Let $I$ be the centre of the inscribed circle of the non-isosceles triangle $ABC$, and let the circle touch the sides $BC, CA, AB$ at the points $A_1, B_1, C_1$ respectively. Prove that the centres of the circumcircles of $\vartriangle AIA_1,\vartriangle BIB_1$ and $\vartriangle CIC_1$ are collinear.

Triangle $\vartriangle ABC$ has side lengths $AB = 8$, $BC = 15$, and $CA = 17$. Circles $\omega_1$ and $\omega_2$ are externally tangent to each other and within $\vartriangle ABC$. The radius of circle $\omega_2$ is four times the radius of circle $\omega_1$. Circle $\omega_1$ is tangent to $\overline{AB}$ and $\overline{BC}$, and circle $\omega_2$ is tangent to $\overline{BC}$ and $\overline{CA}$. Compute the radius of circle $\omega_1$.

$(BEL 4)$ Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O.$

4. A triangle $A B C$ with angle $A$ equal to $60^{\circ}$ is given. Its incircle is tangent to side $A B$ at point $D$, while its excircle tangent to side $A C$, is tangent to the extension of side $A B$ at point $E$. Prove that the perpendicular to side $A C$, passing through point $D$, meets the incircle again at a point equidistant from points $E$ and $C$. (The excircle is the circle tangent to one side of the triangle and to the extensions of two other sides.) Azamat Mardanov

In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$? [asy] import olympiad; size(180); real r = 3, s = 5, t = sqrt(r*r+s*s); defaultpen(linewidth(0.6) + fontsize(10)); pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0); draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D)); label("$A$",A,SW); label("$B$", B, NW); label("$C$",C,NE); label("$D$",D,SE); label("$P$",P,S); [/asy] $\textbf{(A) }3\sqrt 5 \qquad \textbf{(B) }10 \qquad \textbf{(C) }6\sqrt 5 \qquad \textbf{(D) }20\qquad \textbf{(E) }25$

There are $n$ fleas on an infinite sheet of triangulated paper. Initially the fleas are in different small triangles, all of which are inside some equilateral triangle consisting of $n^2$ small triangles. Once a second each flea jumps from its original triangle to one of the three small triangles having a common vertex but no common side with it. For which natural numbers $n$ does there exist an initial configuration such that after a finite number of jumps all the $n$ fleas can meet in a single small triangle?

The tetrahedron $ S.ABC$ has the faces $ SBC$ and $ ABC$ perpendicular. The three angles at $ S$ are all $ 60^{\circ}$ and $ SB \equal{} SC \equal{} 1$. Find the volume of the tetrahedron.

Given five points inside an equilateral triangle of side length $2$, show that there are two points whose distance from each other is at most $ 1$.

Let be $ABCD$ a rectangle with $|AB|=a\ge b=|BC|$. Find points $P,Q$ on the line $BD$ such that $|AP|=|PQ|=|QC|$. Discuss the solvability with respect to the lengths $a,b$.

Find the angle $BCA$ in the quadrilateral of the figure. [img]https://cdn.artofproblemsolving.com/attachments/0/2/974e23be54125cde8610a78254b59685833b5b.png[/img]

Incircle of triangle $ABC$ tangent to $AB$ and $AC$ on $E$ and $F$ respectively. If $X$ is the midpoint of $EF$, prove $\angle BXC > 90^{\circ}$

Let $A$ be a point in the plane, and $\ell$ a line not passing through $A$. Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is [i]not[/i] marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane. (i) Can Evan construct* the reflection of $A$ over $\ell$? (ii) Can Evan construct the foot of the altitude from $A$ to $\ell$? *To construct a point, Evan must have an algorithm which marks the point in finitely many steps. [i]Proposed by Zack Chroman[/i]

In a unit cube $ABCD - EFGH$, an equilateral triangle $BDG$ cuts out a circle from the circumsphere of the cube. Find the area of the circle.

Let $ABC$ be a triangle with incentre $I$. The circle through $B$ tangent to $AI$ at $I$ meets side $AB$ again at $P$. The circle through $C$ tangent to $AI$ at $I$ meets side $AC$ again at $Q$. Prove that $PQ$ is tangent to the incircle of $ABC.$

