A rectangular box has width $12$ inches, length $16$ inches, and height $\tfrac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$.

Suppose that $a$ and $b$ are two distinct positive real numbers such that $\lfloor na\rfloor$ divides $\lfloor nb\rfloor$ for any positive integer $n$. Prove that $a$ and $b$ are positive integers.

The parallel sides of a trapezoid are $ 3$ and $ 9$. The non-parallel sides are $ 4$ and $ 6$. A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm); pair A = origin; pair B = (2.25,0); pair C = (2,1); pair D = (1,1); pair E = waypoint(A--D,0.25); pair F = waypoint(B--C,0.25); draw(A--B--C--D--cycle); draw(E--F); label("6",midpoint(A--D),NW); label("3",midpoint(C--D),N); label("4",midpoint(C--B),NE); label("9",midpoint(A--B),S);[/asy]$ \textbf{(A)}\ 4: 3\qquad \textbf{(B)}\ 3: 2\qquad \textbf{(C)}\ 4: 1\qquad \textbf{(D)}\ 3: 1\qquad \textbf{(E)}\ 6: 1$

A square and a circle have equal perimeters. The ratio of the area of the circle to the area of the square is: $\textbf{(A) }\frac{4}{\pi}\qquad\textbf{(B) }\frac{\pi}{\sqrt{2}}\qquad\textbf{(C) }\frac{4}{1}\qquad\textbf{(D) }\frac{\sqrt{2}}{\pi}\qquad \textbf{(E) }\frac{\pi}{4}$

If $a_1,a_2,a_3,\dots$ is a sequence of positive numbers such that $a_{n+2}=a_na_{n+1}$ for all positive integers $n$, then the sequence $a_1,a_2,a_3,\dots$ is a geometric progression $\textbf{(A) }\text{for all positive values of }a_1\text{ and }a_2\qquad$ $\textbf{(B) }\text{if and only if }a_1=a_2\qquad$ $\textbf{(C) }\text{if and only if }a_1=1\qquad$ $\textbf{(D) }\text{if and only if }a_2=1\qquad $ $\textbf{(E) }\text{if and only if }a_1=a_2=1$

The lines $AD , BE$ and $CF$ intersect in $S$ within a triangle $ABC$ . It is given that $AS: DS = 3: 2$ and $BS: ES = 4: 3$ . Determine the ratio $CS: FS$ . [asy] unitsize (1 cm); pair A, B, C, D, E, F, S; A = (0,0); B = (5,0); C = (1,4); S = (14*A + 15*B + 6*C)/35; D = extension(A,S,B,C); E = extension(B,S,C,A); F = extension(C,S,A,B); draw(A--B--C--cycle); draw(A--D); draw(B--E); draw(C--F); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, N); dot("$D$", D, NE); dot("$E$", E, W); dot("$F$", F, dir(270)); dot("$S$", S, NE); [/asy]

Let $E$ be the midpoint of side $[AB]$ of square $ABCD$. Let the circle through $B$ with center $A$ and segment $[EC]$ meet at $F$. What is $|EF|/|FC|$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ \dfrac{3}{2} \qquad\textbf{(C)}\ \sqrt{5}-1 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \sqrt{3} $

For positive integers $n$, let $f_2(n)$ denote the number of divisors of $n$ which are perfect squares, and $f_3(n)$ denotes the number of positive divisors which are perfect cubes. Prove that for each positive integer $k$ there exists a positive integer $n$ for which $\frac{f_2(n)}{f_3(n)}=k$.

A [i]kite[/i] is a quadrilateral whose diagonals are perpendicular. Let kite $ABCD$ be such that $\angle B = \angle D = 90^\circ$. Let $M$ and $N$ be the points of tangency of the incircle of $ABCD$ to $AB$ and $BC$ respectively. Let $\omega$ be the circle centered at $C$ and tangent to $AB$ and $AD$. Construct another kite $AB^\prime C^\prime D^\prime$ that is similar to $ABCD$ and whose incircle is $\omega$. Let $N^\prime$ be the point of tangency of $B^\prime C^\prime$ to $\omega$. If $MN^\prime \parallel AC$, then what is the ratio of $AB:BC$?

