Let $ABCD$ be a square with unit sides. Which interior point $P$ will the expression $\sqrt2 \cdot AP + BP + CP$ have a minimum value, and what is this minimum?

Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches and the area of triangle $CEF$ is $200$ square inches. Then the number of inches in $BE$ is: [asy] size(6cm); pair A = (0, 0), B = (1, 0), C = (1, 1), D = (0, 1), E = (1.3, 0), F = (0, 0.7); draw(A--B--C--D--cycle); draw(F--C--E--B); label("$A$", A, SW); label("$B$", B, S); label("$C$", C, N); label("$D$", D, NW); label("$E$", E, SE); label("$F$", F, W); //Credit to MSTang for the asymptote [/asy] $\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 20$

Let $O$ be the circumcenter of the triangle $ABC$. Points $D,E$ are on sides $AC,AB$ and points $P,Q,R,S$ are given in plane such that $P,C$ and $R,C$ are on different sides of $AB$ and pints $Q,B$ and $S,B$ are on different sides of $AC$ such that $R,S$ lie on circumcircle of $DAP,EAQ$ and $\triangle BCE \sim \triangle ADQ , \triangle CBD \sim \triangle AEP$(In that order), $\angle ARE=\angle ASD=\angle BAC$, If $RS\| PQ$ prove that $RE ,DS$ are concurrent on $AO$. [i]Proposed by Alireza Dadgarnia[/i]

For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

The circle $\gamma$ passing through the vertex $A$ of triangle $ABC$ intersects its sides $AB$ and $AC$ for the second time at points $X$ and $Y$, respectively. Also, the circle $\gamma$ intersects side $BC$ at points $D$ and $E$ so that $AD = AE$. Prove that the points $B, X, Y, C$ lie on the same circle. [i]Proposed by Mykhailo Shtandenko[/i]

Let $ABCD$ be an isosceles trapezoid, $AD=BC$, $AB \parallel CD$. The diagonals of the trapezoid intersect at the point $O$, and the point $M$ is the midpoint of the side $AD$. The circle circumscribed around the triangle $BCM$ intersects the side $AD$ at the point $K$. Prove that $OK \parallel AB$.

Let $ABC$ be an acute-angled triangle. In $ABC$, $AB \neq AB$, $K$ is the midpoint of the the median $AD$, $DE \perp AB$ at $E$, $DF \perp AC$ at $F$. The lines $KE$, $KF$ intersect the line $BC$ at $M$, $N$, respectively. The circumcenters of $\triangle DEM$, $\triangle DFN$ are $O_1, O_2$, respectively. Prove that $O_1 O_2 \parallel BC$.

Let $ABCD$ be a quadrilateral inscribed in a circle $k$. Let the lines $AC\cap BD=O$, $AD\cap BC=P$, and $AB\cap CD=Q$. Line $QO$ intersects $k$ in points $M$ and $N$. Prove that $PM$ and $PN$ are tangent to $k$.

In triangle $ABC$ points $B_1$ and $C_1$ are on $AB$ and $AC$ respectively and $P{}$ is a point on the segment $B_1C_1$. Find the greatest possible value of $\frac{\min\{S(BPB_1),S(CPC_1)\}}{S(ABC)}$, where $S(XYZ)$ is the area o the triangle $ABC$.

Let $L_1$ and $L_2$ be two parallel lines and $L_3$ a line perpendicular to $L_1$ and $L_2$ at $H$ and $P$, respectively. Points $Q$ and $R$ lie on $L_1$ such that $QR = PR$ ($Q \ne H$). Let $d$ be the diameter of the circle inscribed in the triangle $PQR$. Point $T$ lies $L_2$ in the same semiplane as $Q$ with respect to line $L_3$ such that $\frac{1}{TH}= \frac{1}{d}- \frac{1}{PH}$ . Let $X$ be the intersection point of $PQ$ and $TH$. Find the locus of the points $X$ as $Q$ varies on $L_1$.

A pentagon has vertices labelled $A, B, C, D, E$ in that order counterclockwise, such that $AB$, $ED$ are parallel and $\angle EAB = \angle ABD = \angle ACD = \angle CDA$. Furthermore, suppose that$ AB = 8$, $AC = 12$, $AE = 10$. If the area of triangle $CDE$ can be expressed as $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are integers so that $b$ is square free, and $a, c$ are relatively prime, find $a + b + c$.

In an acute traingle $ABC$ with $AB< BC$ let $BH_b$ be its altitude, and let $O$ be the circumcenter. A line through $H_b$ parallel to $CO$ meets $BO$ at $X$. Prove that $X$ and the midpoints of $AB$ and $AC$ are collinear.

