A line $ l$ is tangent to a circle $ S$ at $ A$. For any points $ B,C$ on $ l$ on opposite sides of $ A$, let the other tangents from $ B$ and $ C$ to $ S$ intersect at a point $ P$. If $ B,C$ vary on $ l$ so that the product $ AB \cdot AC$ is constant, find the locus of $ P$.

What is the largest number of parts into which $n$ planes can divide space? We assume that the set of planes is non-degenerate in the sense that any three planes intersect in one point and no four planes have a common point (and for n=2 it is necessary to require that the planes are not parallel).

Prove that among the vertices of any convex nonagon, three can be found forming an obtuse triangle, none of whose sides coincide with the sides of the nonagon. [i] Proposed by A. Yuran [/i]

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 $\angle AOB$.

For dessert, Melinda eats a spherical scoop of ice cream with diameter $2$ inches. She prefers to eat her ice cream in cube-like shapes, however. She has a special machine which, given a sphere placed in space, cuts it through the planes $x = n$, $y = n$, and $z = n$ for every integer $n$ (not necessarily positive). Melinda centers the scoop of ice cream uniformly at random inside the cube $0 \le x, y,z \le 1$, and then cuts it into pieces using her machine. What is the expected number of pieces she cuts the ice cream into?

Let $ABCD$ be a convex quadrilateral for which $DA = AB$ and $CA = CB$. Set $I_0 = C$ and $J_0 = D$, and for each nonnegative integer $n$, let $I_{n+1}$ and $J_{n+1}$ denote the incenters of $\triangle I_nAB$ and $\triangle J_nAB$, respectively. Suppose that \[ \angle DAC = 15^{\circ}, \quad \angle BAC = 65^{\circ} \quad \text{and} \quad \angle J_{2013}J_{2014}I_{2014} = \left( 90 + \frac{2k+1}{2^n} \right)^{\circ} \] for some nonnegative integers $n$ and $k$. Compute $n+k$. [i]Proposed by Evan Chen[/i]

Let $A_0B_0C_0$ be a triangle with area equal to $\sqrt 2$. We consider the excenters $A_1$,$B_1$ and $C_1$ then we consider the excenters ,say $A_2,B_2$ and $C_2$,of the triangle $A_1B_1C_1$. By continuing this procedure ,examine if it is possible to arrive to a triangle $A_nB_nC_n$ with all coordinates rational.