Points $K$ and $L$ are chosen on the sides $AB$ and $AC$ of an equilateral triangle $ABC$ such that $BK = AL$. Segments $BL$ and $CK$ intersect at $P$. Determine the ratio $\frac{AK}{KB}$ for which the segments $AP$ and $CK$ are perpendicular.

Find all integers of the form $\frac{m-6n}{m+2n}$ where $m,n$ are nonzero rational numbers satisfying $m^3=(27n^2+1)(m+2n)$.

In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$ say $C_1$, and then the midpoint of $A_1D$ which is $A_2$. He continues in this fashion, indefinitely. The total length of his path if $AB=5$ and $BC=12$ is of the form $a + b\sqrt{c}$. Find $\displaystyle\frac{abc}{4}$.

Let $C',B',A'$ be points respectively on sides $AB,AC,BC$ of $\triangle ABC$ satisfying $ \tfrac{AB'}{B'C}= \tfrac{BC'}{C'A}=\tfrac{CA'}{A'B}=k$. Prove that the ratio of the area of the triangle formed by the lines $AA',BB',CC'$ over the area of $\triangle ABC$ is $\tfrac{(k-1)^2}{(k^2+k+1)}$.

The sequence $a_1, a_2, \ldots$ is geometric with $a_1=a$ and common ratio $r$, where $a$ and $r$ are positive integers. Given that $\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r)$.

Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure at right). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$. [asy] size(200); pair A=origin, B=(7,0), C=(3.2,15), D=midpoint(B--C), F=(3,0), P=intersectionpoint(C--F, A--D), ex=B+40*dir(B--P), E=intersectionpoint(B--ex, A--C); draw(A--B--C--A--D^^C--F^^B--E); pair point=P; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$P$", P, dir(0));[/asy]

A homogeneous solid body is made by joining a base of a circular cylinder of height $h$ and radius $r,$ and the base of a hemisphere of radius $r.$ This body is placed with the hemispherical end on a horizontal table, with the axis of the cylinder in a vertical position, and then slightly oscillated. It is intuitively evident that if $r$ is large as compared to $h$, the equilibrium will be stable; but if $r$ is small compared to $h$, the equilibrium will be unstable. What is the critical value of the ratio $r\slash h$ which enables the body to rest in neutral equilibrium in any position?

Let $\omega$ be a circle with center $O$ and diameter $AB$. A circle with center at $B$ intersects $\omega$ at C and $AB$ at $D$. The line $CD$ intersects $\omega$ at a point $E$ ($E\ne C$). The intersection of lines $OE$ and $BC$ is $F$. (a) Prove that triangle $OBF$ is isosceles. (b) If $D$ is the midpoint of $OB$, find the value of the ratio $\frac{FB}{BD}$.

Alex has a piece of cheese. He chooses a positive number a and cuts the piece into several pieces one by one. Every time he choses a piece and cuts it in the same ratio $1 : a$. His goal is to divide the cheese into two piles of equal masses. Can he do it if $(a) a$ is irrational? $(b) a$ is rational, $a \neq 1?$

Points $A, B, C$ and $D$ are located on line $\ell$ so that $\frac{AB}{BC}=\frac{AC}{CD}=\lambda $. A certain circle is tangent to line $\ell$ at point $C$. A line is drawn through $A$ that intersects this circle at points $M$ and $N$ such that the bisector perpendiculars to segments $BM$ and $DN$ intersect at point $Q$ on line $\ell$ . In what ratio does point $Q$ divide segment $AD$?

