In the following, the abbreviation $g \cap h$ will mean the point of intersection of two lines $g$ and $h$. Let $ABCDE$ be a convex pentagon. Let $A^{\prime}=BD\cap CE$, $B^{\prime}=CE\cap DA$, $C^{\prime}=DA\cap EB$, $D^{\prime}=EB\cap AC$ and $E^{\prime}=AC\cap BD$. Furthermore, let $A^{\prime\prime}=AA^{\prime}\cap EB$, $B^{\prime\prime}=BB^{\prime}\cap AC$, $C^{\prime\prime}=CC^{\prime}\cap BD$, $D^{\prime\prime}=DD^{\prime}\cap CE$ and $E^{\prime\prime}=EE^{\prime}\cap DA$. Prove that: \[ \frac{EA^{\prime\prime}}{A^{\prime\prime}B}\cdot\frac{AB^{\prime\prime}}{B^{\prime\prime}C}\cdot\frac{BC^{\prime\prime}}{C^{\prime\prime}D}\cdot\frac{CD^{\prime\prime}}{D^{\prime\prime}E}\cdot\frac{DE^{\prime\prime}}{E^{\prime\prime}A}=1. \] Darij

In a triangle $ABC$ let $K$ be a point on the median $BM$ such that $CK=CM$. It appears that $\angle CBM = 2 \angle ABM$. Prove that $BC=MK$.

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

Let $ABCD$ be an isosceles trapezoid with $AB = 1$, $BC = DA = 5$, $CD = 7$. Let $P$ be the intersection of diagonals $AC$ and $BD$, and let $Q$ be the foot of the altitude from $D$ to $BC$. Let $PQ$ intersect $AB$ at $R$. Compute $\sin \angle RP D$

[b]p1.[/b] A function $f$ defined on the set of positive numbers satisfies the equality $$f(xy) = f(x) + f(y), x, y > 0.$$ Find $f(2007)$ if $f\left( \frac{1}{2007} \right) = 1$. [b]p2.[/b] The plane is painted in two colors. Show that there is an isosceles right triangle with all vertices of the same color. [b]p3.[/b] Show that the number of ways to cut a $2n \times 2n$ square into $1\times 2$ dominoes is divisible by $2$. [b]p4.[/b] Two mirrors form an angle. A beam of light falls on one mirror. Prove that the beam is reflected only finitely many times (even if the angle between mirrors is very small). [b]p5.[/b] A sequence is given by the recurrence relation $a_{n+1} = (s(a_n))^2 +1$, where $s(x)$ is the sum of the digits of the positive integer $x$. Prove that starting from some moment the sequence is periodic. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A circle $\omega$ touched lines $a$ and $b$ at points $A$ and $B$ respectively. An arbitrary tangent to the circle meets $a$ and $b$ at $X$ and $Y$ respectively. Points $X'$ and $Y'$ are the reflections of $X$ and $Y$ about $A$ and $B$ respectively. Find the locus of projections of the center of the circle to the lines $X'Y'$.

The plane angles at vertex $D$ of the pyramid $ABCD$ are equal to $\alpha$,$\beta$ and $\gamma$ ($\angle CDB = a$). An arbitrary point $M$ is taken on edge $CB$. A ball is inscribed in each of the pyramids $ABDM$ and $ACDM$. Let us draw through $D$ a plane distinct from $BCD$, tangent to both balls and not intersecting the segment connecting the centers of the balls. Let this plane intersect the segment $AM$ at point $P$. What is $\angle ADP$ equal to?

The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

Let $ABC$ and $A_{1}B_{1}C_{1}$ be two triangles. Prove that $\frac{a}{a_{1}}+\frac{b}{b_{1}}+\frac{c}{c_{1}}\leq\frac{3R}{2r_{1}}$, where $a = BC$, $b = CA$, $c = AB$ are the sidelengths of triangle $ABC$, where $a_{1}=B_{1}C_{1}$, $b_{1}=C_{1}A_{1}$, $c_{1}=A_{1}B_{1}$ are the sidelengths of triangle $A_{1}B_{1}C_{1}$, where $R$ is the circumradius of triangle $ABC$ and $r_{1}$ is the inradius of triangle $A_{1}B_{1}C_{1}$.

A triangle has area $30$, one side of length $10$, and the median to that side of length $9$. Let $\theta$ be the acute angle formed by that side and the median. What is $\sin{\theta}$? $ \textbf{(A)}\ \frac{3}{10}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{9}{20}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{9}{10} $

Half the volume of a 12 foot high cone-shaped pile is grade A ore while the other half is grade B ore. The pile is worth \$62. One-third of the volume of a similarly shaped 18 foot pile is grade A ore while the other two-thirds is grade B ore. The second pile is worth \$162. Two-thirds of the volume of a similarly shaped 24 foot pile is grade A ore while the other one-third is grade B ore. What is the value in dollars (\$) of the 24 foot pile?

