Johann and Nicole are playing a game on the coordinate plane. First, Johann draws any polygon $\mathcal{S}$ and then Nicole can shift $\mathcal{S}$ to wherever she wants. Johann wins if there exists a point with coordinates $(x, y)$ in the interior of $\mathcal{S}$, where $x$ and $y$ are coprime integers. Otherwise, Nicole wins. Determine who has a winning strategy.

Joining $15^3 = 3375$ cubes of $1$ cm$^3$, bodies with a volume of $3375$ cm$^3$ can be built. Indicate how two bodies $A$ and $B$ are constructed with $3375$ cubes each and such that the lateral surface of $B$ is $10$ times the lateral surface of $A$.

For $n$ distinct positive integers all their $n(n-1)/2$ pairwise sums are considered. For each of these sums Ivan has written on the board the number of original integers which are less than that sum and divide it. What is the maximum possible sum of the numbers written by Ivan?

Quadrilateral $ABCD$ is inscribed in a circle such that the midpoints of its sides also lie on a (different) circle. Let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{CD}$ respectively, and let $P$ be the foot of the perpendicular from the intersection of $\overline{AC}$ and $\overline{BD}$ onto $\overline{BC}$. If the side lengths of $ABCD$ are $1$, $3$, $\sqrt 2$, and $2\sqrt 2$ in some order, compute the greatest possible area of the circumcircle of triangle $MNP$. [i]Proposed by Connor Gordon[/i]

Compute the sum of all real solutions $\alpha$ (in radians) to the equation \[ |\sin\alpha|=\left\lfloor \frac{\alpha}{20} \right\rfloor. \]

A train car held $6000$ pounds of mud which was $88$ percent water. Then the train car sat in the sun, and some of the water evaporated so that now the mud is only $82$ percent water. How many pounds does the mud weigh now?

In the following diagram, let $ABCD$ be a square and let $M,N,P$ and $Q$ be the midpoints of its sides. Prove that \[S_{A'B'C'D'} = \frac 15 S_{ABCD}.\] [asy] import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttzz = rgb(0,0.2,0.6); pen qqzzff = rgb(0,0.6,1); draw((0,4)--(4,4),qqttzz+linewidth(1.6pt)); draw((4,4)--(4,0),qqttzz+linewidth(1.6pt)); draw((4,0)--(0,0),qqttzz+linewidth(1.6pt)); draw((0,0)--(0,4),qqttzz+linewidth(1.6pt)); draw((0,4)--(2,0),qqzzff+linewidth(1.2pt)); draw((2,4)--(4,0),qqzzff+linewidth(1.2pt)); draw((0,2)--(4,4),qqzzff+linewidth(1.2pt)); draw((0,0)--(4,2),qqzzff+linewidth(1.2pt)); dot((0,4),ds); label("$A$", (0.07,4.12), NE*lsf); dot((0,0),ds); label("$D$", (-0.27,-0.37), NE*lsf); dot((4,0),ds); label("$C$", (4.14,-0.39), NE*lsf); dot((4,4),ds); label("$B$", (4.08,4.12), NE*lsf); dot((2,4),ds); label("$M$", (2.08,4.12), NE*lsf); dot((4,2),ds); label("$N$", (4.2,1.98), NE*lsf); dot((2,0),ds); label("$P$", (1.99,-0.49), NE*lsf); dot((0,2),ds); label("$Q$", (-0.48,1.9), NE*lsf); dot((0.8,2.4),ds); label("$A'$", (0.81,2.61), NE*lsf); dot((2.4,3.2),ds); label("$B'$", (2.46,3.47), NE*lsf); dot((3.2,1.6),ds); label("$C'$", (3.22,1.9), NE*lsf); dot((1.6,0.8),ds); label("$D'$", (1.14,0.79), NE*lsf); clip((-4.44,-11.2)--(-4.44,6.41)--(16.48,6.41)--(16.48,-11.2)--cycle); [/asy] [$S_{X}$ denotes area of the $X.$]

Let $AA_1$, $CC_1$ be the altitudes of $\Delta ABC$, and $P$ be an arbitrary point of side $BC$. Point $Q$ on the line $AB$ is such that $QP = PC_1$, and point $R$ on the line $AC$ is such that $RP = CP$. Prove that $QA_1RA$ is a cyclic quadrilateral.

For a $n\times n$ complex valued matrix $A$, show that the following two conditions are equivalent. (i) There exists a $n\times n$ complex valued matrix $B$ such that $AB-BA=A$. (ii) There exists a positive integer $k$ such that $A^k = O$. ($O$ is the zero matrix.)

Let $C_1$ and $C_2$ be two circles intersecting at $A $ and $B$. Let $S$ and $T $ be the centres of $C_1 $ and $C_2$, respectively. Let $P$ be a point on the segment $AB$ such that $ |AP|\ne |BP|$ and $P\ne A, P \ne B$. We draw a line perpendicular to $SP$ through $P$ and denote by $C$ and $D$ the points at which this line intersects $C_1$. We likewise draw a line perpendicular to $TP$ through $P$ and denote by $E$ and F the points at which this line intersects $C_2$. Show that $C, D, E,$ and $F$ are the vertices of a rectangle.