Let $ABC$ be a triangle with circumcenter $O$ and let $X$, $Y$, $Z$ be the midpoints of arcs $BAC$, $ABC$, $ACB$ on its circumcircle. Let $G$ and $I$ denote the centroid of $\triangle XYZ$ and the incenter of $\triangle ABC$. Given that $AB = 13$, $BC = 14$, $CA = 15$, and $\frac {GO}{GI} = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Evan Chen[/i]

In a plane three points $P,Q,R,$ not on a line, are given. Let $k, l, m$ be positive numbers. Construct a triangle $ABC$ whose sides pass through $P, Q,$ and $R$ such that $P$ divides the segment $AB$ in the ratio $1 : k$, $Q$ divides the segment $BC$ in the ratio $1 : l$, and $R$ divides the segment $CA$ in the ratio $1 : m.$

Prove that every positive integer can be written as a finite sum of distinct integral powers of the golden ratio.

Two circles $k_1$ and $k_2$ intersect at two points $A$ and $B$. Some line through the point $B$ meets the circle $k_1$ at a point $C$ (apart from $B$), and the circle $k_2$ at a point $E$ (apart from $B$). Another line through the point $B$ meets the circle $k_1$ at a point $D$ (apart from $B$), and the circle $k_2$ at a point $F$ (apart from $B$). Assume that the point $B$ lies between the points $C$ and $E$ and between the points $D$ and $F$. Finally, let $M$ and $N$ be the midpoints of the segments $CE$ and $DF$. Prove that the triangles $ACD$, $AEF$ and $AMN$ are similar to each other.

Given a non-isoceles triangle $ABC$ inscribes a circle $(O,R)$ (center $O$, radius $R$). Consider a varying line $l$ such that $l\perp OA$ and $l$ always intersects the rays $AB,AC$ and these intersectional points are called $M,N$. Suppose that the lines $BN$ and $CM$ intersect, and if the intersectional point is called $K$ then the lines $AK$ and $BC$ intersect. $1$, Assume that $P$ is the intersectional point of $AK$ and $BC$. Show that the circumcircle of the triangle $MNP$ is always through a fixed point. $2$, Assume that $H$ is the orthocentre of the triangle $AMN$. Denote $BC=a$, and $d$ is the distance between $A$ and the line $HK$. Prove that $d\leq\sqrt{4R^2-a^2}$ and the equality occurs iff the line $l$ is through the intersectional point of two lines $AO$ and $BC$.

As illustrated below, we can dissect every triangle $ABC$ into four pieces so that piece 1 is a triangle similar to the original triangle, while the other three pieces can be assembled into a triangle also similar to the original triangle. Determine the ratios of the sizes of the three triangles and verify that the construction works. [asy] import olympiad;size(350);defaultpen(linewidth(0.7)+fontsize(10)); path p=origin--(13,0)--(9,8)--cycle; path p2=rotate(180)*p, p3=shift(-26,0)*scale(2)*p, p4=shift(-27,-24)*scale(3)*p, p1=shift(-53,-24)*scale(4)*p; pair A=(-53,-24), B=(-8,16), C=(12,-24), D=(-17,8), E=(-1,-24), F=origin, G=(-13,0), H=(-9,-8); label("1", centroid(A,D,E)); label("2", centroid(F,G,H)); label("3", (-10,6)); label("4", (0,-15)); draw(p2^^p3^^p4); filldraw(p1, white, black); pair point = centroid(F,G,H); label("$\mathbf{A}$", A, dir(point--A)); label("$\mathbf{B}$", B, dir(point--B)); label("$\mathbf{C}$", C, dir(point--C)); label("$\mathbf{D}$", D, dir(point--D)); label("$\mathbf{E}$", E, dir(point--E)); label("$\mathbf{F}$", F, dir(point--F)); label("$\mathbf{G}$", G, dir(point--G)); label("$\mathbf{H}$", H, dir(point--H)); real x=90; draw(shift(x)*p1); label("1", centroid(shift(x)*A,shift(x)*D,shift(x)*E)); draw(shift(130,0)*p4); draw(shift(130,0)*shift(-27,-24)*p); draw(shift(130,0)*shift(-1,-24)*p3); label("2", shift(130,0)*shift(-27,-24)*centroid(F,(9,8),(13,0))); label("3", shift(130,0)*shift(-1,-24)*(-10,6)); label("4", shift(130,0)*(0,-15)); label("Piece 2 rotated $180^\circ$", (130,10));[/asy]