Two circles $C_1$ and $C_2$ intersect at two distinct points $P, Q$ in a plane. Let a line passing through $P$ meet the circles $C_1$ and $C_2$ in $A$ and $B$ respectively. Let $Y$ be the midpoint of $AB$ and let $QY$ meet the cirlces $C_1$ and $C_2$ in $X$ and $Z$ respectively. Show that $Y$ is also the midpoint of $XZ$.

Show that among any nine complex numbers whose affixes in the complex plane lie on the unit circle, there are at least two of them such that the modulus of their sum is greater than $ \sqrt 2. $ [i]Ion Tecu[/i]

$ABCD$ is convex quadrilateral with $AB=CD$. $AC$ and $BD$ intersect in $O$. $X,Y,Z,T$ are midpoints of $BC,AD,AC,BD$. Prove, that circumcenter of $OZT$ lies on $XY$.

In regular tetrahedron $ABCD$, $E\in AB,F\in CD$, satisfying: $\frac{|AE|}{|EB|}=\frac{|CF|}{|FD|}=\lambda(\lambda\in R_+)$. Note that $f(\lambda)=\alpha_{\lambda}+\beta_{\lambda}$, where $\alpha_{\lambda}=<EF,AC>,\alpha_{\lambda}=<EF,BD>$. $\text{(A)}$ $f(\lambda)$ increases in $(0,+\infty)$ $\text{(B)}$ $f(\lambda)$ decreases in $(0,+\infty)$ $\text{(C)}$ $f(\lambda)$ increases in $(0,1)$, decreases in $(1,+\infty)$ $\text{(D)}$ $f(\lambda)$ is a fixed value in $(0,+\infty)$

Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of a rectangle.

The circles in the figure have their centers at $C$ and $D$ and intersect at $A$ and $B$. Let $\angle ACB =60$, $\angle ADB =90^o$ and $DA = 1$ . Find the length of $CA$. [img]https://cdn.artofproblemsolving.com/attachments/0/1/950a55984283091d15083fadcf35e8b95cb229.png[/img]

As shown in the figure below a regular dodecahedron (the polyhedron consisting of 12 congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring? [asy] import graph; unitsize(4.5cm); pair A = (0.082, 0.378); pair B = (0.091, 0.649); pair C = (0.249, 0.899); pair D = (0.479, 0.939); pair E = (0.758, 0.893); pair F = (0.862, 0.658); pair G = (0.924, 0.403); pair H = (0.747, 0.194); pair I = (0.526, 0.075); pair J = (0.251, 0.170); pair K = (0.568, 0.234); pair L = (0.262, 0.449); pair M = (0.373, 0.813); pair N = (0.731, 0.813); pair O = (0.851, 0.461); path[] f; f[0] = A--B--C--M--L--cycle; f[1] = C--D--E--N--M--cycle; f[2] = E--F--G--O--N--cycle; f[3] = G--H--I--K--O--cycle; f[4] = I--J--A--L--K--cycle; f[5] = K--L--M--N--O--cycle; draw(f[0]); axialshade(f[1], white, M, gray(0.5), (C+2*D)/3); draw(f[1]); filldraw(f[2], gray); filldraw(f[3], gray); axialshade(f[4], white, L, gray(0.7), J); draw(f[4]); draw(f[5]); [/asy] $\textbf{(A) } 125 \qquad \textbf{(B) } 250 \qquad \textbf{(C) } 405 \qquad \textbf{(D) } 640 \qquad \textbf{(E) } 810$

In triangle $ABC$ with $AB>BC$, $BM$ is a median and $BL$ is an angle bisector. The line through $M$ and parallel to $AB$ intersects $BL$ at point $D$, and the line through $L$ and parallel to $BC$ intersects $BM$ at point $E$. Prove that $ED$ is perpendicular to $BL$.

The quadrilateral $ABCD$ has perpendicular diagonals that meet at $O$. The incenters of triangles $ABC, BCD, CDA, DAB$ form a quadrilateral with perimeter $P$. Show that the sum of the inradii of the triangles $AOB, BOC, COD, DOA$ is less than or equal to $\frac{P} {2}$.

A right-angled triangle $ABC$ is given. Its catheus $AB$ is the base of a regular triangle $ADB$ lying in the exterior of $ABC$, and its hypotenuse $AC$ is the base of a regular triangle $AEC$ lying in the interior of $ABC$. Lines $DE$ and $AB$ meet at point $M$. The whole configuration except points $A$ and $B$ was erased. Restore the point $M$.

Let $a,b$ be the smaller sides of a right triangle. Let $c$ be the hypothenuse and $h$ be the altitude from the right angle. Fint the maximal value of $\frac{c+h}{a+b}$.

Is there a $1987$-gon with consecutive sides lengths $1, 2, 3,..., 1986, 1987$, in which you can fit a circle?