A circle touches the sides of an angle with vertex $A$ at points $B$ and $C.$ A line passing through $A$ intersects this circle in points $D$ and $E.$ A chord $BX$ is parallel to $DE.$ Prove that $XC$ passes through the midpoint of the segment $DE.$

$ABC$ is a triangle. The point A' lies on the side opposite to $A$ and $BA'/BC = k$, where $1/2 < k < 1$. Similarly, $B'$ lies on the side opposite to $B$ with $CB'/CA = k$, and $C'$ lies on the side opposite to $C$ with $AC'/AB = k$. Show that the perimeter of $A'B'C'$ is less than $k$ times the perimeter of $ABC$.

Triangle $ABC$ lies in the first quadrant. Points $A$, $B$, and $C$ are reflected across the line $y=x$ to points $A'$, $B'$, and $C'$, respectively. Assume that none of the vertices of the triangle lie on the line $y=x$. Which of the following statements is [u][i]not[/i][/u] always true? $(A)$ Triangle $A'B'C'$ lies in the first quadrant. $(B)$ Triangles $ABC$ and $A'B'C'$ have the same area. $(C)$ The slope of line $AA'$ is $-1$. $(D)$ The slopes of lines $AA'$ and $CC'$ are the same. $(E)$ Lines $AB$ and $A'B'$ are perpendicular to each other.

Given $\triangle ABC$ inscribed in $(O)$ an let $I$ and $I_a$ be it's incenter and $A$-excenter ,respectively. Tangent lines to $(O)$ at $C,B$ intersect the angle bisector of $A$ at $M,N$ ,respectively. Second tangent lines through $M,N$ intersect $(O)$ at $X,Y$. Prove that $XYII_a$ is cyclic.

A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$, $BC = 2$. The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$. How large, in degrees, is $\angle ABM$? [asy] size(180); pathpen = rgb(0,0,0.6)+linewidth(1); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6) + linewidth(0.7) + linetype("4 4"), dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1); pair A=(0,1), B=(0,0), C=(2,0), D=(2,1), E=A/2, F=(2,.5), M=(1/3^.5,1), N=reflect(B,M)*A; D(B--M--D("N",N,NE)--B--D("C",C,SE)--D("D",D,NE)--M); D(D("M",M,plain.N)--D("A",A,NW)--D("B",B,SW),dd); D(D("E",E,W)--D("F",F,plain.E),dd); [/asy]

Let $ABC$ be an acute triangle such that $AB < AC$. Let $\omega$ be the circumcircle of $ABC$ and assume that the tangent to $\omega$ at $A$ intersects the line $BC$ at $D$. Let $\Omega$ be the circle with center $D$ and radius $AD$. Denote by $E$ the second intersection point of $\omega$ and $\Omega$. Let $M$ be the midpoint of $BC$. If the line $BE$ meets $\Omega$ again at $X$, and the line $CX$ meets $\Omega$ for the second time at $Y$, show that $A, Y$, and $M$ are collinear. [i]Proposed by Nikola Velov, North Macedonia[/i]

Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.

The altitudes $BB_1$ and $CC_1$ of an acute triangle $ABC$ intersect in point $H$. Let $B_2$ and $C_2$ be points on the segments $BH$ and $CH$, respectively, such that $BB_2=B_1H$ and $CC_2=C_1H$. The circumcircle of the triangle $B_2HC_2$ intersects the circumcircle of the triangle $ABC$ in points $D$ and $E$. Prove that the triangle $DEH$ is right-angled.

Circle I is circumscribed about a given square and circle II is inscribed in the given square. If $r$ is the ratio of the area of circle $I$ to that of circle $II$, then $r$ equals: $\text{(A)} \ \sqrt 2 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \sqrt 3 \qquad \text{(D)} \ 2\sqrt 2 \qquad \text{(E)} \ 2\sqrt 3$

Perpendiculars are drawn from an interior point $M$ of the equilateral triangle $ABC$ to its sides , intersecting them at points $D, E$ and $F$ . Find the locus of all points $M$ such that $DEF$ is a right triangle . (J . Tabov , Sofia)

Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$), $\omega_1$ and $\omega_2$ be the circumcircles of triangles $AEM$ and $CDM.$ It is known that the circles $\omega_1$ and $\omega_2$ are tangent. Find the ratio in which the circle $\omega_2$ divides $AC.$

