Given a triangle $ABC$ with symmedians $BE,CF(E,F$ are on the sides $CA,AB,$ respectively$)$ intersecting at [i]Lemoine[/i] point $L.$ Prove that: $AB=AC$ in each case: a) $LB=LC.$ b) $BE=CF.$

In $\vartriangle ABC$, the external angle bisector of $\angle BAC$ intersects line $BC$ at $D$. $E$ is a point on ray $\overrightarrow{AC}$ such that $\angle BDE = 2\angle ADB$. If $AB = 10$, $AC = 12$, and $CE = 33$, compute $\frac{DB}{DE}$ .

How many sides of the convex polygon can equal its longest diagonal?

Let ABC be a triangle with angle ACB=60. Let AA' and BB' be altitudes and let T be centroid of the triangle ABC. If A'T and B'T intersect triangle's circumcircle in points M and N respectively prove that MN=AB.

$ABC$ is a triangle with $\hat{A}<\hat{C}$, and $D$ is the point on $BC$ such that $B\hat{A}D=A\hat{C}B$. The perpendicular bisectors of $AD$ and $AC$ intersect in the point $E$. Prove that $B\hat{A}E=90^\circ$.

This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base. [img]https://1.bp.blogspot.com/-lPpfHqfMMRY/XzcBIiF-n2I/AAAAAAAAMW8/nPs_mLe5C8srcxNz45Wg-_SqHlRAsAmigCLcBGAsYHQ/s0/2005%2BMohr%2Bp1.png[/img]

Let $ABC$ be a triangle in which $AB=AC$ and let $I$ be its incenter. It is known that $BC=AB+AI$. Let $D$ be a point on line $BA$ extended beyond $A$ such that $AD=AI$. Prove that $DAIC$ is a cyclic quadrilateral.

Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.

For triangle $\vartriangle ABC$, define its $A$-excircle to be the circle that is externally tangent to line segment $BC$ and extensions of $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, and define the $B$-excircle and $C$-excircle likewise. Then, define the $A$-[i]veryexcircle [/i] to be the unique circle externally tangent to both the $A$-excircle as well as the extensions of $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, but that shares no points with line $\overleftrightarrow{BC}$, and define the $B$-veryexcircle and $C$-veryexcircle likewise. Compute the smallest integer $N \ge 337$ such that for all $N_1 \ge N$, the area of a triangle with lengths $3N^2_1$ , $3N^2_1 + 1$, and $2022N_1$ is at most $\frac{1}{22022}$ times the area of the triangle formed by connecting the centers of its three veryexcircles. If your submitted estimate is a positive number $E$ and the true value is $A$, then your score is given by $\max \left(0, \left\lfloor 25 \min \left( \frac{E}{A}, \frac{A}{E}\right)^3\right\rfloor \right)$.

Two rays with common endpoint $O$ forms a $30^\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB = 1$. The maximum possible length of $OB$ is $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \dfrac{1+\sqrt{3}}{\sqrt{2}} \qquad \textbf{(C)}\ \sqrt{3} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \dfrac{4}{\sqrt{3}}$

Five lighthouses are located, in order, at points $A, B, C, D$, and $E$ along the shore of a circular lake with a diameter of $10$ miles. Segments $AD$ and $BE$ are diameters of the circle. At night, when sitting at $A$, the lights from $B, C, D$, and $E$ appear to be equally spaced along the horizon. The perimeter in miles of pentagon $ABCDE$ can be written $m +\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. Let $\ell$ be the tangent to $\omega$ parallel to $BC$ and distinct from $BC$. Let $D$ be the intersection of $\ell$ and $AC$, and let $M$ be the midpoint of $\overline{ID}$. Prove that $\angle AMD = \angle DBC$.

Given acute angled traingle $ABC$ and altitudes $AA_1$, $BB_1$, $CC_1$. Let $M$ midpoint of $BC$. $P$ point of intersection of circles $(AB_1C_1)$ and $(ABC)$ . $T$ is point of intersection of tangents to $(ABC)$ at $B$ and $C$. $S$ point of intersection of $AT$ and $(ABC)$. Prove that $P,A_1,S$ and midpoint of $MT$ collinear.

