Baron Munchausen plays billiards on a table with the shape of an equilateral triangle. He claims to have shot a ball from one of the sides of this table so that it passed through a certain point three times in three different directions and then returned to the original point on the side. Can that be true, assuming that the usual law of reflection holds? (Μ Evdokimov)

Two isosceles trapezoids are inscribed in a circle in such a way that each side of each trapezoid is parallel to a certain side of the other trapezoid . Prove that the diagonals of one trapezoid are equal to the diagonals of the other.

In the regular hexagon $ABCDEF$, a point $X$ was marked on the diagonal $AD$ such that $\angle AEX = 65^\circ$. What is the degree measure of the angle $\angle XCD$? [i]A.V.Smirnov, I.A.Efremov[/i]

Let $ C_1$ be a circle and $ P$ be a fixed point outside the circle $ C_1$. Quadrilateral $ ABCD$ lies on the circle $ C_1$ such that rays $ AB$ and $ CD$ intersect at $ P$. Let $ E$ be the intersection of $ AC$ and $ BD$. (a) Prove that the circumcircle of triangle $ ADE$ and the circumcircle of triangle $ BEC$ pass through a fixed point. (b) Find the the locus of point $ E$.

Let $ABC$ be a triangle with $AB<AC<BC$.Let $O$ be the center of it's circumcircle and $D$ be the center of minor arc $\overarc{AB}$.Line $AD$ intersects $BC$ at $E$ and the circumcircle of $BDE$ intersects $AB$ at $Z$ ,($Z\not=B$).The circumcircle of $ADZ$ intersects $AC$ at $H$ ,($H\not=A$),prove that $BE=AH$.

All edges of a tetrahedron $ABCD$ are equal. The tetrahedron $ABCD$ is inscribed in a sphere. $CC'$ and $DD'$ are diameters. Find the angle between the planes $ABC$' and $ACD'$. (A Zaslavskiy)

Two circles $\alpha$ and $\beta$ intersect at points $P$ and $Q$. The lines connecting a point $R$ on $\beta$ with $P$ and $Q$ intersect $\alpha$ again at $S$ and $T$ respectively. Prove that $ST$ is parallel to the line tangent to $\beta$ at $R$.

Let $ A_1A_2$ be the external tangent line to the nonintersecting cirlces $ \omega_1(O_1)$ and $ \omega_2(O_2)$,$ A_1\in\omega_1$,$ A_2\in\omega_2$.Points $ K$ is the midpoint of $ A_1A_2$.And $ KB_1$ and $ KB_2$ are tangent lines to $ \omega_1$ and $ \omega_2$,respectvely($ B_1\neq A_1$,$ B_2\neq A_2$).Lines $ A_1B_1$ and $ A_2B_2$ meet in point $ L$,and lines $ KL$ and $ O_1O_2$ meet in point $ P$. Prove that points $ B_1,B_2,P$ and $ L$ are concyclic.

A circle is inscribed in quadrilateral $ABCD,$ tangent to $\overline{AB}$ at $P$ and to $\overline{CD}$ at $Q.$ Given that $AP=19, PB=26, CQ=37,$ and $QD=23,$ find the square of the radius of the circle.

Given a tetrahedron $VABC$ with edges $VA$, $VB$ and $VC$ perpendicular any two of them. The sum of the lengths of the tetrahedron's edges is $3p$. Find the maximal volume of $VABC$.