We want to cover a rectangular $5 \times 137$ with the following figures, prove that this is impossible. \[\text{Squars are the same and all are } \Huge{1 \times 1}\] [asy] import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); draw((2,4)--(0,4),linewidth(2pt)); draw((0,4)--(0,0),linewidth(2pt)); draw((0,0)--(2,0),linewidth(2pt)); draw((2,0)--(2,1),linewidth(2pt)); draw((2,1)--(0,1),linewidth(2pt)); draw((1,0)--(1,4),linewidth(2pt)); draw((2,4)--(2,3),linewidth(2pt)); draw((2,3)--(0,3),linewidth(2pt)); draw((0,2)--(1,2),linewidth(2pt)); label("(1)", (0.56,-1.54), SE*lsf); draw((4,2)--(4,1),linewidth(2pt)); draw((7,2)--(7,1),linewidth(2pt)); draw((4,2)--(7,2),linewidth(2pt)); draw((4,1)--(7,1),linewidth(2pt)); draw((6,0)--(6,3),linewidth(2pt)); draw((5,3)--(5,0),linewidth(2pt)); draw((5,0)--(6,0),linewidth(2pt)); draw((5,3)--(6,3),linewidth(2pt)); label("(2)", (5.13,-1.46), SE*lsf); draw((9,0)--(9,3),linewidth(2pt)); draw((10,3)--(10,0),linewidth(2pt)); draw((12,3)--(12,0),linewidth(2pt)); draw((11,0)--(11,3),linewidth(2pt)); draw((9,2)--(12,2),linewidth(2pt)); draw((12,1)--(9,1),linewidth(2pt)); draw((9,3)--(10,3),linewidth(2pt)); draw((11,3)--(12,3),linewidth(2pt)); draw((12,0)--(11,0),linewidth(2pt)); draw((9,0)--(10,0),linewidth(2pt)); label("(3)", (10.08,-1.48), SE*lsf); draw((14,1)--(17,1),linewidth(2pt)); draw((15,2)--(17,2),linewidth(2pt)); draw((15,2)--(15,0),linewidth(2pt)); draw((15,0)--(14,0)); draw((14,1)--(14,0),linewidth(2pt)); draw((16,2)--(16,0),linewidth(2pt)); label("(4)", (15.22,-1.5), SE*lsf); draw((14,0)--(16,0),linewidth(2pt)); draw((17,2)--(17,1),linewidth(2pt)); draw((19,3)--(19,0),linewidth(2pt)); draw((20,3)--(20,0),linewidth(2pt)); draw((20,3)--(19,3),linewidth(2pt)); draw((19,2)--(20,2),linewidth(2pt)); draw((19,1)--(20,1),linewidth(2pt)); draw((20,0)--(19,0),linewidth(2pt)); label("(5)", (19.11,-1.5), SE*lsf); dot((0,0),ds); dot((0,1),ds); dot((0,2),ds); dot((0,3),ds); dot((0,4),ds); dot((1,4),ds); dot((2,4),ds); dot((2,3),ds); dot((1,3),ds); dot((1,2),ds); dot((1,1),ds); dot((2,1),ds); dot((2,0),ds); dot((1,0),ds); dot((5,0),ds); dot((6,0),ds); dot((5,1),ds); dot((6,1),ds); dot((5,2),ds); dot((6,2),ds); dot((5,3),ds); dot((6,3),ds); dot((7,2),ds); dot((7,1),ds); dot((4,1),ds); dot((4,2),ds); dot((9,0),ds); dot((9,1),ds); dot((9,2),ds); dot((9,3),ds); dot((10,0),ds); dot((11,0),ds); dot((12,0),ds); dot((10,1),ds); dot((10,2),ds); dot((10,3),ds); dot((11,1),ds); dot((11,2),ds); dot((11,3),ds); dot((12,1),ds); dot((12,2),ds); dot((12,3),ds); dot((14,0),ds); dot((15,0),ds); dot((16,0),ds); dot((15,1),ds); dot((14,1),ds); dot((16,1),ds); dot((15,2),ds); dot((16,2),ds); dot((17,2),ds); dot((17,1),ds); dot((19,0),ds); dot((20,0),ds); dot((19,1),ds); dot((20,1),ds); dot((19,2),ds); dot((20,2),ds); dot((19,3),ds); dot((20,3),ds); clip((-0.41,-10.15)--(-0.41,8.08)--(21.25,8.08)--(21.25,-10.15)--cycle); [/asy]

Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. The sum of the squares of their ages is $\textbf{(A) }42\qquad\textbf{(B) }46\qquad\textbf{(C) }122\qquad\textbf{(D) }290\qquad \textbf{(E) }326$

Let $ABCD$ be a trapezoid with $AD \parallel BC$, $AD < BC$. Let $E$ be a point on the side $AB$ and $F$ be point on the side $CD$. The circle $(AEF)$ intersects the segment $AD$ again at $A_1$ and the circle $(CEF)$ intersects these segment $BC$ again at $C_1$. Prove that the lines $A_1 C_1$, $BD$ and $EF$ are concurrent. [i]Proposed by A. Kuznetsov[/i]

There was a tub on the plane, with its upper base greater that the lower one. The tub was overturned. Prove that the area of its visible shade did decrease. (The tub is a frustum of a right circular cone: its bases are two discs in parallel planes, such that their centers lie on a line perpendicular to these planes. The visible shade is the total shade besides the shade under the tub. Consider the sun rays as parallel.)

Are there convex $n$-gonal ($n \ge 4$) and triangular pyramids such that the four trihedral angles of the $n$-gonal pyramid are equal trihedral angles of a triangular pyramid? [hide=original wording] Существуют ли выпуклая n-угольная (n>= 4) и треугольная пирамиды такие, что четыре трехгранных угла n-угольной пирамиды равны трехгранным углам треугольной пирамиды?[/hide]

Restore a triangle $ABC$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.

