Let $O, H$ be circumcenter and orthogonal center of triangle $ABC$. $M,N$ are midpoints of $BH$ and $CH$. $BB'$ is diagonal of circumcircle. If $HONM$ is a cyclic quadrilateral, prove that $B'N=\frac12AC$.

Draw the smaller number of line segments connecting points of the figure such that the new figure obtained to have exactly: [img]https://cdn.artofproblemsolving.com/attachments/d/1/098e03714904573a1eacd2d3dc28b4e8c42c7c.png[/img] i) one axis of symmetry ii) two axes of symmetry iii) four axes of symmetry Draw a new figure, at each case.

Given an isosceles triangle $ABC$ (with $AB=BC$). A point $X$ is chosen on a side $AC$. Some circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through the circumcentre of triangle $ABC$. [b]proposed by Sergej Berlov[/b]

The sum of the base-$ 10$ logarithms of the divisors of $ 10^n$ is $ 792$. What is $ n$? $ \textbf{(A)}\ 11\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 13\qquad \textbf{(D)}\ 14\qquad \textbf{(E)}\ 15$

Let $ABC$ be a triangle with $AB = AC$. The incircle touches $BC$, $AC$ and $AB$ at $D$, $E$ and $F$ respectively. Let $P$ be a point on the arc $\overarc{EF}$ that does not contain $D$. Let $Q$ be the second point of intersection of $BP$ and the incircle of $ABC$. The lines $EP$ and $EQ$ meet the line $BC$ at $M$ and $N$, respectively. Prove that the four points $P, F, B, M$ lie on a circle and $\frac{EM}{EN} = \frac{BF}{BP}$.

Let $\omega_P$ and $\omega_Q$ be two circles of radius $1$, intersecting in points $A$ and $B$. Let $P$ and $Q$ be two regular $n$-gons (for some positive integer $n\ge4$) inscribed in $\omega_P$ and $\omega_Q$, respectively, such that $A$ and $B$ are vertices of both $P$ and $Q$. Suppose a third circle $\omega$ of radius $1$ intersects $P$ at two of its vertices $C$, $D$ and intersects $Q$ at two of its vertices $E$, $F$. Further assume that $A$, $B$, $C$, $D$, $E$, $F$ are all distinct points, that $A$ lies outside of $\omega$, and that $B$ lies inside $\omega$. Show that there exists a regular $2n$-gon that contains $C$, $D$, $E$, $F$ as four of its vertices.

Can a figure have a greater than $1$ and finite number of centers of symmetry?

Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy? [i]Proposed by A. Golovanov[/i]

Let $a, b, c \geq 0$ and satisfy \[ a^2+b^2+c^2 +abc = 4 . \] Show that \[ 0 \le ab + bc + ca - abc \leq 2. \]

Let $ABC$ be an acute scalene triangle with $O$ as its circumcenter. Point $P$ lies inside triangle $ABC$ with $\angle PAB = \angle PBC$ and $\angle PAC = \angle PCB$. Point $Q$ lies on line $BC$ with $QA = QP$. Prove that $\angle AQP = 2\angle OQB$.

Let $ABC$ be a triangle inscribed in a circle of center $O$ and radius $R$. Let $I$ be the incenter of $ABC$, and let $r$ be the inradius of the same triangle, $O\neq I$, and let $G$ be its centroid. Prove that $IG\perp BC$ if and only if $b=c$ or $b+c=3a$.

A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$. The diagonals $AC$, $BD$ of $ABCD$ meet at $P$. Circumcircles of $\triangle ABP$ and $\triangle CDP$ meet at $P$ and $Q$ ($O,P,Q$ are pairwise distinct). Show that $\angle OQP=90^{\circ}$.

Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy? [i]Proposed by A. Golovanov[/i]

Let $ABC$ be an acute triangle and $A_1, B_1$ and $C_1$, points on the sides $BC, CA$ and $AB$, respectively, such that $CB_1 = A_1B_1$ and $BC_1 = A_1C_1$. Let $D$ be the symmetric of $A_1$ with respect to $B_1C_1, O$ and $O_1$ are the circumcenters of triangles $ABC$ and $A_1B_1C_1$, respectively. If $A \ne D, O \ne O_1$ and $AD$ is perpendicular to $OO_1$, prove that $AB = AC$.

Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$, and then written in nondecreasing order as $x_1\le x_2\le\cdots\le x_{2012}$. If the probability that $x_{i+1} - x_i \le \frac{1}{2011}$ for $i=1,2,\ldots,2011$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$, find the remainder when $m+n$ is divided by 1000. [i]Victor Wang.[/i]

Let $ABCD$ be a convex quadrilateral with $A,B,C,D$ concyclic. Assume $\angle ADC$ is acute and $\frac{AB}{BC}=\frac{DA}{CD}$. Let $\Gamma$ be a circle through $A$ and $D$, tangent to $AB$, and let $E$ be a point on $\Gamma$ and inside $ABCD$. Prove that $AE\perp EC$ if and only if $\frac{AE}{AB}-\frac{ED}{AD}=1$.

Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

Erick stands in the square in the 2nd row and 2nd column of a 5 by 5 chessboard. There are \$1 bills in the top left and bottom right squares, and there are \$5 bills in the top right and bottom left squares, as shown below. \[\begin{tabular}{|p{1em}|p{1em}|p{1em}|p{1em}|p{1em}|} \hline \$1 & & & & \$5 \\ \hline & E & & &\\ \hline & & & &\\ \hline & & & &\\ \hline \$5 & & & & \$1 \\ \hline \end{tabular}\] Every second, Erick randomly chooses a square adjacent to the one he currently stands in (that is, a square sharing an edge with the one he currently stands in) and moves to that square. When Erick reaches a square with money on it, he takes it and quits. The expected value of Erick's winnings in dollars is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Consider a cube $ABCDA'B'C'D'$ (where $ABCD$ is a square and $AA' \parallel BB' \parallel CC' \parallel DD'$) and a point $P$ on the line $AA'$. Construct center $S$ of a sphere which has plane $ABB'$ as a plane of symmetry, $P$ lies on the sphere and $p = AB$, $q = A'D'$ are its tangent lines. Discuss conditions of solvability with respect to different position of the point $P$ (on line $AA'$).

A $4\times4$ grid has its 16 cells colored arbitrarily in three colors. A [i]swap[/i] is an exchange between the colors of two cells. Prove or disprove that it always takes at most three swaps to produce a line of symmetry, regardless of the grid's initial coloring. [i]Proposed by Matthew Babbitt[/i]

Let $ABC$ be a isosceles triangle, with $AB=AC$. A line $r$ that pass through the incenter $I$ of $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. Let $F$ and $G$ be points on $BC$ such that $BF=CE$ and $CG=BD$. Show that the angle $\angle FIG$ is constant when we vary the line $r$.

Let $ABCD$ be a rectangle. Determine the set of all points $P$ from the region between the parallel lines $AB$ and $CD$ such that $\angle APB=\angle CPD$.

Starting with two points A and B, some circles and points are constructed as shown in the figure:[list][*]the circle with centre A through B [*]the circle with centre B through A [*]the circle with centre C through A [*]the circle with centre D through B [*]the circle with centre E through A [*]the circle with centre F through A [*]the circle with centre G through A[/list] [i][size=75](I think the wording is not very rigorous, you should assume intersections from the drawing)[/size][/i] Show that $M$ is the midpoint of $AB$. [img]https://cdn.artofproblemsolving.com/attachments/d/4/2352ab21cc19549f0381e88ddde9dce4299c2e.png[/img]

Let $ABCD$ be a convex quadrilateral with $A,B,C,D$ concyclic. Assume $\angle ADC$ is acute and $\frac{AB}{BC}=\frac{DA}{CD}$. Let $\Gamma$ be a circle through $A$ and $D$, tangent to $AB$, and let $E$ be a point on $\Gamma$ and inside $ABCD$. Prove that $AE\perp EC$ if and only if $\frac{AE}{AB}-\frac{ED}{AD}=1$.