[b][b]p1.[/b][/b] Find the perimeter of a regular hexagon with apothem $3$. [b]p2.[/b] Concentric circles of radius $1$ and r are drawn on a circular dartboard of radius $5$. The probability that a randomly thrown dart lands between the two circles is $0.12$. Find $r$. [b]p3.[/b] Find all ordered pairs of integers $(x, y)$ with $0 \le x \le 100$, $0 \le y \le 100$ satisfying $$xy = (x - 22) (y + 15) .$$ [b]p4.[/b] Points $A_1$,$A_2$,$...$,$A_{12}$ are evenly spaced around a circle of radius $1$, but not necessarily in order. Given that chords $A_1A_2$, $A_3A_4$, and $A_5A_6$ have length $2$ and chords $A_7A_8$ and $A_9A_{10}$ have length $2 sin (\pi / 12)$, find all possible lengths for chord $A_{11}A_{12}$. [b]p5.[/b] Let $a$ be the number of digits of $2^{1998}$, and let $b$ be the number of digits in $5^{1998}$. Find $a + b$. [b]p6.[/b] Find the volume of the solid in $R^3$ defined by the equations $$x^2 + y^2 \le 2$$ $$x + y + |z| \le 3.$$ [b]p7.[/b] Positive integer $n$ is such that $3n$ has $28$ positive divisors and $4n$ has $36$ positive divisors. Find the number of positive divisors of $n$. [b]p8.[/b] Define functions $f$ and $g$ by $f (x) = x +\sqrt{x}$ and $g (x) = x + 1/4$. Compute $$g(f(g(f(g(f(g(f(3)))))))).$$ (Your answer must be in the form $a + b \sqrt{ c}$ where $a$, $b$, and $c$ are rational.) [b]p9.[/b] Sequence $(a_1, a_2,...)$ is defined recursively by $a_1 = 0$, $a_2 = 100$, and $a_n = 2a_{n-1}-a_{n-2}-3$. Find the greatest term in the sequence $(a_1, a_2,...)$. [b]p10.[/b] Points $X = (3/5, 0)$ and $Y = (0, 4/5)$ are located on a Cartesian coordinate system. Consider all line segments which (like $\overline{XY}$ ) are of length 1 and have one endpoint on each axis. Find the coordinates of the unique point $P$ on $\overline{XY}$ such that none of these line segments (except $\overline{XY}$ itself) pass through $P$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let be a point $ P $ in the interior of a parallelogram $ ABCD $ such that $ \angle PAD=\angle PCD. $ Prove that the bisectors of $ \angle BAD $ and $ \angle BPD $ are parallel.

Show that for an arbitrary tetrahedron there are two planes such that the ratio of the areas of the projections of the tetrahedron onto the two planes is not less than $\sqrt2$.

Let $ABC$ be a triangle and $M$ be the midpoint of $AB$. If it is known that $\angle CAM + \angle MCB = 90^o$, show that triangle $ABC$ is isosceles or right.

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

In a right triangle, the point $O$ is the center of the circumcircle. Another circle of smaller radius centered at the point $O$ touches the larger leg and the altitude drawn from the top of the right angle. Find the acute angles of a right triangle and the ratio of the radii of the circumscribed and smaller circles.

The figure shows a $60^o$ angle in which are placed $2007$ numbered circles (only the first three are shown in the figure). The circles are numbered according to size. The circles are tangent to the sides of the angle and to each other as shown. Circle number one has radius $1$. Determine the radius of circle number $2007$. [img]https://1.bp.blogspot.com/-1bsLIXZpol4/Xzb-Nk6ospI/AAAAAAAAMWk/jrx1zVYKbNELTWlDQ3zL9qc_22b2IJF6QCLcBGAsYHQ/s0/2007%2BMohr%2Bp4.png[/img]

Let $ABC$ be a triangle with incenter $I$ and excenters $I_a$, $I_b$, and $I_c$ opposite $A$, $B$, and $C$, respectively. Let $D$ be an arbitrary point on the circumcircle of $\triangle{ABC}$ that does not lie on any of the lines $II_a$, $I_bI_c$, or $BC$. Suppose the circumcircles of $\triangle{DII_a}$ and $\triangle{DI_bI_c}$ intersect at two distinct points $D$ and $F$. If $E$ is the intersection of lines $DF$ and $BC$, prove that $\angle{BAD} = \angle{EAC}$. [i]Proposed by Zach Chroman[/i]

Let $ABC$ be a triangle in which $AB=AC$. Suppose the orthocentre of the triangle lies on the incircle. Find the ratio $\frac{AB}{BC}$.

In a right-angled triangle, $a$ and $b$ denote the lengths of the two catheti. A circle with radius $r$ has the center on the hypotenuse and touches both catheti. Show that $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$.

On the extension of chord $AB$ of a circle centroid at $O$ a point $X$ is taken and tangents $XC$ and $XD$ to the circle are drawn from it with $C$ and $D$ lying on the circle, let $E$ be the midpoint of the line segment $CD$. If $\angle OEB = 140^o$ then determine with proof the magnitude of $\angle AOB$.

Let $ABC$ be a right triangle with right angle at $ B$ and $\angle C=30^o$. If $M$ is midpoint of the hypotenuse and $I$ the incenter of the triangle, show that $ \angle IMB=15^o$.

Let $ABCD$ be a convex quadrilateral with the property that $AB$ extended and $CD$ extended intersect at a right angle. Prove that $AC\cdot BD>AD\cdot BC$.

If $R$ and $S$ are two rectangles with integer sides such that the perimeter of $R$ equals the area of $S$ and the perimeter of $S$ equals the area of $R$, then we call $R$ and $S$ a friendly pair of rectangles. Find all friendly pairs of rectangles.

There are five research labs on Mars. Is it always possible to divide Mars to five connected congruent regions such that each region contains exactly on research lab. [img]http://i37.tinypic.com/f2iq8g.png[/img]