Let $H$ be the orthocenter of a given triangle $ABC$. Let $BH$ and $AC$ meet at a point $E$, and $CH$ and $AB$ meet at $F$. Suppose that $X$ is a point on the line $BC$. Also suppose that the circumcircle of triangle $BEX$ and the line $AB$ intersect again at $Y$, and the circumcircle of triangle $CFX$ and the line $AC$ intersect again at $Z$. Show that the circumcircle of triangle $AYZ$ is tangent to the line $AH$. [i]Proposed by usjl[/i]

Let $ABCD$ be a cyclic quadrilateral, let the angle bisectors at $A$ and $B$ meet at $E$, and let the line through $E$ parallel to side $CD$ intersect $AD$ at $L$ and $BC$ at $M$. Prove that $LA + MB = LM$.

Let $a$ be a real number. 1) Prove that there is a triangle with side lengths $\sqrt{a^2-a + 1}$, $\sqrt{a^2+a + 1}$, and $\sqrt{4a^2 + 3}$. 2) Prove that the area of this triangle does not depend on $a$.

Prove that it is not possible to cover a $6\times 6$ square board with eighteen $2\times 1$ rectangles, in such a way that each of the lines going along the interior gridlines cuts at least one of the rectangles. Show also that it is possible to cover a $6\times 5$ rectangle with fifteen $2\times 1 $ rectangles so that the above condition is fulfilled.

Circle $\Omega$ with center $O$ is the circumcircle of an acute triangle $\triangle ABC$ with $AB<BC$ and orthocenter $H$. On the line $BO$ there is point $D$ such that $O$ is between $B$ and $D$ and $\angle ADC= \angle ABC$ . The semi-line starting at $H$ and parallel to $BO$ wich intersects segment $AC$ , intersects $\Omega$ at $E$. Prove that $BH=DE$.

Circles $\omega_1,\omega_2,\omega_3$ with centers $O_1,O_2,O_3$, respectively, are externally tangent to each other. The circle $\omega_1$ touches $\omega_2$ at $P_1$ and $\omega_3$ at $P_2$. For any point $A$ on $\omega_1$, $A_1$ denotes the point symmetric to $A$ with respect to $O_1$. Show that the intersection points of $AP_2$ with $\omega_3$, $A_1P_3$ with $\omega_2$, and $AP_3$ with $A_1P_2$ lie on a line.

Determine the set of points $P$ in the plane of a square $ABCD$ for which \[\max (PA, PC)=\frac1{\sqrt2}(PB+PD).\] [i]Titu Andreescu and I.V. Maftei[/i]

Prove that it is impossible for seven distinct straight lines to be situated in the euclidean plane so as to have at least six points where exactly three of these lines intersect and at least four points where exactly two of these lines intersect.

Let $a$ be a positive number. Assume that the parameterized curve $C: \ x=t+e^{at},\ y=-t+e^{at}\ (-\infty <t< \infty)$ is touched to $x$ axis. (1) Find the value of $a.$ (2) Find the area of the part which is surrounded by two straight lines $y=0, y=x$ and the curve $C.$

Find the locus of centers of regular triangles such that three given points $A, B, C$ lie respectively on three lines containing sides of the triangle.

On the side $BC$ of the parallelogram $ABCD$, points $E$ and $F$ are given ($E$ lies between $B$ and $F$) and the diagonals $AC, BD$ meet at $O$. If it's known that $AE, DF$ are tangent to the circumcircle of $\triangle AOD$, prove that they're tangent to the circumcircle of $\triangle EOF$ as well.

In the configuration below, $\theta$ is measured in radians, $C$ is the center of the circle, $BCD$ and $ACE$ are line segments and $AB$ is tangent to the circle at $A$. [asy] defaultpen(fontsize(10pt)+linewidth(.8pt)); pair A=(0,-1), E=(0,1), C=(0,0), D=dir(10), F=dir(190), B=(-1/sin(10*pi/180))*dir(10); fill(Arc((0,0),1,10,90)--C--D--cycle,mediumgray); fill(Arc((0,0),1,190,270)--B--F--cycle,mediumgray); draw(unitcircle); draw(A--B--D^^A--E); label("$A$",A,S); label("$B$",B,W); label("$C$",C,SE); label("$\theta$",C,SW); label("$D$",D,NE); label("$E$",E,N); [/asy] A necessary and sufficient condition for the equality of the two shaded areas, given $0 < \theta < \frac{\pi}{2}$, is $ \textbf{(A)}\ \tan \theta = \theta\qquad\textbf{(B)}\ \tan \theta = 2\theta\qquad\textbf{(C)}\ \tan \theta = 4\theta\qquad\textbf{(D)}\ \tan 2\theta = \theta\qquad \\ \textbf{(E)}\ \tan \frac{\theta}{2} = \theta$

Let $ABC$ be an acute triangle, with median $AM$, height $AH$ and internal angle bisector $AL$. Suppose that $B, H, L, M, C$ are collinear in that order, and $LH<LM$. Prove that $BC>2AL$.

a) Fold a $10 \times 10$ square from a $1 \times 118$ rectangular strip. b) Fold a $10 \times 10$ square from a $1 \times (100+9\sqrt3)$ rectangular strip (approximately $1\times 115.58$). The strip can be bent, but not torn.

Two triangles $ABC, A’B’C’$ have the same orthocenter $H$ and the same circumcircle with center $O$. Letting $PQR$ be the triangle formed by $AA’, BB’, CC’$, prove that the circumcenter of $PQR$ lies on $OH$.

