_C$ determine a triangle $XYZ$. Prove that the incenter of $XYZ$, the circumcenter of $XYZ$ and $I$ are collinear.
The plane is covered with network of regular congruent disjoint hexagons. Prove that there cannot exist a square which has its four vertices in the vertices of the hexagons.
[i]Gabriel Nagy[/i]
The vertices of two acute-angled triangles lie on the same circle. The Euler circle (nine-point circle) of one of the triangles passes through the midpoints of two sides of the other triangle. Prove that the triangles have the same Euler circle.
EDIT by pohoatza (in concordance with Luis' PS): [hide=Alternate/initial version ]Let $ABC$ be a triangle with circumcenter $\Gamma$ and nine-point center $\gamma$. Let $X$ be a point on $\Gamma$ and let $Y$, $Z$ be on $\Gamma$ so that the midpoints of segments $XY$ and $XZ$ are on $\gamma$. Prove that the midpoint of $YZ$ is on $\gamma$.[/hide]
We want to place $2012$ pockets, including variously colored balls, into $k$ boxes such that
[b]i)[/b] For any box, all pockets in this box must include a ball with the same color
or
[b]ii)[/b] For any box, all pockets in this box must include a ball having a color which is not included in any other pocket in this box
Find the smallest value of $k$ for which we can always do this placement whatever the number of balls in the pockets and whatever the colors of balls.
Let $C_1$ be a circle with center $O$ and let $B$ and $C$ be points in $C_1$ such that $BOC$ is an equilateral triangle. Let $D$ be the midpoint of the minor arc $BC$ of $C_1$. Let $C_2$ be the circle with center $C$ that passes through $B$ and $O$. Let $E$ be the second intersection of $C_1$ and $C_2$. The parallel to $DE$ through $B$ intersects $C_1$ for second time in $A$. Let $C_3$ be the circumcircle of triangle $AOC$. The second intersection of $C_2$ and $C_3$ is $F$. Show that $BE$ and $BF$ trisect the angle $\angle ABC$.
In trapezoid $ ABCD$ with $ \overline{BC}\parallel\overline{AD}$, let $ BC\equal{}1000$ and $ AD\equal{}2008$. Let $ \angle A\equal{}37^\circ$, $ \angle D\equal{}53^\circ$, and $ m$ and $ n$ be the midpoints of $ \overline{BC}$ and $ \overline{AD}$, respectively. Find the length $ MN$.
Let $C$ be the curve expressed by $x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).$
(1) Express $y$ in terms of $x$.
(2) Find the area of the figure $D$ enclosed by the $x$-axis and $C$.
(3) Find the volume of the solid generated by a rotation of $D$ about the $y$-axis.
Find the sum of all integers $n$ with $2 \le n \le 999$ and the following property: if $x$ and $y$ are randomly selected without replacement from the set $\left\{ 1,2,\dots,n \right\}$, then $x+y$ is even with probability $p$, where $p$ is the square of a rational number.
[i]Proposed by Ivan Koswara[/i]
Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
$ \textbf{(A)}\ 2520\qquad\textbf{(B)}\ 2880\qquad\textbf{(C)}\ 3120\qquad\textbf{(D)}\ 3250\qquad\textbf{(E)}\ 3750 $
Let $ABC$ be a triangle and let $BB_1,CC_1$ be respectively the bisectors of $\angle{B},\angle{C}$ with $B_1$ on $AC$ and $C_1$ on $AB$, Let $E,F$ be the feet of perpendiculars drawn from $A$ onto $BB_1,CC_1$ respectively. Suppose $D$ is the point at which the incircle of $ABC$ touches $AB$. Prove that $AD=EF$
We match sets $ M$ of points in the coordinate plane to sets $ M*$ according to the rule that $ (x*,y*) \in M*$ if and only if $ x \cdot x* \plus{} y \cdot y* \leq 1$ whenever $ (x,y) \in M.$ Find all triangles $ Q$ such that $ Q*$ is the reflection of $ Q$ in the origin.