Let $\vartriangle ABC$ be an acute triangle with $AB > AC$. Let $P$ be the foot of the altitude from $C$ to $AB$ and let $Q$ be the foot of the altitude from $B$ to $AC$. Let $X$ be the intersection of $PQ$ and $BC$. Let the intersection of the circumcircles of triangle $\vartriangle AXC$ and triangle $\vartriangle PQC$ be distinct points: $C$ and $Y$ . Prove that $PY$ bisects $AX$.

In a triangle $ABC, X$ and $Y$ are points on the segments $AB$ and $AC$, respectively, such that $AX : XB = 1 : 2$ and $AY :YC = 2:1$. If the area of triangle $AXY$ is $10$, then what is the area of triangle $ABC$?

If in a quadrilateral $ABCD$ whose vertices lie on a circle of radius $1$, holds $$|AB| \cdot |BC| \cdot |CD|\cdot |DA| \ge 4$$, then $ABCD$ is a square. Prove it. [hide=Hint given in contest] You can use Ptolemy's formula $|AB| \cdot |CD| + |BC|\cdot |AD|= |AC| \cdot|BD|$[/hide]

Given is a right triangle $ABC$ with perimeter $2$, with $\angle B=90^o$ . Point $S$ is the center of the excircle to the side $AB$ of the triangle and $H$ is the intersection of the heights of the triangle $ABS$ . Determine the smallest possible length of the segment $HS $.

Find all positive integer bases $b \ge 9$ so that the number \[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \] is a perfect cube in base 10 for all sufficiently large positive integers $n$. [i]Proposed by Yang Liu[/i]

A rectangular piece of paper has the side lengths $12$ and $15$. A corner is bent about as shown in the figure. Determine the area of the gray triangle. [img]https://1.bp.blogspot.com/-HCfqWF0p_eA/XzcIhnHS1rI/AAAAAAAAMYg/KfY14frGPXUvF-H6ZVpV4RymlhD_kMs-ACLcBGAsYHQ/s0/1993%2BMohr%2Bp2.png[/img]

Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.

Let \(ABCDE\) be a regular pentagon. Let \(P\) be a variable point on the interior of segment \(AB\) such that \(PA\ne PB\). The circumcircles of \(\triangle PAE\) and \(\triangle PBC\) meet again at \(Q\). Let \(R\) be the circumcenter of \(\triangle DPQ\). Show that as \(P\) varies, \(R\) lies on a fixed line. [i]Proposed by Karthik Vedula[/i]

Find all triangles such that its angles form an arithmetic sequence and the corresponding sides form a geometric sequence.

Let $ABC$ be an acute triangle. Points $B'$ and $C'$ are located on the interior of sides $AB$ and $AC$, respectively. Let $M$ denote the second intersection of the circumcircles of triangles $ABC$ and $AB'C'$, while let $N$ denote the second intersection of the circumcircles of triangles $ABC'$ and $AB'C$. Reflect $M$ across lines $AB$ and $AC$, and let $l$ denote the line through the reflections. a) Prove that the line through $M$ perpendicular to $AM$, the line $AK$, and $l$ are either concurrent or all parallel. b) Show that if the three lines are concurrent at $S$, then triangles $SBC'$ and $SCB'$ have equal areas. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

Equilateral triangle $ABC$ has $AD = DB = FG = AE = EC = 4$ and $BF = GC = 2$. From $D$ and $G$ are drawn perpendiculars to $EF$ intersecting at $H$ and $I$, respectively. The three polygons $ECGI$, $FGI$, and $BFHD$ are rearranged to $EANL$, $MNK$, and $AMJD$ so that the rectangle $HLKJ$ is formed. Find its area. [img]https://cdn.artofproblemsolving.com/attachments/d/4/7e6667f0f0544b6fbc860f8d86c8ceaaf85cc1.png[/img]

5. Let $\triangle$ABC be a triangle with circumcentre [i]O[/i]. The perpendicular line from [i]O[/i] to [i]BC[/i] intersects line [i]BC[/i] at [i]M[/i] and line [i]AC[/i] at [i]P[/i], and the perpendicular line from [i]O[/i] to [i]AC[/i] intersects line [i]AC[/i] at [i]N[/i] and line [i]BC[/i] at [i]Q[/i]. Let [i]D[/i] be the intersection point of lines [i]PQ[/i] and [i]MN[/i]. construct the parallelogram [i]PCQJ[/i] with [i]PJ[/i] || [i]CQ[/i] and [i]QJ[/i] || [i]CP[/i]. Prove the following: a) The points [i]A[/i], [i]B[/i], [i]O[/i], [i]P[/i], [i]Q[/i], [i]J[/i] are all on the same circle. b) line [i]OD[/i] is perpendicular to line [i]CJ[/i].

