Triangle $ABC$ has side lengths $13$, $14$ and $15$. Let $k, k_A,k_B,k_C$ be four circles of radius $ r$ inside the triangle such that $k_A$ is tangent to sides $AB$ and $AC$, $k_B$ is tangent to sides $BA$ and $BC$, $k_C$ is tangent to sides $CA$ and $CB$, and $k$ is externally tangent to circles $k_A$, $k_B$ and $k_C$. Let $r = m/n$ where $m$ and $n$ are coprime. Find $m + n$.

Let $A, B, C, D$ be points chosen on a circle, in that order. Line $BD$ is reflected over lines $AB$ and $DA$ to obtain lines $\ell_1$ and $\ell_2$ respectively. If lines $\ell_1$, $\ell_2$, and $AC$ meet at a common point and if $AB = 4$, $BC = 3$, $CD = 2$, compute the length $DA$.

Let $ABCDEF$ be a regular hexagon and $M\in (DE)$, $N\in(CD)$ such that $m (\widehat {AMN}) = 90^\circ$ and $AN = CM \sqrt {2}$. Find the value of $\frac{DM}{ME}$.

Prove that the diagonals of a convex quadrilateral are perpendicular if and only if the sum of the squares of one pair of opposite sides is equal to the sum of the squares of the other pair.

Point $D$ on side $BC$ of acute triangle ABC is such that $AB=AD$. The circumcircle of triangle $ABD$ intersects side $AC$ at points $A$ and $K$. Line $DK$ intersects the perpendicular drawn from $B$ on $AC$, at the point $L$. Prove that $CL= BC$

Let $ABCD$ be a cyclic quadrilateral, and angle bisectors of $\angle BAD$ and $\angle BCD$ meet at point $I$. Show that if $\angle BIC = \angle IDC$, then $I$ is the incenter of triangle $ABD$.

In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?

We have a scalene acute triangle $ABC$ (triangle with no two equal sides) and a point $D$ on side $BC$. Pick a point $E$ on side $AB$ and a point $F$ on side $AC$ such that $\angle DEB=\angle DFC$. Lines $DF,\, DE$ intersect $AB,\, AC$ at points $M,\, N$, respectively. Denote $(I_1),\, (I_2)$ by the circumcircles of triangles $DEM,\, DFN$ in that order. The circle $(J_1)$ touches $(I_1)$ internally at $D$ and touches $AB$ at $K$, circle $(J_2)$ touches $(I_2)$ internally at $D$ and touches $AC$ at $H$. $P$ is the intersection of $(I_1),\, (I_2)$ different from $D$. $Q$ is the intersection of $(J_1),\, (J_2)$ different from $D$. a. Prove that all points $D,\, P,\, Q$ lie on the same line. b. The circumcircles of triangles $AEF,\, AHK$ intersect at $A,\, G$. $(AEF)$ also cut $AQ$ at $A,\, L$. Prove that the tangent at $D$ of $(DQG)$ cuts $EF$ at a point on $(DLG)$.

Find the relation of the black part length and the white part length for the main diagonal of the a) $100\times 99$ chess-board; b) $101\times 99$ chess-board.

On the coordinate plane, Let $C$ be the graph of $y=(\ln x)^2\ (x>0)$ and for $\alpha >0$, denote $L(\alpha)$ be the tangent line of $C$ at the point $(\alpha ,\ (\ln \alpha)^2).$ (1) Draw the graph. (2) Let $n(\alpha)$ be the number of the intersection points of $C$ and $L(\alpha)$. Find $n(\alpha)$. (3) For $0<\alpha <1$, let $S(\alpha)$ be the area of the region bounded by $C,\ L(\alpha)$ and the $x$-axis. Find $S(\alpha)$. 2010 Tokyo Institute of Technology entrance exam, Second Exam.

Let $ABC$ be an acute-angled triangle. Let $A_1$ be the midpoint of the arc $BC$ which contain the point $A$. Let $A_2$ and $A_3$ be the foot(s) of the perpendicular(s) of the point $A_1$ to the lines $AB$ and $AC$, respectively. Define $B_2,B_3,C_2,C_3$ analogously. a) Prove that the line $A_2A_3$ cuts $BC$ in the midpoint. b) Prove that the lines $A_2A_3,B_2B_3$ and $C_2C_3$ are concurrents.

Suppose that $ABC$ is a triangle with $AB = 6, BC = 12$, and $\angle B = 90^{\circ}$. Point $D$ lies on side $BC$, and point $E$ is constructed on $AC$ such that $\angle ADE = 90^{\circ}$. Given that $DE = EC = \frac{a\sqrt{b}}{c}$ for positive integers $a, b,$ and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, find $a+ b+c$. [i]Proposed by Andrew Wu[/i]

Circle $k$ passes through $A$ and intersects the sides of $\Delta ABC$ in $P,Q$, and $L$. Prove that: $\frac{S_{PQL}}{S_{ABC}}\leq \frac{1}{4} (\frac{PL}{AQ})^2$.

Let $ABCD$ be a trapezoid such that $AB \parallel CD$, $|AB|<|CD|$, and $\text{Area}(ABC)=30$. Let the line through $B$ parallel to $AD$ meet $[AC]$ at $E$. If $|AE|:|EC|=3:2$, then what is the area of trapezoid $ABCD$? $ \textbf{(A)}\ 45 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 72 \qquad\textbf{(D)}\ 80 \qquad\textbf{(E)}\ 90 $

Title is self explanatory. Pick two points on the unit sphere. What is the expected distance between them?

A triangle has side lengths $7, 9,$ and $12$. What is the area of the triangle?

$\triangle ABC$ has side lengths $AB=15$, $BC=34$, and $CA=35$. Let the circumcenter of $ABC$ be $O$. Let $D$ be the foot of the perpendicular from $C$ to $AB$. Let $R$ be the foot of the perpendicular from $D$ to $AC$, and let $W$ be the perpendicular foot from $D$ to $BC$. Find the area of quadrilateral $CROW$.

Let us consider all rectangles with sides of length $a,b$ both of which are whole numbers. Do more of these rectangles have perimeter $2000$ or perimeter $2002$?

$\Delta ABC$ is right-angled at $B$, with $AB=1$ and $BC=3$. $E$ is the foot of perpendicular from $B$ to $AC$. $BA$ and $BE$ are produced to $D$ and $F$ respectively such that $D$, $F$, $C$ are collinear and $\angle DAF=\angle BAC$. Find the length of $AD$.

In coordinate system, there are six points $P_i(x_i,y_i)(i=1,2,\cdots,6)$, satisfying: (1) $x_i,y_i\in\{-2,-1,0,1,2\}$. (2) For any three points, they are not collinear. Prove that there exists a triangle $\triangle P_iP_jP_k(1\leq i<j<k\leq6)$, its area is not larger than $2$.

What is the area of a triangle with side lengths $17$, $25$, and $26$? [i]2019 CCA Math Bonanza Lightning Round #3.2[/i]

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.

We are given a polyhedron with at least $5$ vertices, such that exactly $3$ edges meet in each of the vertices. Prove that we can assign a rational number to every vertex of the given polyhedron such that the following conditions are met: $(i)$ At least one of the numbers assigned to the vertices is equal to $2020$. $(ii)$ For every polygonal face, the product of the numbers assigned to the vertices of that face is equal to $1$.

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$