In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$. Prove that $|AR|\cdot |BQ|=|P I|^2$

Given a triangle $ABC$ for which $\angle BAC \neq 90^{\circ}$, let $B_1, C_1$ be variable points on $AB,AC$, respectively. Let $B_2,C_2$ be the points on line $BC$ such that a spiral similarity centered at $A$ maps $B_1C_1$ to $C_2B_2$. Denote the circumcircle of $AB_1C_1$ by $\omega$. Show that if $B_1B_2$ and $C_1C_2$ concur on $\omega$ at a point distinct from $B_1$ and $C_1$, then $\omega$ passes through a fixed point other than $A$. [i]Proposed by Max Jiang[/i]

The diagram below shows a large triangle with area $72$. Each side of the triangle has been trisected, and line segments have been drawn between these trisection points parallel to the sides of the triangle. Find the area of the shaded region. [asy] size(4cm); pair A,B1,B2,B3,C1,C2,C3,M,I,J; A=origin; dotfactor=4; B1=dir(49); B2=2*B1; B3=3*B1; C1=1.35*dir(127); C2=2*C1; C3=3*C1; M=(B2+C2)/2; I=B1+C2; J=C1+B2; pair d[] = {A,B1,B2,B3,C1,C2,C3,M,I,J}; filldraw(C1--B1--B2--J--I--C2--cycle,rgb(.76,.76,.76)); draw(A--C3--B3--cycle); draw(C1--J^^C2--B2^^B1--I); for(int i=0;i<10;++i){ dot(d[i]); } [/asy]

When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$ Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$ 30 points

Given a right angled triangle $ABC$ in which $\angle ACB = 90^o$. Let $D$ be an inner point of $AB$, and let $E$ be an inner point of $AC$. It is known that $\angle ADE = 90^o$, and that the length of the segment $AD$ is $8$, the length of the segment $DE$ is $15$, and the length of segment $CE$ is $3$. What is the area of triangle $ABC$?

How many positive integers $n<10^6$ are there such that $n$ is equal to twice of square of an integer and is equal to three times of cube of an integer? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of the above} $

On the sides of triangle $ABC$, isosceles right-angled triangles $AUB, CVB$, and $AWC$ are placed. These three triangles have their right angles at vertices $U, V$ , and $W$, respectively. Triangle $AUB$ lies completely inside triangle $ABC$ and triangles $CVB$ and $AWC$ lie completely outside $ABC$. See the figure. Prove that quadrilateral $UVCW$ is a parallelogram. [asy] import markers; unitsize(1.5 cm); pair A, B, C, U, V, W; A = (0,0); B = (2,0); C = (1.7,2.5); U = (B + rotate(90,A)*(B))/2; V = (B + rotate(90,C)*(B))/2; W = (C + rotate(90,A)*(C))/2; draw(A--B--C--cycle); draw(A--W, StickIntervalMarker(1,1,size=2mm)); draw(C--W, StickIntervalMarker(1,1,size=2mm)); draw(B--V, StickIntervalMarker(1,2,size=2mm)); draw(C--V, StickIntervalMarker(1,2,size=2mm)); draw(A--U, StickIntervalMarker(1,3,size=2mm)); draw(B--U, StickIntervalMarker(1,3,size=2mm)); draw(rightanglemark(A,U,B,5)); draw(rightanglemark(B,V,C,5)); draw(rightanglemark(A,W,C,5)); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, N); dot("$U$", U, NE); dot("$V$", V, NE); dot("$W$", W, NW); [/asy]

The lengths of the altitudes of a triangle are positive integers, and the length of the radius of the incircle is a prime number. Find the lengths of the sides of the triangle.

Let $X_0$ be the interior of a triangle with side lengths $3, 4,$ and $5$. For all positive integers $n$, define $X_n$ to be the set of points within $1$ unit of some point in $X_{n-1}$. The area of the region outside $X_{20}$ but inside $X_{21}$ can be written as $a\pi + b$, for integers $a$ and $b$. Compute $100a + b$.

Find the greatest possible distance between any two points inside or along the perimeter of an equilateral triangle with side length $2$. [i]Proposed by Alex Li[/i]

Does a parallelogram exist such that all pairwise meets of bisectors of its angles are situated outside it?

Let $A$ be a solid $a\times b\times c$ rectangular brick, where $a,b,c>0$. Let $B$ be the set of all points which are a distance of at most one from some point of $A$. Express the volume of $B$ as a polynomial in $a,b,$ and $c$.

Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$

Let $ABC$ be a right triangle with $\angle C = 90^\circ$. Let $P_A$, $P_B$, and $P_C$ be regular pentagons with side lengths $BC$, $CA$, and $AB$, respectively. Prove that $[P_A]+[P_B]=[P_C]$. [i]Proposed by CaptainFlint[/i]

The space diagonal (interior diagonal) of a cube has length $6$. Find the $\textit{surface area}$ of the cube.

The circumcentre of an acute triangle $ABC$ is $O$. Line $AC$ intersects the circumcircle of $AOB$ at a point $X$, in addition to the vertex $A$. Prove that the line $XO$ is perpendicular to the line $BC$.

The triangle $ABC$ has perimeter $36$, and the length of $BC$ is $9$. Point $M$ is the midpoint of $AC$, and $I$ is the incenter. Find the angle $MIC$.

Let $ABC$ be a triangle, and let $D$ be the foot of the $A-$altitude. Points $P, Q$ are chosen on $BC$ such that $DP = DQ = DA$. Suppose $AP$ and $AQ$ intersect the circumcircle of $ABC$ again at $X$ and $Y$. Prove that the perpendicular bisectors of the lines $PX$, $QY$, and $BC$ are concurrent. [i]Proposed by Pranjal Srivastava[/i]

$A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP<DP$.

Let $P$ and $Q$ be points taken inside of triangle $ABC$ such that $\angle APB=\angle AQC$ and $\angle APC=\angle AQB$. Circumcircle of $APQ$ intersects $AB$ and $AC$ second time at $K$ and $L$ respectively. Prove that $B,C,L,K$ are concyclic.

$A(-1,0),B(1,0)$. $D(x,0)$ is a point on $AB$. $CD\perp AB$, and $C$ is a point on unit circle. When $x\in$________, segments $AD,BD,CD$ can be three sides of a acute triangle.

Consider 2 concentric circle radii $ R$ and $ r$ ($ R > r$) with centre $ O.$ Fix $ P$ on the small circle and consider the variable chord $ PA$ of the small circle. Points $ B$ and $ C$ lie on the large circle; $ B,P,C$ are collinear and $ BC$ is perpendicular to $ AP.$ [b]i.)[/b] For which values of $ \angle OPA$ is the sum $ BC^2 \plus{} CA^2 \plus{} AB^2$ extremal? [b]ii.)[/b] What are the possible positions of the midpoints $ U$ of $ BA$ and $ V$ of $ AC$ as $ \angle OPA$ varies?

How many ways are there to fill an $8\times8\times8$ cube with $1\times1\times8$ sticks? Rotations and reflections are considered distinct. [i]Team #2[/i]

All faces of a convex polyhedron are parallelograms. Can the polyhedron have exactly 1992 faces?

Let there be a quadrilateral $ABCD$ such that $\angle CAD=45, \angle ACD=30, \angle BAC=\angle BCA=15$. Find $\angle DBC$.