Let $ABC$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $BC$ such that $\angle{CAX} = \angle{YAB}$. Suppose that: $1)$ $K$ and $S$ are the feet of the perpendiculars from from $B$ to the lines $AX$ and $AY$ respectively. $2)$ $T$ and $L$ are the feet of the perpendiculars from $C$ to the lines $AX$ and $AY$ respectively. Prove that $KL$ and $ST$ intersect on the line $BC$.

. In the isosceles triangle $ABC$ with $AC = BC$ we denote by $D$ the foot of the altitude through $C$. The midpoint of $CD$ is denoted by $M$. The line $BM$ intersects $AC$ in $E$. Prove that the length of $AC$ is three times that of $CE$.

The figure shows an arbitrary (green) triangle in the center. White squares were built on its sides to the outside. Some of their vertices were connected by segments, white squares were built on them again to the outside, and so on. In the spaces between the squares, triangles and quadrilaterals were formed, which were painted in different colors. Prove that [list=a] [*]all colored quadrilaterals are trapezoids; [*]the areas of all polygons of the same color are equal; [*]the ratios of the bases of one-color trapezoids are equal; [*]if $S_0=1$ is the area of the original triangle, and $S_i$ is the area of the colored polygons at the $i^{\text{th}}$ step, then $S_1=1$, $S_2=5$ and for $n\geqslant 3$ the equality $S_n=5S_{n-1}-S_{n-2}$ is satisfied. [/list] [i]Proposed by F. Nilov[/i] [center][img width="40"]https://i.ibb.co/n8gt0pV/Screenshot-2023-03-09-174624.png[/img][/center]

A circle and an ellipse with foci $F_{1}$, $F_{2}$ lying inside it are given. Construct a chord $AB$ of the circle touching the ellipse and such that $AF_{1}F_{2}B$ is a cyclic quadrilateral. Proposed by: A.Zaslavsky

In the plane we are given the circles $\Gamma$ and $\Delta$ tangent to each other and $\Gamma$ contains $\Delta$. The radius of $\Gamma$ is $R$ and of $\Delta$ is $\frac{R}{2}$. Prove that for each positive integer $n \geq 3$, the equation: \[ (p(1) - p(n))^2 = (n-1)^2 \cdot (2 \cdot (p(1) + p(n)) - (n-1)^2 - 8) \] is the necessary and sufficient condition for $n$ to exist $n$ distinct circles $\Upsilon_1, \Upsilon_2, \ldots, \Upsilon_n$ such that all these circles are tangent to $\Gamma$ and $\Delta$ and $\Upsilon_i$ is tangent to $\Upsilon_{i+1}$, and $\Upsilon_1$ has radius $\frac{R}{p(1)}$ and $\Upsilon_n$ has radius $\frac{R}{p(n)}$.

Given an isosceles triangle $ABC$ ($AB = AC$), the inscribed circle $\omega$ touches its sides $AB$ and $AC$ at points $K$ and $L$, respectively. On the extension of the side of the base $BC$, towards $B$, an arbitrary point $M$. is chosen. Line $M$ intersects $\omega$ at the point $N$ for the second time, line $BN$ intersects the second point $\omega$ at the point $P$. On the line $PK$, there is a point $X$ such that $K$ lies between $P$ and $X$ and $KX = KM$. Determine the locus of the point $X$.

Perimeter of triangle $ABC$ is $1$. Circle $\omega$ touches side $BC$, continuation of side $AB$ at $P$ and continuation of side $AC$ in $Q$. Line through midpoints $AB$ and $AC$ intersects circumcircle of $APQ$ at $X$ and $Y$. Find length of $XY$.

Faith has four different integer length segments. It turned out that any three of them can form a triangle. What is the smallest total length of this set of segments?

