Triangle $\vartriangle BMT$ has $BM = 4$, $BT = 6$, and $MT = 8$. Point $A$ lies on line $\overleftrightarrow{BM}$ and point $Y$ lies on line $\overleftrightarrow{BT}$ such that $\overline{AY}$ is parallel to $\overline{MT}$ and the center of the circle inscribed in triangle $\vartriangle BAY$ lies on $\overline{MT}$. Compute $AY$ .

For a given triangle $ ABC$, let $ X$ be a variable point on the line $ BC$ such that $ C$ lies between $ B$ and $ X$ and the incircles of the triangles $ ABX$ and $ ACX$ intersect at two distinct points $ P$ and $ Q.$ Prove that the line $ PQ$ passes through a point independent of $ X$.

The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at a point $X$. The circumcircles of triangles $ABX$ and $CDX$ meet at a point $Y$ (apart from $X$). Let $O$ be the center of the circumcircle of the quadrilateral $ABCD$. Assume that the points $O$, $X$, $Y$ are all distinct. Show that $OY$ is perpendicular to $XY$.

Let $ABCD$ be a convex quadrilateral such that $\vartriangle ABC$ is an equilateral triangle. Let $P$ be a point inside the quadrilateral such that $\vartriangle APD$ is an equilateral triangle and $\angle PCD = 30^o$. Suppose $CP = 6$ and $CD = 8$. The area of the triangle formed by $P$, the midpoint of $\overline{BC}$, and the midpoint of $\overline{AB}$ can be expressed in simplest radical form as $\frac{a+b\sqrt{c}}{d}$ , where $a$, $b$, $c$, and $d$ are positive integers with $gcd(a, b, d) = 1$ and with $c$ not divisible by the square of any prime. Compute $a + b + c + d$.

A regular $n$-gon is inscribed in a circle of radius $1$. Let $a_1,\cdots,a_{n-1}$ be the distances of one of the vertices of the polygon to all the other vertices. Prove that \[(5-a_1^2)\cdots(5-a_{n-1}^2)=F_n^2\] where $F_n$ is the $n^{th}$ term of the Fibonacci sequence $1,1,2,\cdots$

In triangle $ABC$ with incenter $I$ and circumcircle $\omega$, the tangent through $C$ to $\omega$ intersects $AB$ at point $D$. The angle bisector of $\angle CDB$ intersects $AI$ and $BI$ at $E$ and $F$, respectively. Let $M$ be the midpoint of $[EF]$. Prove that line $MI$ passes through the midpoint of arc $ACB$ of $w$ .

In triangle $ABC$, the angle $\angle BCA$ is obtuse and $\angle BAC = 2\angle ABC\,.$ The line through $B$ and perpendicular to $BC$ intersects line $AC$ in $D$. Let $M$ be the midpoint of $AB$. Prove that $\angle AMC=\angle BMD$. source : http://cage.ugent.be/~hvernaev/Olympiade/PMO982.pdf

Let $ABCD$ be a rectangle with $5AD <2 AB$ . On the side $AB$ consider the points $S$ and $T$ such that $AS = ST = TB$. Let $M, N$ and $P$ be the projections of points $A, S$ and $T$ on lines $DS, DT$ and $DB$ respectively .Prove that the points $M, N$, and $P$ are collinear if and only if $15 AD^2 = 2 AB^2$.

Let $ABCD$ be quadrilateral inscribed in a circle. Let $M$ be the midpoint of the segment $BD$. If the tangents of the circle at $ B$, and at $D$ are also concurrent with the extension of $AC$, prove that $\angle AMD = \angle CMD$.

The straight line containing the centers of the circumscribed and inscribed circles of triangle $ABC$ intersects rays $BA$ and $BC$ and forms an angle with the altitude to side $BC$ equal to half the angle $\angle BAC$. What is angle $\angle ABC$?

In convex quadrilateral $ABCD, AB \perp AD, AD = DC$. Let $ E$ be a point on side $BC$, and $F$ be a point on the extension of $DE$ such that $\angle ABF = \angle DEC>90^{\circ}$. Let $O$ be the circumcenter of $\triangle CDE$, and $P$ be a point on the side extension of $FO$ satisfying $FB =FP$. Line BP intersects AC at point Q. Prove that $\angle AQB =\angle DPF.$

Let $DEF$ be a triangle and H the foot of the altitude from $D$ to $EF$. If $DE = 60$, $DF = 35$, and $DH = 21$, what is the difference between the minimum and the maximum possible values for the area of $DEF$?