Let $ABC$ be an equilateral triangle. On the small arc $AB{}$ of its circumcircle $\Omega$, consider the point $N{}$ such that the small arc $NB$ measures $30^\circ{}$. The perpendiculars from $N{}$ onto $AC$ and $AB$ intersect $\Omega$ again at $P{}$ and $Q{}$ respectively. Let $H_1,H_2$ and $H_3$ be the orthocenters of the triangles $NAB, QBC$ and $CAP$ respectively. [list=a] [*]Prove that the triangle $NPQ$ is equilateral. [*]Prove that the triangle $H_1H_2H_3$ is equilateral. [/list]

The isoelectric point of glycine is the pH at which it has zero charge. Its charge is $-\frac13$ at pH $3.55$, while its charge is $\frac12$ at pH $9.6$. Charge increases linearly with pH. What is the isoelectric point of glycine?

In an inscribed quadrilateral $ABCD$, we have $BC=CD$ but $AB\neq AD$. Points $I$ and $J$ are the incenters of triangles $ABC$ and $ACD$ respectively. Point $K$ was chosen on segment $AC$ so that $IK=JK$. Points $M$ and $N$ are the incenters of triangles $AIK$ and $AJK$. Prove that the perpendicular to $CD$ at $D$ and the perpendicular to $KI$ at $I$ intersect on the circumcircle of $MAN$.

Equiangular hexagon $ ABCDEF$ has side lengths $ AB \equal{} CD \equal{} EF \equal{} 1$ and $ BC \equal{} DE \equal{} FA \equal{} r$. The area of $ \triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $ r$? $ \textbf{(A)}\ \frac {4\sqrt {3}}{3} \qquad \textbf{(B)}\ \frac {10}{3} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ \frac {17}{4} \qquad \textbf{(E)}\ 6$

A unit square is cut into four pieces that can be arranged to make an isosceles triangle as shown below. What is the perimeter of the triangle? Express your answer in simplest radical form. [asy] filldraw((0, 3)--(-1, 3)--(-2, 2)--(-1, 1)--cycle,lightgrey); filldraw((0, 3)--(1, 3)--(2, 2)--(1, 1)--cycle,lightgrey); filldraw((0, 4)--(-1, 3)--(1, 3)--cycle,grey); draw((-1, 1)--(0,0)--(1, 1)); filldraw((4,1)--(3,2)--(2,0)--(3,0)--cycle,lightgrey); filldraw((4,1)--(5,2)--(6,0)--(5,0)--cycle,lightgrey); filldraw((4,1)--(3,0)--(5,0)--cycle,grey); draw((3,2)--(4,4)--(5,2)); [/asy]

In the triangle $ABC$, the heights $AA_1$ and $BB_1$ are drawn. On the side $AB$, points $M$ and $K$ are chosen so that $B_1K\parallel BC$ and $A_1 M\parallel AC$. Prove that the angle $AA_1K$ is equal to the angle $BB_1M$.

Let $A,B,C,D$ be four distinct points on a circle such that the lines $(AC)$ and $(BD)$ intersect at $E$, the lines $(AD)$ and $(BC)$ intersect at $F$ and such that $(AB)$ and $(CD)$ are not parallel. Prove that $C,D,E,F$ are on the same circle if, and only if, $(EF)\bot(AB)$.

Let $A_n$ be the number of ways to tile a $4 \times n$ rectangle using $2 \times 1$ tiles. Prove that $A_n$ is divisible by 2 if and only if $A_n$ is divisible by 3.

Let $ABC$ be a isosceles triangle, with $AB=AC$. A line $r$ that pass through the incenter $I$ of $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. Let $F$ and $G$ be points on $BC$ such that $BF=CE$ and $CG=BD$. Show that the angle $\angle FIG$ is constant when we vary the line $r$.

Consider a regular cube with side length $2$. Let $A$ and $B$ be $2$ vertices that are furthest apart. Construct a sequence of points on the surface of the cube $A_1$, $A_2$, $\ldots$, $A_k$ so that $A_1=A$, $A_k=B$ and for any $i = 1,\ldots, k-1$, the distance from $A_i$ to $A_{i+1}$ is $3$. Find the minimum value of $k$.

The incircle of triangle $ABC$ touches $BC$, $CA$, $AB$ at points $A_1$, $B_1$, $C_1$, respectively. The perpendicular from the incenter $I$ to the median from vertex $C$ meets the line $A_1B_1$ in point $K$. Prove that $CK$ is parallel to $AB$.

Given triangle $ABC$, the three altitudes intersect at point $H$. Determine all points $X$ on the side $BC$ so that the symmetric of $H$ wrt point $X$ lies on the circumcircle of triangle $ABC$.