Let $ABCD$ be a convex quadrilateral with $AB=CD$. Let $R,S$ be midpoints of sides $AD,BC$ respectively. Consider rays $AU, DV$ parallel with ray $RS$ and all of them point in the same direction. Show that $\angle BAU=\angle CDV$.

Let $ABCD$ be a square. $E$ and $F$ lie on sides $AB$ and $BC$, respectively, such that $BE = BF$. The line perpendicular to $CE$, which passes through $B$, intersects $CE$ and $AD$ at points $G$ and $H$, respectively. The lines $FH$ and $CE$ intersect at point $P$ and the lines $GF$ and $CD$ intersect at point $Q$. Prove that the line $DP$ is perpendicular to the line $BQ$.

Let points $A$, $B$, $C$ lie on a circle, and line $b$ be the tangent to the circle at point $B$. Perpendiculars $PA_1$ and $PC_1$ are dropped from a point $P$ on line $b$ onto lines $AB$ and $BC$ respectively. Points $A_1$ and $C_1$ lie inside line segments $AB$ and $BC$ respectively. Prove that $A_1C_1$ is perpendicular to $AC$.

In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.

In triangle $ABC,$ we have $\angle A \leq 90^\circ$ and $\angle B = 2 \angle C.$ The interior bisector of the angle $C$ meets the median $AM$ in $D.$ Prove that $\angle MDC \leq 45^\circ.$ When does equality hold? [asy] import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqqqzz = rgb(0,0,0.6); pen ffqqtt = rgb(1,0,0.2); pen qqwuqq = rgb(0,0.39,0); draw((3.14,5.81)--(3.23,0.74),ffqqtt+linewidth(2pt)); draw((3.23,0.74)--(9.73,1.32),ffqqtt+linewidth(2pt)); draw((9.73,1.32)--(3.14,5.81),ffqqtt+linewidth(2pt)); draw((9.73,1.32)--(3.19,2.88)); draw((3.14,5.81)--(6.71,1.05)); pair parametricplot0_cus(real t){ return (0.7*cos(t)+5.8,0.7*sin(t)+2.26); } draw(graph(parametricplot0_cus,-0.9270442638657642,-0.23350086562377703)--(5.8,2.26)--cycle,qqwuqq); dot((3.14,5.81),ds); label("$A$", (2.93,6.09),NE*lsf); dot((3.23,0.74),ds); label("$B$", (2.88,0.34),NE*lsf); dot((9.73,1.32),ds); label("$C$", (10.11,1.04),NE*lsf); dot((3.19,2.88),ds); label("$X$", (2.77,2.94),NE*lsf); dot((6.71,1.05),ds); label("$M$", (6.75,0.5),NE*lsf); dot((5.8,2.26),ds); label("$D$", (5.89,2.4),NE*lsf); label("$\alpha$", (6.52,1.65),NE*lsf); clip((-3.26,-11.86)--(-3.26,8.36)--(20.76,8.36)--(20.76,-11.86)--cycle); [/asy]

Let ABC with orthocenter $H$ and circumcenter $O$ be an acute scalene triangle satisfying $AB = AM$ where $M$ is the midpoint of $BC$. Suppose $Q$ and $K$ are points on $(ABC)$ distinct from A satisfying $\angle AQH = 90$ and $\angle BAK = \angle CAM$. Let $N$ be the midpoint of $AH$. • Let $I$ be the intersection of $B\text{-midline}$ and $A\text{-altitude}$ Prove that $IN = IO$. • Prove that there is point $P$ on the symmedian lying on circle with center $B$ and radius $BM$ such that $(APN)$ is tangent to $AB$. [i]Proposed by Krutarth Shah[/i]

Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex). Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.

Equilateral triangle $ABC$ has side length $20$. Let $PQRS$ be a square such that $A$ is the midpoint of $\overline{RS}$ and $Q$ is the midpoint of $\overline{BC}$. Compute the area of $PQRS$.