The inscribed circle $\omega$ of an equilateral $\Delta ABC$ is tangent to its sides $AB$,$BC$ and $CA$ in points $D$,$E$, and $F$, respectively. Point $H$ is the foot of the altitude from $D$ to $EF$. Let $AH\cap BC=X,BH\cap CA=Y$. It is known that $XY\cap AB=T$. Let $D$ be the center of the circumscribed circle of $\Delta BYX$. Prove that $OH\perp CT$.

Points $P$ and $Q$ lie respectively on sides $AB$ and $AC$ of a triangle $ABC$ and $BP=CQ$. Segments $BQ$ and $CP$ cross at $R$. Circumscribed circles of triangles $BPR$ and $CQR$ cross again at point $S$ different from $R$. Prove that point $S$ lies on the bisector of angle $BAC$.

A cubical cake with edge length $ 2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $ M$ is the midpoint of a top edge. The piece whose top is triangle $ B$ contains $ c$ cubic inches of cake and $ s$ square inches of icing. What is $ c\plus{}s$? [asy]unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle); draw((1,1)--(-1,0)); pair P=foot((1,-1),(1,1),(-1,0)); draw((1,-1)--P); draw(rightanglemark((-1,0),P,(1,-1),4)); label("$M$",(-1,0),W); label("$C$",(-0.1,-0.3)); label("$A$",(-0.4,0.7)); label("$B$",(0.7,0.4));[/asy]$ \textbf{(A)}\ \frac{24}{5} \qquad \textbf{(B)}\ \frac{32}{5} \qquad \textbf{(C)}\ 8\plus{}\sqrt5 \qquad \textbf{(D)}\ 5\plus{}\frac{16\sqrt5}{5} \qquad \textbf{(E)}\ 10\plus{}5\sqrt5$

Let $ABCD$ be an isosceles trapezoid with $AB=AD=BC, AB//CD, AB>CD$. Let $E= AC \cap BD$ and $N$ symmetric to $B$ wrt $AC$. Prove that the quadrilateral $ANDE$ is cyclic.

Let $\triangle ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. Let $D$ be a point on the circumcircle of $ABC$ such that $AD \perp BC$. Suppose that $AB = 6, DB = 2$, and the ratio $\tfrac{\text{area}(\triangle ABC)}{\text{area}(\triangle HBC)}=5.$ Then, if $OA$ is the length of the circumradius, then $OA^2$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Problems 7, 8 and 9 are about these kites. To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. The kites look like the one in the diagram. For her small kite Genevieve draws the kite on a one-inch grid. For the large kite she triples both the height and width of the entire grid. [asy] for (int a = 0; a < 7; ++a) { for (int b = 0; b < 8; ++b) { dot((a,b)); } } draw((3,0)--(0,5)--(3,7)--(6,5)--cycle);[/asy] What is the number of square inches in the area of the small kite? $ \text{(A)}\ 21\qquad\text{(B)}\ 22\qquad\text{(C)}\ 23\qquad\text{(D)}\ 24\qquad\text{(E)}\ 25 $

A right triangle of the following condition is given: the three side lengths are all positive integers and the length of the shortest segment is $141$. For the triangle that has the minimum area while satisfying the condition, find the lengths of the other two sides.

On a plane, Bob chooses 3 points $A_0$, $B_0$, $C_0$ (not necessarily distinct) such that $A_0B_0+B_0C_0+C_0A_0=1$. Then he chooses points $A_1$, $B_1$, $C_1$ (not necessarily distinct) in such a way that $A_1B_1=A_0B_0$ and $B_1C_1=B_0C_0$. Next he chooses points $A_2$, $B_2$, $C_2$ as a permutation of points $A_1$, $B_1$, $C_1$. Finally, Bob chooses points $A_3$, $B_3$, $C_3$ (not necessarily distinct) in such a way that $A_3B_3=A_2B_2$ and $B_3C_3=B_2C_2$. What are the smallest and the greatest possible values of $A_3B_3+B_3C_3+C_3A_3$ Bob can obtain?

