Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$. (Slovakia)

Given a cyclic quadrilateral $ABCD$. Suppose $E, F, G, H$ are respectively the midpoint of the sides $AB, BC, CD, DA$. The line passing through $G$ and perpendicular on $AB$ intersects the line passing through $H$ and perpendicular on $BC$ at point $K$. Prove that $\angle EKF = \angle ABC$.

Let $ABC$ an acute triangle and $H$ its orthocenter. Let $E$ and $F$ be the intersection of lines $BH$ and $CH$ with $AC$ and $AB$ respectively, and let $D$ be the intersection of lines $EF$ and $BC$. Let $\Gamma_1$ be the circumcircle of $AEF$, and $\Gamma_2$ the circumcircle of $BHC$. The line $AD$ intersects $\Gamma_1$ at point $I \neq A$. Let $J$ be the feet of the internal bisector of $\angle{BHC}$ and $M$ the midpoint of the arc $\stackrel{\frown}{BC}$ from $\Gamma_2$ that contains the point $H$. The line $MJ$ intersects $\Gamma_2$ at point $N \neq M$. Show that the triangles $EIF$ and $CNB$ are similar.

Points $D$ and $E$ lie on the lines $BC$ and $AC$ respectively so that $B$ is between $C$ and $D$, $C$ is between $A$ and $E$, $BC = BD$ and $\angle BAD = \angle CDE$. It is known that the ratio of the perimeters of the triangles $ABC$ and $ADE$ is $2$. Find the ratio of the areas of these triangles.

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]

What fraction of the Earth's volume lies above the $45$ degrees north parallel? You may assume the Earth is a perfect sphere. The volume in question is the smaller piece that we would get if the sphere were sliced into two pieces by a plane.

A triangle $ ABC$ is given. Let $ B_1$ be the reflection of $ B$ across the line $ AC$, $ C_1$ the reflection of $ C$ across the line $ AB$, and $ O_1$ the reflection of the circumcentre of $ ABC$ across the line $ BC$. Prove that the circumcentre of $ AB_1C_1$ lies on the line $ AO_1$. [i]Proposed by A. Akopyan[/i]

Given an acute-angled $\vartriangle AB$C with altitude $AH$ ( $\angle BAC > 45^o > \angle AB$C). The perpendicular bisector of $AB$ intersects $BC$ at point $D$. Let $K$ be the midpoint of $BF$, where $F$ is the foot of the perpendicular from $C$ on $AD$. Point $H'$ is the symmetric to $H$ wrt $K$. Point $P$ lies on the line $AD$, such that $H'P \perp AB$. Prove that $AK = KP$.

The triangle $ABC$ satisfies $AB=10$ and has angles $\angle{A}=75^{\circ}$, $\angle{B}=60^{\circ}$, and $\angle C = 45^{\circ}$. Let $I_A$ be the center of the excircle opposite $A$, and let $D$, $E$ be the circumcenters of triangle $BCI_A$ and $ACI_A$ respectively. If $O$ is the circumcenter of triangle $ABC$, then the area of triangle $EOD$ can be written as $\tfrac{a\sqrt{b}}{c}$ for square-free $b$ and coprime $a,c$. Find the value of $a+b+c$.

Given a fixed circle $C$ and a line L through the center $O$ of $C$. Take a variable point $P$ on $L$ and let $K$ be the circle with center $P$ through $O$. Let $T$ be the point where a common tangent to $C$ and $K$ meets $K$. What is the locus of $T$?

Diameter $AB$ divides a circle into two semicircles. Points $P_1$ , $P_2$, $...$, $P_n$ are given on one of the semicircles in this order. How should a point C be chosen on the other semicircle in order to maximize the sum of the areas of triangles $CP_1P_2$, $CP_2P_3$, $...$,$CP_{n-1}P_n$?

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $\ell$ be a line passing through two sides of triangle $ABC$. Line $\ell$ cuts triangle $ABC$ into two figures, a triangle and a quadrilateral, that have equal perimeter. What is the maximum possible area of the triangle?

Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Let $D$ be a point on the line $AC$ such that $AB = AD$ and $A$ lies between $C$ and $D$. Suppose that there are two points $E \ne F$ on the circumcircle of the triangle $DBC$ such that $AE = AF = BC$. Prove that the line $EF$ passes through the circumcenter of $ABC$.

Let $M$ be the midpoint of the side $AB$ of a triangle $ABC$. A circle through point $C$ that has a point of tangency to the line $AB$ at point $A$ and a circle through point $C$ that has a point of tangency to the line $AB$ at point $B$ intersect the second time at point $N$. Prove that $|CM|^2 + |CN|^2 - |MN|^2 = |CA|^2 + |CB|^2 - |AB|^2$.

A circle with radius $1$ is circumscribed by a rhombus. What is the minimum possible area of this rhombus?

Restore a triangle $ABC$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.

