In regular triangular pyramid $P-ABC$, $PO$ is its height, $M$ is the midpoint of $PO$. Draw the plane that passes $AM$ and parallel to $BC$. Now the triangular pyramid is divided into two parts. Find the ratio of their volume.

Squares $S_1$ and $S_2$ are inscribed in right triangle $ABC$, as shown in the figures below. Find $AC + CB$ if area$(S_1) = 441$ and area$(S_2) = 440$. [asy] size(250); real a=15, b=5; real x=a*b/(a+b), y=a/((a^2+b^2)/(a*b)+1); pair A=(0,b), B=(a,0), C=origin, X=(y,0), Y=(0, y*b/a), Z=foot(Y, A, B), W=foot(X, A, B); draw(A--B--C--cycle); draw(W--X--Y--Z); draw(shift(-(a+b), 0)*(A--B--C--cycle^^(x,0)--(x,x)--(0,x))); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$A$", (A.x-a-b,A.y), dir(point--A)); label("$B$", (B.x-a-b,B.y), dir(point--B)); label("$C$", (C.x-a-b,C.y), dir(point--C)); label("$S_1$", (x/2-a-b, x/2)); label("$S_2$", intersectionpoint(W--Y, X--Z)); dot(A^^B^^C^^(-a-b,0)^^(-b,0)^^(-a-b,b));[/asy]

Forty slips are placed into a hat, each bearing a number $ 1$, $ 2$, $ 3$, $ 4$, $ 5$, $ 6$, $ 7$, $ 8$, $ 9$, or $ 10$, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let $ p$ be the probability that all four slips bear the same number. Let $ q$ be the probability that two of the slips bear a number $ a$ and the other two bear a number $ b\not\equal{} a$. What is the value of $ q/p$? $ \textbf{(A)}\ 162\qquad \textbf{(B)}\ 180\qquad \textbf{(C)}\ 324\qquad \textbf{(D)}\ 360\qquad \textbf{(E)}\ 720$

In $\triangle ABC$, medians $\overline{AD}$ and $\overline{CE}$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC?$ [asy] unitsize(75); pathpen = black; pointpen=black; pair A = MP("A", D((0,0)), dir(200)); pair B = MP("B", D((2,0)), dir(-20)); pair C = MP("C", D((1/2,1)), dir(100)); pair D = MP("D", D(midpoint(B--C)), dir(30)); pair E = MP("E", D(midpoint(A--B)), dir(-90)); pair P = MP("P", D(IP(A--D, C--E)), dir(150)*2.013); draw(A--B--C--cycle); draw(A--D--E--C); [/asy] $\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 13.5 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 14.5 \qquad \textbf{(E)}\ 15 $

Given triangle $ABC$ and point $P$ on the circumcircle of triangle $ABC$. Suppose the line $CP$ intersects line $AB$ at point $E$ and line $BP$ intersect line $AC$ at point $F$. Suppose also the perpendicular bisector of $AB$ intersects $AC$ at point $K$ and the perpendicular bisector of $AC$ intersects $AB$ at point $J$. Prove that $$\left( \frac{CE}{BF}\right)^2= \frac{AJ \cdot JE }{ AK \cdot KF}$$

A wheel with a rubber tire has an outside diameter of $ 25$ in. When the radius has been decreased a quarter of an inch, the number of revolutions in one mile will: $ \textbf{(A)}\ \text{be increased about }2\% \qquad\textbf{(B)}\ \text{be increased about }1\%$ $ \textbf{(C)}\ \text{be increased about }20\% \qquad\textbf{(D)}\ \text{be increased about }\frac {1}{2}\% \qquad\textbf{(E)}\ \text{remain the same}$

Let $ABC$ be a triangle and $I$ its incentre. Let $\varrho_1$ and $\varrho_2$ be the inradii of triangles $IAB$ and $IAC$ respectively. (a) Show that there exists a function $f: ( 0, \pi ) \mapsto \mathbb{R}$ such that \[ \frac{ \varrho_1}{ \varrho_2} = \frac{f(C)}{f(B)} \] where $B = \angle ABC$ and $C = \angle BCA$ (b) Prove that \[ 2 ( \sqrt{2} -1 ) < \frac{ \varrho_1} { \varrho_2} < \frac{ 1 + \sqrt{2}}{2} \]

Let $ABC$ be a right triangle. It is known that there are points $D$ on the side $AC$ and $E$ on the side $BC$ such that $AB = AD = BE$ and $BD$ is perpendicular to $DE$. Calculate the ratios $\frac{AB}{BC}$ and $\frac{BC}{CA}$.