You have a $7\times 7$ board divided into $49$ boxes. Mateo places a coin in a box. a) Prove that Mateo can place the coin so that it is impossible for Emi to completely cover the $48$ remaining squares, without gaps or overlaps, using $15$ $3\times1$ rectangles and a cubit of three squares, like those in the figure. [img]https://cdn.artofproblemsolving.com/attachments/6/9/a467439094376cd95c6dfe3e2ad3e16fe9f124.png[/img] b) Prove that no matter which square Mateo places the coin in, Emi will always be able to cover the 48 remaining squares using $14$ $3\times1$ rectangles and two cubits of three squares.

Let $S_n$ be the area of the figure enclosed by a curve $y=x^2(1-x)^n\ (0\leq x\leq 1)$　and the $x$-axis. Find $\lim_{n\to\infty} \sum_{k=1}^n S_k.$

The touching point of the excircle with the side of a triangle and the base of the altitude to this side are symmetric wrt the base of the corresponding bisector. Prove that this side is equal to one third of the perimeter.

The angles $A$ and $B$ of base of the isosceles triangle $ABC$ are equal to $40^o$. Inside $\vartriangle ABC$, $P$ is such that $\angle PAB = 30^o$ and $\angle PBA = 20^o$. Calculate, without table, $\angle PCA$.

Points $K, L, M$ and $N$ lying on the sides $AB, BC, CD$ and $DA$ of a square $ABCD$ are vertices of another square. Lines $DK$ and $N M$ meet at point $E$, and lines $KC$ and $LM$ meet at point $F$ . Prove that $EF\parallel AB$.

In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? ${ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 27\qquad\textbf{(E)}\ 30 $

Given a triangle $ABC$ and points $D, E, F$, which are points of contact of the inscribed circle to the sides of the triangle. i) Prove that $\frac{2pr}{R} \le DE + EF + DF \le p$ ($p$ is the semiperimeter, $r$ and $R$ are respectively the radius of the inscribed and circumscribed circle of $\vartriangle ABC$). ii). Find out when equality is achieved.

A $ 9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $ C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $ \sqrt {13}$, ii) the closed broken line $ C_1C_2\ldots C_{96}C_1$ has a centre of symmetry? [i]Bulgaria[/i]

$d$ and $n$ are positive integers such that $d \mid n$. The n-number sets $(x_1, x_2, \cdots x_n)$ satisfy the following condition: (1) $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n$ (2) $d \mid (x_1+x_2+ \cdots x_n)$ Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy $x_n=n$.

A bounded planar region of area $S$ is covered by a finite family $F$ of closed discs. Prove that $F$ contains a subfamily consisting of pairwise disjoint discs, of joint area not less than $S/9$.

Let $CD$ be a chord of a circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$ perpendicular to $CD$ at $N$ with $AN > NB$. A circle $\Gamma_2$ centered at $C$ with radius $CN$ intersects $\Gamma_1$ at points $P$ and $Q$. The line $PQ$ intersects $CD$ at $M$ and $AC$ at $K$; and the extension of $NK$ meets $\Gamma_2$ at $L$. Prove that $PQ$ is perpendicular to $AL$

In triangle $ABC$, side $AC$ is $8$ cm. Two segments are drawn parallel to $AC$ that have their ends on $AB$ and $BC$ and that divide the triangle into three parts of equal area. What is the length of the parallel segment closest to $AC$?

Point $O$ is the intersection point of the heights of an acute triangle $\vartriangle ABC$. Prove that the three circles which pass: a) through $O, A, B$, b) through $O, B, C$, and c) through $O, C, A$, are equal

Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.

A circle of radius $2$ is inscribed in equilateral triangle $ABC$. The altitude from $A$ to $BC$ intersects the circle at a point $D$ not on $BC$. $BD$ intersects the circle at a point $E$ distinct from $D$. Find the length of $BE$.

If $AG_a,BG_b$, and $CG_c$ are symmedians in $\Delta ABC$ ($G_a\in BC,G_b\in AC,G_c\in AB$), is it possible for $\Delta G_a G_b G_c$ to be equilateral when $\Delta ABC$ is not equilateral?

A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.