[b]p1.[/b] Find all prime factors of $8051$. [b]p2.[/b] Simplify $$[\log_{xyz}(x^z)][1 + \log_x y + \log_x z],$$ where $x = 628$, $y = 233$, $z = 340$. [b]p3.[/b] In prokaryotes, translation of mRNA messages into proteins is most often initiated at start codons on the mRNA having the sequence AUG. Assume that the mRNA is single-stranded and consists of a sequence of bases, each described by a single letter A,C,U, or G. Consider the set of all pieces of bacterial mRNA six bases in length. How many such mRNA sequences have either no A’s or no U’s? [b]p4.[/b] What is the smallest positive $n$ so that $17^n + n$ is divisible by $29$? [b]p5.[/b] The legs of the right triangle shown below have length $a = 255$ and $b = 32$. Find the area of the smaller rectangle (the one labeled $R$). [img]https://cdn.artofproblemsolving.com/attachments/c/d/566f2ce631187684622dfb43f36c7e759e2f34.png[/img] [b]p6.[/b] A $3$ dimensional cube contains ”cubes” of smaller dimensions, ie: faces ($2$-cubes),edges ($1$-cubes), and vertices ($0$-cubes). How many 3-cubes are in a $5$-cube? PS. You had better use hide for answers.

Given a non-right triangle $ABC$ with $BC>AC>AB$. Two points $P_1 \neq P_2$ on the plane satisfy that, for $i=1,2$, if $AP_i, BP_i$ and $CP_i$ intersect the circumcircle of the triangle $ABC$ at $D_i, E_i$, and $F_i$, respectively, then $D_iE_i \perp D_iF_i$ and $D_iE_i = D_iF_i \neq 0$. Let the line $P_1P_2$ intersects the circumcircle of $ABC$ at $Q_1$ and $Q_2$. The Simson lines of $Q_1$, $Q_2$ with respect to $ABC$ intersect at $W$. Prove that $W$ lies on the nine-point circle of $ABC$.

In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear circles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\omega_C$ have a common point $X$ other than $I$, and that $\angle AXO = \angle OXA'$. [i]Proposed by Sammy Luo[/i]

In $\Delta ABC$ $(AB>BC)$ $BM$ and $BL$ $(M,L\in AC)$ are a median and an angle bisector respectively. Let the line through $M$, parallel to $AB$, intersect $BL$ in point $D$ and the line through $L$, parallel to $BC$, intersect $BM$ in point $E$. Prove that $DE\perp BL$.

Regular hexagon $ABCDEF$ is inscribed in rectangle $PQRS$ with $AB = 1$, A and $B$ on side $PQ$, $C$ on side $QR$, $D$ and $E$ on side $RS$, and $F$ on side $SP$. What is the area of $PQRS$?

Let $\displaystyle ABC$ and $\displaystyle DBC$ be isosceles triangle with the base $\displaystyle BC$. We know that $\displaystyle \measuredangle ABD = \frac{\pi}{2}$. Let $\displaystyle M$ be the midpoint of $\displaystyle BC$. The points $\displaystyle E,F,P$ are chosen such that $\displaystyle E \in (AB)$, $\displaystyle P \in (MC)$, $\displaystyle C \in (AF)$, and $\displaystyle \measuredangle BDE = \measuredangle ADP = \measuredangle CDF$. Prove that $\displaystyle P$ is the midpoint of $\displaystyle EF$ and $\displaystyle DP \perp EF$.

Let us denote successively $r$ and $r_a$ the radii of the inscribed circle and the exscribed circle wrt to side BC of triangle $ABC$. Prove that if it is true that $r+r_a=|BC|$ , then the triangle $ABC$ is a right one

Two circles $\omega_1$ and $\omega_2$ are said to be $\textit{orthogonal}$ if they intersect each other at right angles. In other words, for any point $P$ lying on both $\omega_1$ and $\omega_2$, if $\ell_1$ is the line tangent to $\omega_1$ at $P$ and $\ell_2$ is the line tangent to $\omega_2$ at $P$, then $\ell_1\perp \ell_2$. (Two circles which do not intersect are not orthogonal.) Let $\triangle ABC$ be a triangle with area $20$. Orthogonal circles $\omega_B$ and $\omega_C$ are drawn with $\omega_B$ centered at $B$ and $\omega_C$ centered at $C$. Points $T_B$ and $T_C$ are placed on $\omega_B$ and $\omega_C$ respectively such that $AT_B$ is tangent to $\omega_B$ and $AT_C$ is tangent to $\omega_C$. If $AT_B = 7$ and $AT_C = 11$, what is $\tan\angle BAC$?

We will say that a plane is [i]well-colored[/i] in several colors if it is divided into convex polygons with an area of at least $1/1000$ and each polygon is colored in one color. Points lying on the border of several polygons can be colored in any of their colors. Are there convex is a $9$-gon $R$ and a good coloring of the plane in $7$ colors such that in any polygon obtained from $R$ by a translate to any vector, all colors occupy the same area ($1/7$ of the area of $R$)?