In a triangle $ABC$, the median $AD$ divides $\angle{BAC}$ in the ratio $1:2$. Extend $AD$ to $E$ such that $EB$ is perpendicular $AB$. Given that $BE=3,BA=4$, find the integer nearest to $BC^2$.

Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\frac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game? $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{2}{3}\qquad\textbf{(D) }\frac{3}{4}\qquad\textbf{(E) }\frac{4}{5}$

Let $M, N$ be points inside the angle $\angle BAC$ usch that $\angle MAB\equiv \angle NAC$. If $M_{1}, M_{2}$ and $N_{1}, N_{2}$ are the projections of $M$ and $N$ on $AB, AC$ respectively then prove that $M, N$ and $P$ the intersection of $M_{1}N_{2}$ with $N_{1}M_{2}$ are collinear.

Let $k$ be a circle with centre $M$ and let $AB$ be a diameter of $k$. Furthermore, let $C$ be a point on $k$ such that $AC = AM$. Let $D$ be the point on the line $AC$ such that $CD = AB$ and $C$ lies between $A$ and $D$. Let $E$ be the second intersection of the circumcircle of $BCD$ with line $AB$ and $F$ be the intersection of the lines $ED$ and $BC$. The line $AF$ cuts the segment $BD$ in $X$. Determine the ratio $BX/XD$.

Given six real numbers $a$, $b$, $c$, $x$, $y$, $z$ such that $0 < b-c < a < b+c$ and $ax + by + cz = 0$. What is the sign of the sum $ayz + bzx + cxy$ ?

Let be two natural numbers $ m,n\ge 2, $ two increasing finite sequences of real numbers $ \left( a_i \right)_{1\le i\le n} ,\left( b_j \right)_{1\le j\le m} , $ and the set $$ \left\{ a_i+b_j| 1\le i\le n,1\le j\le m \right\} . $$ Show that the set above has $ n+m-1 $ elements if and only if the two sequences above are arithmetic progressions and these have the same ratio.

Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA=5$ and $CD=2$. Point $E$ lies on side $CA$ such that $\angle DEF = 60^\circ$. The area of triangle $DEF$ is $14\sqrt{3}$. The two possible values of the length of side $AB$ are $p \pm q\sqrt{r}$, where $p$ and $q$ are rational, and $r$ is an integer not divisible by the square of a prime. Find $r$.

Acute triangle $ABP$, where $AB > BP$, has altitudes $BH$, $PQ$, and $AS$. Let $C$ denote the intersection of lines $QS$ and $AP$, and let $L$ denote the intersection of lines $HS$ and $BC$. If $HS = SL$ and $HL$ is perpendicular to $BC$, find the value of $\frac{SL}{SC}$.

A cyclic pentagon is given. Prove that the ratio of its area to the sum of the diagonals is not greater than the quarter of the circumradius.

An isosceles triangle with $AB=AC$ has an inscribed circle $O$, which touches its sides $BC,CA,AB$ at $K,L,M$ respectively. The lines $OL$ and $KM$ intersect at $N$; the lines $BN$ and $CA$ intersect at $Q$. Let $P$ be the foot of the perpendicular from $A$ on $BQ$. Suppose that $BP=AP+2\cdot PQ$. Then, what values can the ratio $\frac{AB}{BC}$ assume?