In the triangle $ABC$, $\angle ABC = \angle ACB = 78^o$. On the sides $AB$ and $AC$, respectively, the points $D$ and $E$ are chosen such that $\angle BCD = 24^o$, $\angle CBE = 51^o$. Find the measure of angle $\angle BED$.

A rectangle $ABCD$ and a point $P$ are given. Lines passing through $A$ and $B$ and perpendicular to $PC$ and $PD$ respectively, meet at a point $Q$. Prove that $PQ \perp AB$.

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$. Circle $\omega_1$ with center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\omega_2$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\omega_1$ and $\omega_2$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.

In a triangle $ABC$, let $D$ and $E$ trisect $BC$, so $BD = DE = EC$. Let $F$ be the point on $AB$ such that $\frac{AF}{F B}= 2$, and $G$ on $AC$ such that $\frac{AG}{GC} =\frac12$ . Let $P$ be the intersection of $DG$ and $EF$, and extend $AP$ to intersect $BC$ at a point $X$. Find $\frac{BX}{XC}$

[b]p1.[/b] Consider the following function $f(x) = \left(\frac12 \right)^x - \left(\frac12 \right)^{x+1}$. Evaluate the infinite sum $f(1) + f(2) + f(3) + f(4) +...$ [b]p2.[/b] Find the area of the shape bounded by the following relations $$y \le |x| -2$$ $$y \ge |x| - 4$$ $$y \le 0$$ where |x| denotes the absolute value of $x$. [b]p3.[/b] An equilateral triangle with side length $6$ is inscribed inside a circle. What is the diameter of the largest circle that can fit in the circle but outside of the triangle? [b]p4.[/b] Given $\sin x - \tan x = \sin x \tan x$, solve for $x$ in the interval $(0, 2\pi)$, exclusive. [b]p5.[/b] How many rectangles are there in a $6$ by $6$ square grid? [b]p6.[/b] Find the lateral surface area of a cone with radius $3$ and height $4$. [b]p7.[/b] From $9$ positive integer scores on a $10$-point quiz, the mean is $ 8$, the median is $ 8$, and the mode is $7$. Determine the maximum number of perfect scores possible on this test. [b]p8.[/b] If $i =\sqrt{-1}$, evaluate the following continued fraction: $$2i +\frac{1}{2i +\frac{1}{2i+ \frac{1}{2i+...}}}$$ [b]p9.[/b] The cubic polynomial $x^3-px^2+px-6$ has roots $p, q$, and $r$. What is $(1-p)(1-q)(1-r)$? [b]p10.[/b] (Variant on a Classic.) Gilnor is a merchant from Cutlass, a town where $10\%$ of the merchants are thieves. The police utilize a lie detector that is $90\%$ accurate to see if Gilnor is one of the thieves. According to the device, Gilnor is a thief. What is the probability that he really is one? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let equilateral triangle $ABC$ with side length $6$ be inscribed in a circle and let $P$ be on arc $AC$ such that $AP \cdot P C = 10$. Find the length of $BP$.

Let $ABC$ be an acute triangle, with $\angle A > 60^\circ$, and let $H$ be it's orthocenter. Let $M$ and $N$ be points on $AB$ and $AC$, respectively, such that $\angle HMB = \angle HNC = 60^\circ$. Also, let $O$ be the circuncenter of $HMN$ and $D$ be a point on the semiplane determined by $BC$ that contains $A$ in such a way that $DBC$ is equilateral. Prove that $H$, $O$ and $D$ are collinear.

$A, B, C$ are collinear with $B$ betweeen $A$ and $C$. $K_{1}$ is the circle with diameter $AB$, and $K_{2}$ is the circle with diameter $BC$. Another circle touches $AC$ at $B$ and meets $K_{1}$ again at $P$ and $K_{2}$ again at $Q$. The line $PQ$ meets $K_{1}$ again at $R$ and $K_{2}$ again at $S$. Show that the lines $AR$ and $CS$ meet on the perpendicular to $AC$ at $B$.