Is it possible to number the $8$ vertices of a cube from $1$ to $8$ in such a way that the value of the sum on every edge is different?

Points $A, M_1, M_2$ and $C$ are on a line in this order. Let $k_1$ the circle with center $M_1$ passing through $A$ and $k_2$ the circle with center $M_2$ passing through $C$. The two circles intersect at points $E$ and $F$. A common tangent of $k_1$ and $k_2$, touches $k_1$ at $B$ and $k_2$ at $D$. Show that the lines $AB, CD$ and $EF$ intersect at one point.

Suppose $ABC$ is a triangle, and define $B_1$ and $C_1$ such that $\triangle AB_1C$ and $\triangle AC_1B$ are isosceles right triangles on the exterior of $\triangle ABC$ with right angles at $B_1$ and $C_1$, respectively. Let $M$ be the midpoint of $\overline{B_1C_1}$; if $B_1C_1 = 12$, $BM = 7$ and $CM = 11$, what is the area of $\triangle ABC$?

A figure consisting of five equal-sized squares is placed as shown in a rectangle of size $7\times 8$ units. Find the side length of the squares. [img]https://cdn.artofproblemsolving.com/attachments/e/e/cbc2b7b0693949790c1958fb1449bdd15393d8.png[/img]

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$.

In a regular $2n$-gonal prism, bases $A_1A_2\cdots A_{2n}$ and $B_1B_2\cdots B_{2n}$ have circumradii equal to $R$. If the length of the lateral edge $A_1B_1$ varies, the angle between the line $A_1B_{n+1}$ and the plane $A_1A_3B_{n+2}$ is maximal for $A_1B_1=2R\cos\frac\pi{2n}$.

The graph of a function $f: \mathbb{R}\to\mathbb{R}$ has two has at least two centres of symmetry. Prove that $f$ can be represented as sum of a linear and periodic funtion.

In regular hexagon $ABCDEF$ with side length $2$, let $P$, $Q$, $R$, and $S$ be the feet of the altitudes from $A$ to $BC$, $EF$, $CF$, and $BE$, respectively. Find the area of quadrilateral $PQRS$.

Let $ABC$ be a triangle and $D$ be a point on segment $BC$. If $\triangle ABD$ is equilateral and $\angle ACB = 14^{\circ}$, what is $\angle{DAC}$? $\textbf{(A) }26^{\circ}\qquad\textbf{(B) }34^{\circ}\qquad\textbf{(C) }46^{\circ}\qquad\textbf{(D) }50^{\circ}\qquad\textbf{(E) }54^{\circ}$

$(CZS 5)$ A convex quadrilateral $ABCD$ with sides $AB = a, BC = b, CD = c, DA = d$ and angles $\alpha = \angle DAB, \beta = \angle ABC, \gamma = \angle BCD,$ and $\delta = \angle CDA$ is given. Let $s = \frac{a + b + c +d}{2}$ and $P$ be the area of the quadrilateral. Prove that $P^2 = (s - a)(s - b)(s - c)(s - d) - abcd \cos^2\frac{\alpha +\gamma}{2}$

Let $\Gamma_1$ and $\Gamma_2$ be two circles with different radii, with $\Gamma_1$ the smaller one. The two circles meet at distinct points $A$ and $B$. $C$ and $D$ are two points on the circles $\Gamma_1$ and $\Gamma_2$, respectively, and such that $A$ is the midpoint of $CD$. $CB$ is extended to meet $\Gamma_2$ at $F$, while $DB$ is extended to meet $\Gamma_1$ at $E$. The perpendicular bisector of $CD$ and the perpendicular bisector of $EF$ meet at $P$. (a) Prove that $\angle{EPF} = 2\angle{CAE}$. (b) Prove that $AP^2 = CA^2 + PE^2$.

The circles $(R, r)$ and $(P, \rho)$, where $r > \rho$, touch externally at $A$. Their direct common tangent touches $(R, r)$ at B and $(P, \rho)$ at $C$. The line $RP$ meets the circle $(P, \rho)$ again at $D$ and the line $BC$ at $E$. If $|BC| = 6|DE|$, prove that: [b](a)[/b] the lengths of the sides of the triangle $RBE$ are in an arithmetic progression, and [b](b)[/b] $|AB| = 2|AC|.$