Let $D$ be the interior of the circle $C$ and let $A \in C$. Show that the function $f : D \to \mathbb R, f(M)=\frac{|MA|}{|MM'|}$ where $M' = AM \cap C$, is strictly convex; i.e., $f(P) <\frac{f(M_1)+f(M_2)}{2}, \forall M_1,M_2 \in D, M_1 \neq M_2$ where $P$ is the midpoint of the segment $M_1M_2.$

Let $f:[0,1]\rightarrow \mathbb{R}$ a differentiable function such that $f(0)=f(1)=0$ and $|f'(x)|\le 1,\ \forall x\in [0,1]$. Prove that: \[\left|\int_0 ^1f(t)dt\right|<\frac{1}{4}\]

Let $P$ and $Q$ be points on line $\ell$ with $PQ = 12$. Two circles, $\omega$ and ­$\Omega$, are both tangent to $\ell$ at $P$ and are externally tangent to each other. A line through $Q$ intersects $\omega$ at $A$ and $B$, with $A$ closer to $Q$ than $B$, such that $AB = 10$. Similarly, another line through $Q$ intersects ­ ­$\Omega$ at $C$ and $D$, with $C$ closer to $Q$ than $D$, such that $CD = 7$. Find the ratio $AD/BC$.

In a parallelogram $ABCD$ with $\angle A < 90$, the circle with diameter $AC$ intersects the lines $CB$ and $CD$ again at $E$ and $F$ , and the tangent to this circle at $A$ meets the line $BD$ at $P$ . Prove that the points $P$, $E$, $F$ are collinear.

Let $ABC$ be a triangle and $D,E$ be points on $BC,CA$ such that $AD,BE$ are angle bisectors of $\triangle ABC$. Let $F,G$ be points on the circumcircle of $\triangle ABC$ such that $AF||DE$ and $FG||BC$. Prove that $\frac{AG}{BG}= \frac{AB+AC}{AB+BC}$.

Four semicircles of radius $1$ are placed in a square, as shown below. The diameters of these semicircles lie on the sides of the square and each semicircle touches a vertex of the square. Find the absolute difference between the shaded area and the "hatched" area. [asy] import patterns; add("hatch",hatch(1.2mm)); add("checker",checker(2mm)); real r = 1 + sqrt(3); filldraw((0,0)--(r,0)--(r,r)--(0,r)--cycle,gray(0.4),linewidth(1.5)); fill((1,0)--(r,1)--(r-1,r)--(0,r-1)--cycle,white); fill((1,0)--(r,1)--(r-1,r)--(0,r-1)--cycle,pattern("hatch")); filldraw(arc((1,0),1,0,180)--(0,0)--cycle,white,linewidth(1.5)); filldraw(arc((r,1),1,90,270)--(r,0)--cycle,white,linewidth(1.5)); filldraw(arc((r-1,r),1,180,360)--(r,r)--cycle,white,linewidth(1.5)); filldraw(arc((0,r-1),1,270,450)--(0,r)--cycle,white,linewidth(1.5)); [/asy] [i]Proposed by Connor Gordon[/i]

Let $AH_A, BH_B, CH_C$ be the altitudes of triangle $ABC$. Prove that if $\frac{H_BC}{AC} = \frac{H_CA}{AB}$, then the line symmetric to $BC$ with respect to line $H_BH_C$ is tangent to the circumscribed circle of triangle $H_BH_CA$. [i](Proposed by Mykhailo Bondarenko)[/i]

Let $ S$ be a set of $ 100$ points in the plane. The distance between every pair of points in $ S$ is different, with the largest distance being $ 30$. Let $ A$ be one of the points in $ S$, let $ B$ be the point in $ S$ farthest from $ A$, and let $ C$ be the point in $ S$ farthest from $ B$. Let $ d$ be the distance between $ B$ and $ C$ rounded to the nearest integer. What is the smallest possible value of $ d$?

Circle $\odot O$ with diameter $\overline{AB}$ has chord $\overline{CD}$ drawn such that $\overline{AB}$ is perpendicular to $\overline{CD}$ at $P$. Another circle $\odot A$ is drawn, sharing chord $\overline{CD}$. A point $Q$ on minor arc $\overline{CD}$ of $\odot A$ is chosen so that $\text{m} \angle AQP + \text{m} \angle QPB = 60^\circ$. Line $l$ is tangent to $\odot A$ through $Q$ and a point $X$ on $l$ is chosen such that $PX=BX$. If $PQ = 13$ and $BQ = 35$, find $QX$. [i]Proposed by Aaron Lin[/i]

The internal bisectors of angles $\angle DAB$ and $\angle BCD$ of a quadrilateral $ABCD$ intersect at the point $X_1$, and the external bisectors of these angles intersect at the point $X_2$. The internal bisectors of angles $\angle ABC$ and $\angle CDA$ intersect at the point $Y_1$, and the external bisectors of these angles intersect at the point $Y_2$. Prove that the angle between the lines $X_1X_2$ and $Y_1Y_2$ equals the angle between the diagonals $AC$ and $BD$. [i](A. Voidelevich)[/i]

let $ABC$ be a right angled triangle with inradius $1$ find the minimum area of triangle $ABC$

Suppose there are $4n$ line segments of unit length inside a circle of radius $n$. Furthermore, a straight line $L$ is given. Prove that there exists a straight line $L'$ that is either parallel or perpendicular to $L$ and that $L'$ cuts at least two of the given line segments.