Three mutually tangent spheres of radius $ 1$ rest on a horizontal plane. A sphere of radius $ 2$ rests on them. What is the distance from the plane to the top of the larger sphere? $ \textbf{(A)}\ 3 \plus{} \frac {\sqrt {30}}{2} \qquad \textbf{(B)}\ 3 \plus{} \frac {\sqrt {69}}{3} \qquad \textbf{(C)}\ 3 \plus{} \frac {\sqrt {123}}{4}\qquad \textbf{(D)}\ \frac {52}{9}\qquad \textbf{(E)}\ 3 \plus{} 2\sqrt2$

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

$ABCD$ is a cyclic quadrilateral and $AC$ intersects $BD$ at $E$. $M, N$ are the midpoints of $AB, CD$, respectively. $\odot(AMN)$ meets $\odot(ABCD)$ again at $P$. $\odot(CMN)$ meets $\odot(ABCD)$ again at $Q$. $\odot(PEQ)$ meets $BD$ again at $T$. Prove that $M,N,T$ are colinear. [i]Proposed by chengbilly[/i]

Let $a, b, c$ be the sides and $R$ be the circumradius of a triangle. Prove that \[\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\ge\frac{1}{R^2}.\]

On the side $AB$ of the parallelogram $ABCD$ we take the points $X$ and $Y$ such that the points $A$, $X$, $Y$ and $B$ appear in this order. The lines $DX$ and $CY$ intersect at the point $Z$. Suppose that the area of the triangle $\triangle XYZ$ is equal to the sum of the areas of the triangles $\triangle AXD$ and $\triangle CYB$. Prove that the area of the quadrilateral $XYCD$ is equal to $3$ times the area of the triangle $\triangle XYZ$.

Two congruent rectangles are located as shown in the figure. Find the area of the shaded part. [img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]

In the right triangle $ABC$ ($m ( \angle A) = 90^o$) $D$ is the foot of the altitude from $A$. The bisectors of the angles $ABD$ and $ADB$ intersect in $I_1$ and the bisectors of the angles $ACD$ and $ADC$ in $I_2$. Find the angles of the triangle if the sum of distances from $I_1$ and $I_2$ to $AD$ is equal to $\frac14$ of the length of $BC$.

A cube $ABCDA_1B_1C_1D_1$ is given. Find the locus of points $E$ on the face $A_1B_1C_1D_1$ for which there exists a line intersecting the lines $AB$, $A_1D_1$, $B_1D$, and $EC$.

While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$. [hide="Clarifications"] [list] [*] The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect. [*] $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''[/list][/hide] [i]Ray Li[/i]

A regular octagon $P$ is given whose incircle $k$ has diameter $1$. About $k$ is circumscribed a regular $16$-gon, which is also inscribed in $P$, cutting from $P$ eight isosceles triangles. To the octagon $P$, three of these triangles are added so that exactly two of them are adjacent and no two of them are opposite to each other. Every $11$-gon so obtained is said to be $P'$. Prove the following statement: Given a finite set $M$ of points lying in $P$ such that every two points of this set have a distance not exceeding $1$, one of the $11$-gons $P'$ contains all of $M$.

Let $ABCD$ be a parallelogram and $P$ be a point on diagonal $BD$ such that $\angle PCB=\angle ACD$. Circumcircle of triangle $ABD$ intersects line $AC$ at points $A$ and $E$. Prove that \[\angle AED=\angle PEB.\]

In triangle $ABC, M$ is the midpoint of $AB$ and $D$ the foot of the bisector of angle $\angle ABC$. If $MD$ and $BD$ are known to be perpendicular, calculate $\frac{AB}{BC}$.

Let $O$ be the center of the circle $\omega$, and let $A$ be a point inside this circle, different from $O$. Find all points $P$ on the circle $\omega$ for which the angle $\angle OPA$ acquires the greatest value.

Let $C_1$ and $C_2$ be two concentric reflective hollow metal spheres of radius $R$ and $R\sqrt3$ respectively. From a point $P$ on the surface of $C_2$, a ray of light is emitted inward at $30^o$ from the radial direction. The ray eventually returns to $P$. How many total reflections off of $C_1$ and $C_2$ does it take?