Show that there exists a triangle $ABC$ such that $a \ne b$ and $a+t_a = b+t_b$, where $t_a,t_b$ are the medians corresponding to $a,b$, respectively. Also prove that there exists a number $k$ such that every such triangle satisfies $a+t_a = b+t_b = k(a+b)$. Finally, find all possible ratios $a : b$ in such triangles.

Let $P$ be a point inside a triangle $ABC$ such that $$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$ Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.

Let $\omega_a, \omega_b, \omega_c$ be the exscribed circles tangent to the sides $a, b, c$ of a triangle $ABC$, respectively, $ I_a, I_b, I_c$ be the centers of these circles, respectively, $T_a, T_b, T_c$ be the points of contact of these circles to the line $BC$, respectively. The lines $T_bI_c$ and $T_cI_b$ intersect at the point $Q$. Prove that the center of the circle inscribed in triangle $ABC$ lies on the line $T_aQ$.

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$

Given a circle $C$ and a point $P$ on its exterior, two tangents to the circle are drawn through $P$, with $A$ and $B$ being the points of tangency. We take a point $Q$ on the minor arc $AB$ of $C$. Let $M$ be the intersection of $AQ$ with the line perpendicular to $AQ$ that goes through $P$, and let $N$ be the intersection of $BQ$ with the line perpendicular to $BQ$ that goes through $P$. Show that, by varying $Q$ on the minor arc $AB$, all of the lines $MN$ pass through the same point.

Given an triangle $ABC$ isosceles at the vertex $A$, let $P$ and $Q$ be the touchpoints with $AB$ and $AC$, respectively with the circle $T$, which is tangent to both and is internally tangent to the circumcircle of $ABC$. Let $R$ and $S$ be the points of the circumscribed circle of $ABC$ such that $AP = AR = AS$ . Prove that $RS$ is tangent to $T$ .

Let $\Gamma$ be the circle of an acute triangle $ABC$ and let $H$ be its orthocenter. The circle $\omega$ with diameter $AH$ cuts $\Gamma$ at point $D$ ($D\ne A$). Let $M$ be the midpoint of the smaller arc $BC$ of $\Gamma$ . Let $N$ be the midpoint of the largest arc $BC$ of the circumcircle of the triangle $BHC$. Prove that there is a circle that passes through the points $D, H, M$ and $N$.

A chord $AB$ is drawn in a circle. On its extensions beyond points $A$ and $B$, points $P$ and $Q$ respectively are taken such that $AP = BQ$. Through $P$ and $Q$ two tangents to the circle are drawn, intersecting at point $M$. Find the locus of points $M$ ($P$ and $Q$ move along a straight line and for any $P$ and $Q$ all possible pairs of tangents are taken, which determine four points from the desired locus of points) .

Let $ABCD$ be a convex quadrilateral. Two squares are constructed such that $AB{}$ and $CD{}$ are their diagonals. Show that if these squares have a common vertex inside $ABCD$, then the squares that have $BC{}$ and $AD{}$ as diagonals also have a common vertex inside $ABCD$.

In the triangle $ABC$ we have $\angle A = 2\angle C$ and $2\angle B = \angle A + \angle C$. The angle bisector of $\angle C$ intersects the segment $AB$ in $E$, let $F$ be the midpoint of $AE$, let $AD$ be the altitude of the triangle $ABC$. The perpendicular bisector of $DF$ intersects $AC$ in $M$. Prove that $AM = CM$.

Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

Prove that if all four vertices of a parallelogram are lattice points and there are some other lattice points in or on the parallelogram, then its area exceeds $1$.

A tortoise walks $60$ meters per hour and a lizard walks at $240$ meters per hour. There is a rectangle $ABCD$ where $AB =60$ and $AD =120$. Both start from the vertex $A$ and in the same direction ($A \to B \to D \to A$), crossing the edge of the rectangle. The lizard has the habit of advancing two consecutive sides of the rectangle, turning to go back one, turning to go forward two, turning to go back one and so on. How many times and in what places do the tortoise and the lizard meet when the tortoise completes its third turn?