Squares $CAKL$ and $CBMN$ are constructed on the sides of acute-angled triangle $ABC$, outside of the triangle. Line $CN$ intersects line segment $AK$ at $X$, while line $CL$ intersects line segment $BM$ at $Y$. Point $P$, lying inside triangle $ABC$, is an intersection of the circumcircles of triangles $KXN$ and $LYM$. Point $S$ is the midpoint of $AB$. Prove that angle $\angle ACS=\angle BCP$.

In acute triangle $ABC (AB>AC)$, $M$ is the midpoint of minor arc $BC$, $O$ is the circumcenter of $(ABC)$ and $AK$ is its diameter. The line parallel to $AM$ through $O$ meets segment $AB$ at $D$, and $CA$ extended at $E$. Lines $BM$ and $CK$ meet at $P$, lines $BK$ and $CM$ meet at $Q$. Prove that $\angle OPB+\angle OEB =\angle OQC+\angle ODC$.

In convex hexagon $ABCDEF$, we know that $\angle BCA = \angle DEC = \angle AFB = \angle CBD = \angle EDF.$ Prove that $AB = CD = EF.$

Find all quadruples of real numbers such that each of them is equal to the product of some two other numbers in the quadruple.

Evaluate $\int_{\frac 12}^1 \sqrt{1-x^2}\ dx.$

A uniform disk, a thin hoop, and a uniform sphere, all with the same mass and same outer radius, are each free to rotate about a fixed axis through its center. Assume the hoop is connected to the rotation axis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the rims, as shown. Rank the objects according to their kinetic energies after a given time $t$, from least to greatest. [asy] size(225); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); filldraw(circle((0,0),1),gray(.7)); draw((0,-1)--(2,-1),EndArrow); label("$\vec{F}$",(1, -1),S); label("Disk",(-1,0),W); filldraw(circle((5,0),1),gray(.7)); filldraw(circle((5,0),0.75),white); draw((5,-1)--(7,-1),EndArrow); label("$\vec{F}$",(6, -1),S); label("Hoop",(6,0),E); filldraw(circle((10,0),1),gray(.5)); draw((10,-1)--(12,-1),EndArrow); label("$\vec{F}$",(11, -1),S); label("Sphere",(11,0),E); [/asy] $ \textbf{(A)} \ \text{disk, hoop, sphere}$ $\textbf{(B)}\ \text{sphere, disk, hoop}$ $\textbf{(C)}\ \text{hoop, sphere, disk}$ $\textbf{(D)}\ \text{disk, sphere, hoop}$ $\textbf{(E)}\ \text{hoop, disk, sphere} $

The student drew a triangle $ABC$ on the board, in which $AB>BC$. On the side $AB$ is taken point $D$ such that $BD = AC$. Let points $E$ and $F$ be the midpoints of the segments $AD$ and $BC$ respectively. Then the whole picture was erased, leaving only dots $E$ and $F$. Restore triangle $ABC$.

Two circles intersect at points $M$ and $N$. Let $A$ be a point on the first circle, distinct from $M,N$. The lines $AM$ and $AN$ meet the second circle again at $B$ and $C$, respectively. Prove that the tangent to the first circle at $A$ is parallel to $BC$.

Points $A$ and $B$ lie on circle $\omega$. Point $P$ lies on the extension of segment $AB$ past $B$. Line $\ell$ passes through $P$ and is tangent to $\omega$. The tangents to $\omega$ at points $A$ and $B$ intersect $\ell$ at points $D$ and $C$ respectively. Given that $AB=7$, $BC=2$, and $AD=3$, compute $BP$.

Let $B$ be a point on a circle $k_1, A \ne B$ be a point on the tangent to the circle at $B$, and $C$ a point not lying on $k_1$ for which the segment $AC$ meets $k_1$ at two distinct points. Circle $k_2$ is tangent to line $AC$ at $C$ and to $k_1$ at point $D$, and does not lie in the same half-plane as $B$. Prove that the circumcenter of triangle $BCD$ lies on the circumcircle of $\vartriangle ABC$

Answer the following questions. (1) By setting $ x\plus{}\sqrt{x^2\minus{}1}\equal{}t$, find the indefinite integral $ \int \sqrt{x^2\minus{}1}\ dx$. (2) Given two points $ P(p,\ q)\ (p>1,\ q>0)$ and $ A(1,\ 0)$ on the curve $ x^2\minus{}y^2\equal{}1$. Find the area $ S$ of the figure bounded by two lines $ OA,\ OP$ and the curve in terms of $ p$. (3) Let $ S\equal{}\frac{\theta}{2}$. Express $ p,\ q$ in terms of $ \theta$.

Prove that a bounded figure cannot have more than one center of symmetry.

Two triangles $ABC$ and $A'B'C'$ are given. The lines $AB$ and $A'B'$ meet at $C_1$ and the lines parallel to them and passing through $C$ and $C'$ meet at $C_2$. The points $A_1,A_2$, $B_1,B_2$ are defined similarly. Prove that $A_1A_2,B_1B_2,C_1C_1$ are either parallel or concurrent.