Let $\vartriangle ABC$ be a triangle with $\angle ABC > \angle BCA \ge 30^o$ . The angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $CA$ and $AB$ at $D$ and $E$ respectively, and $BD$ and $CE$ intersect at $P$. Suppose that $P D = P E$ and the incircle of $\vartriangle ABC$ has unit radius. What is the maximum possible length of $BC$?

[b]p1.[/b] Suppose $x \ne 1$ or $10$ and logarithms are computed to the base $10$. Define $y= 10^{\frac{1}{1-\log x}}$ and $z = ^{\frac{1}{1-\log y}}$ . Prove that $x= 10^{\frac{1}{1-\log z}}$ [b]p2.[/b] If $n$ is an odd number and $x_1, x_2, x_3,..., x_n$ is an arbitrary arrangement of the integers $1, 2,3,..., n$, prove that the product $$(x_1 -1)(x_2-2)(x_3- 3)... (x_n-n)$$ is an even number (possibly negative or zero). [b]p3.[/b] Prove that $\frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot(2n} < \sqrt{\frac{1}{2n + 1}}$ for all integers $n = 1,2,3,...$ [b]p4.[/b] Prove that if three angles of a convex polygon are each $60^o$, then the polygon must be an equilateral triangle. [b]p5.[/b] Find all solutions, real and complex, of $$4 \left(x^2+\frac{1}{x^2} \right)-4 \left( x+\frac{1}{x} \right)-7=0$$ [b]p6.[/b] A man is $\frac38$ of the way across a narrow railroad bridge when he hears a train approaching at $60$ miles per hour. No matter which way he runs he can [u]just [/u] escape being hit by the train. How fast can he run? Prove your assertion. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked $x$? [asy] defaultpen(linewidth(0.7)); path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3); draw(p^^(2,3)--(4,3)^^shift(2,0)*p^^(2,0)--origin); draw(shift(1,0)*p, dashed); label("$x$", (0.3,0.5), E); label("$A$", (1.3,0.5), E); label("$B$", (1.3,1.5), E); label("$C$", (2.3,1.5), E); label("$D$", (2.3,2.5), E); label("$E$", (3.3,2.5), E);[/asy] $ \mathbf{(A)}\; A\qquad \mathbf{(B)}\; B\qquad \mathbf{(C)}\; C\qquad \mathbf{(D)}\; D\qquad \mathbf{(E)}\; E$

The incircle of triangle $ABC$ touches $BC$ at $D$ and $AB$ at $F$, intersects the line $AD$ again at $H$ and the line $CF$ again at $K$. Prove that $\frac{FD\times HK}{FH\times DK}=3$

Let $ABCD$ be an isosceles trapezoid with $AB=5$, $CD = 8$, and $BC = DA = 6$. There exists an angle $\theta$ such that there is only one point $X$ satisfying $\angle AXD = 180^{\circ} - \angle BXC = \theta$. Find $\sin(\theta)^2$.

Diameter $AB$ divides a circle into two semicircles. Points $P_1$ , $P_2$, $...$, $P_n$ are given on one of the semicircles in this order. How should a point C be chosen on the other semicircle in order to maximize the sum of the areas of triangles $CP_1P_2$, $CP_2P_3$, $...$,$CP_{n-1}P_n$?

$ABC$ is a non-isosceles triangle. $T_A$ is the tangency point of incircle of $ABC$ in the side $BC$ (define $T_B$,$T_C$ analogously). $I_A$ is the ex-center relative to the side BC (define $I_B$,$I_C$ analogously). $X_A$ is the mid-point of $I_BI_C$ (define $X_B$,$X_C$ analogously). Show that $X_AT_A$,$X_BT_B$,$X_CT_C$ meet in a common point, colinear with the incenter and circumcenter of $ABC$.

The centre of the circumcircle of quadrilateral $ABCD$ is $O$ and $O$ is not on any of the sides of $ABCD$. $P=AC \cap BD$. The circumecentres of $\triangle{OAB}$, $\triangle{OBC}$, $\triangle{OCD}$ and $\triangle{ODA}$ are $O_1$, $O_2$, $O_3$ and $O_4$ respectively. Prove that $O_1O_3$, $O_2O_4$ and $OP$ are concurrent.

In the coordinate plane a rectangle with vertices $ (0, 0),$ $ (m, 0),$ $ (0, n),$ $ (m, n)$ is given where both $ m$ and $ n$ are odd integers. The rectangle is partitioned into triangles in such a way that [i](i)[/i] each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form $ x \equal{} j$ or $ y \equal{} k,$ where $ j$ and $ k$ are integers, and the altitude on this side has length 1; [i](ii)[/i] each “bad” side (i.e., a side of any triangle in the partition that is not a “good” one) is a common side of two triangles in the partition. Prove that there exist at least two triangles in the partition each of which has two good sides.

In an acute-angled triangle $ABC$ is $AB > BC$ , and the points $A_1$ and $C_1$ are the feet of the altitudes of from the vertices $A$ and $C$. Let $D$ be the second intersection of the circumcircles of triangles $ABC$ and $A_1BC_1$ (different of $B$). Let $Z$ be the intersection of the tangents to the circumcircle of the triangle ABC at the points $A$ and $C$ , and let the lines $ZA$ and $A_1C_1$ intersect at the point $X$, and the lines $ZC$ and $A_1C_1$ intersect at the point $Y$. Prove that the point $D$ lies on the circumcircle of the triangle $XYZ$.