Three congruent equilateral triangles $T_1$, $T_2$, and $T_3$ are stacked from left to right inside rectangle $JHMT$ such that the bottom left vertex of $T_1$ is $T$, the bottom side of $T_1$ lies on $\overline{MT}$, the bottom left vertex of $T_2$ is the midpoint of a side of $T_1$, the bottom left vertex of $T_3$ is the midpoint of a side of $T_2$, and the other two vertices of $T_3$ lie on $\overline{JH}$ and $\overline{HM}$, as shown below. Given that rectangle $JHMT$ has area $2022$, find the area of any one of the triangles $T_1$, $T_2$, or $T_3$. [asy] unitsize(0.111111111111111111cm); real s = sqrt(4044/sqrt(75)); real l = 5s/2; real w = s * sqrt(3); pair J,H,M,T,V1,V2,V3,V4,V5,V6,V7,V8,C1,C2,C3; J = (0,w); H = (l,w); M = (l,0); T = (0,0); V1 = (s,0); V2 = (s/2,s * sqrt(3)/2); V3 = (V1+V2)/2; V4 = (3 * s/4+s,s * sqrt(3)/4); V5 = (3 * s/4+s/2,s * sqrt(3)/4+s * sqrt(3)/2); V6 = (V4+V5)/2; V7 = (l,s * sqrt(3)/4+s * sqrt(3)/4); V8 = (l-s/2,w); C1 = (T+V1+V2)/3; C2 = (V3+V4+V5)/3; C3 = (V6+V7+V8)/3; draw(J--H--M--T--cycle); draw(V1--V2--T); draw(V3--V4--V5--cycle); draw(V6--V7--V8--cycle); label("$J$", J, NW); label("$H$", H, NE); label("$M$", M, SE); label("$T$", T, SW); label("$T_1$", C1); label("$T_2$", C2); label("$T_3$", C3); [/asy]

Let $\omega_1,\omega_2$ be two non-intersecting circles, with circumcenters $O_1,O_2$ respectively, and radii $r_1,r_2$ respectively where $r_1 < r_2$. Let $AB,XY$ be the two internal common tangents of $\omega_1,\omega_2$, where $A,X$ lie on $\omega_1$, $B,Y$ lie on $\omega_2$. The circle with diameter $AB$ meets $\omega_1,\omega_2$ at $P$ and $Q$ respectively. If $$\angle AO_1P+\angle BO_2Q=180^{\circ},$$ find the value of $\frac{PX}{QY}$ (in terms of $r_1,r_2$).

Given four points $A,$ $B,$ $C,$ $D$ on a circle such that $AB$ is a diameter and $CD$ is not a diameter. Show that the line joining the point of intersection of the tangents to the circle at the points $C$ and $D$ with the point of intersection of the lines $AC$ and $BD$ is perpendicular to the line $AB.$

Let $P$ be an arbitrary point on the diagonal $AC$ of cyclic quadrilateral $ABCD$, and $PK, PL, PM, PN, PO$ be the perpendiculars from $P$ to $AB, BC, CD, DA, BD$ respectively. Prove that the distance from $P$ to $KN$ is equal to the distance from $O$ to $ML$.

Circles $\omega_1$ and $\omega_2$ intersect at $X$ and $Y$. Through point $Y$ two lines pass, one of which intersects $\omega_1$ and $\omega_2$ for the second time at $A$ and $B$, and the other at $C$ and $D$. Line $AD$ intersects for the second time circles $\omega_1$ and $\omega_2$ at $P$ and $Q$. It turned out that $YP=YQ$ Prove that the circumcircles of triangles $BCY$ and $PQY$ are tangent to each other.

Given an $ABC$ acute triangle with $O$ the center of the circumscribed circle. Suppose that $\omega$ is a circle that is tangent to the line $AO$ at point $A$ and also tangent to the line $BC$. Prove that $\omega$ is also tangent to the circumcircle of the triangle $BOC$.

Determine the locus of orthocenters of triangles, given the midpoint of a side and the feet of the altitudes drawn on two other sides.

Let $S$ be a circle with radius $2$, let $S_1$ be a circle,with radius $1$ and tangent, internally to $S$ in $B$ and let $S_2$ be a circle, with radius $1$ and tangent to $S_1$ in $A$, but $S_2$ isn't tangent to $S$. If $K$ is the point of intersection of the line $AB$ and the circle $S$, prove that $K$ is in the circle $S_2$.

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.

Suppose convex hexagon $ \text{HEXAGN}$ has $ 120^\circ$-rotational symmetry about a point $ P$—that is, if you rotate it $ 120^\circ$ about $ P$, it doesn't change. If $ PX\equal{}1$, find the area of triangle $ \triangle{GHX}$.

$\text{ABCD}$ is a rectangle, $\text{D}$ is the center of the circle, and $\text{B}$ is on the circle. If $\text{AD}=4$ and $\text{CD}=3$, then the area of the shaded region is between [asy] pair A,B,C,D; A=(0,4); B=(3,4); C=(3,0); D=origin; draw(circle(D,5)); fill((0,5)..(1.5,4.7697)..B--A--cycle,black); fill(B..(4,3)..(5,0)--C--cycle,black); draw((0,5)--D--(5,0)); label("A",A,NW); label("B",B,NE); label("C",C,S); label("D",D,SW); [/asy] $\text{(A)}\ 4\text{ and }5 \qquad \text{(B)}\ 5\text{ and }6 \qquad \text{(C)}\ 6\text{ and }7 \qquad \text{(D)}\ 7\text{ and }8 \qquad \text{(E)}\ 8\text{ and }9$