In a triangle $ABC$, the bisector of the largest angle $\angle A$ meets $BC$ at point $D$. Let $E$ and $F$ be the feet of perpendiculars from $D$ to $AC$ and $AB$, respectively. Let $R$ denote the ratio between the areas of triangles $DEB$ and $DFC$. (a) Prove that, for every real number $r > 0$, one can construct a triangle ABC for which $R$ is equal to $r$. (b) Prove that if $R$ is irrational, then at least one side length of $\vartriangle ABC$ is irrational. (c) Give an example of a triangle $ABC$ with exactly two sides of irrational length, but with rational $R$.

Four congruent spheres are stacked so that each is tangent to the other three. A larger sphere, $R$, contains the four congruent spheres so that all four are internally tangent to $R$. A smaller sphere, $S$, sits in the space between the four congruent spheres so that all four are externally tangent to $S$. The ratio of the surface area of $R$ to the surface area of $S$ can be written $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.

In the figure, $ ABCD$ is a $ 2\times 2$ square, $ E$ is the midpoint of $ \overline{AD}$, and $ F$ is on $ \overline{BE}$. If $ \overline{CF}$ is perpendicular to $ \overline{BE}$, then the area of quadrilateral $ CDEF$ is [asy]defaultpen(linewidth(.8pt)); dotfactor=4; pair A = (0,2); pair B = origin; pair C = (2,0); pair D = (2,2); pair E = midpoint(A--D); pair F = foot(C,B,E); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F); label("$A$",A,N);label("$B$",B,S);label("$C$",C,S);label("$D$",D,N);label("$E$",E,N);label("$F$",F,NW); draw(A--B--C--D--cycle); draw(B--E); draw(C--F); draw(rightanglemark(B,F,C,4));[/asy]$ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3 \minus{} \frac {\sqrt {3}}{2}\qquad \textbf{(C)}\ \frac {11}{5}\qquad \textbf{(D)}\ \sqrt {5}\qquad \textbf{(E)}\ \frac {9}{4}$

In triangle $ABC,$ angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D.$ The lengths of the sides of $\triangle ABC$ are integers, $BD=29^3,$ and $\cos B=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Let $\varphi$ be the positive solution to the equation $$x^2=x+1.$$ For $n\ge 0$, let $a_n$ be the unique integer such that $\varphi^n-a_n\varphi$ is also an integer. Compute $$\sum_{n=0}^{10}a_n.$$

An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black? [asy] unitsize(36); fill((0,0)--(2,0)--(1,sqrt(3))--cycle,gray); draw((0,0)--(2,0)--(1,sqrt(3))--cycle,linewidth(1)); fill((4,0)--(6,0)--(5,sqrt(3))--cycle,gray); fill((5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--cycle,white); draw((5,sqrt(3))--(4,0)--(5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--(5,0)--(6,0)--cycle,linewidth(1)); fill((8,0)--(10,0)--(9,sqrt(3))--cycle,gray); fill((9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--cycle,white); fill((17/2,0)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--cycle,white); fill((9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--cycle,white); fill((19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--cycle,white); draw((9,sqrt(3))--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--(9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--(17/2,0)--(33/4,sqrt(3)/4)--(8,0)--(9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--(9,0)--(19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--(19/2,0)--(10,0)--cycle,linewidth(1)); label("Change 1",(3,3*sqrt(3)/4),N); label("$\Longrightarrow $",(3,5*sqrt(3)/8),S); label("Change 2",(7,3*sqrt(3)/4),N); label("$\Longrightarrow $",(7,5*sqrt(3)/8),S); [/asy] $\text{(A)}\ \frac{1}{1024} \qquad \text{(B)}\ \frac{15}{64} \qquad \text{(C)}\ \frac{243}{1024} \qquad \text{(D)}\ \frac{1}{4} \qquad \text{(E)}\ \frac{81}{256}$

$ABC$ is an equilateral triangle. $D$ and$ D'$ are points on opposite sides of the plane $ABC$ such that the two tetrahedra $ABCD$ and $ABCD'$ are congruent (but not necessarily with the vertices in that order). If the polyhedron with the five vertices $A, B, C, D, D'$ is such that the angle between any two adjacent faces is the same, find $DD'/AB$ .