Two players play alternatively on an infinite square grid. The first player puts an $X$ in an empty cell and the second player puts an $O$ in an empty cell. The first player wins if he gets $11$ adjacent $X$'s in a line - horizontally, vertically or diagonally. Show that the second player can always prevent the first player from winning.

A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? $ \textbf{(A)}\ 5\sqrt{2}-7 \qquad \textbf{(B)}\ 7-4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9} $

In July 1861, $ 366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month? \[ \textbf{(A)}\ \frac{366}{31 \times 24} \qquad \textbf{(B)}\ \frac{366 \times 31}{24} \qquad \textbf{(C)}\ \frac{366 \times 24}{31} \\ \textbf{(D)}\ \frac{31 \times 24}{366} \qquad \textbf{(E)}\ 366 \times 31 \times 24 \]

Solve the equation $ \sqrt{4 \minus{} 3\sqrt{10 \minus{} 3x}} \equal{} x \minus{} 2$.

a) Evaluate \[\lim_{n\to \infty} \underbrace{\sqrt{a+\sqrt{a+\ldots+\sqrt{a+\sqrt{b}}}}}_{n\ \text{square roots}}\] with $a,b>0$. b)Let $(a_n)_{n\ge 1}$ and $(x_n)_{n\ge 1}$ such that $a_n>0$ and \[x_n=\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}},\ \forall n\in \mathbb{N}^*\] Prove that: 1) $(x_n)_{n\ge 1}$ is bounded if and only if $(a_n)_{n\ge 1}$ is bounded. 2) $(x_n)_{n\ge 1}$ is convergent if and only if $(a_n)_{n\ge 1}$ is convergent. [i]Valentin Matrosenco[/i]

If the markings on the number line are equally spaced, what is the number $\text{y}$? [asy] draw((-4,0)--(26,0),Arrows); for(int a=0; a<6; ++a) { draw((4a,-1)--(4a,1)); } label("0",(0,-1),S); label("20",(20,-1),S); label("y",(12,-1),S); [/asy] $\text{(A)}\ 3 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 16$

Find all linear homogeneous differential equations with continuous coefficients (on the whole real line) such that for any solution $ f(t)$ and any real number $ c,f(t\plus{}c)$ is also a solution.

$24$ students attend a mathematical circle. For any team consisting of $6$ students, the teacher considers it to be either [b]GOOD [/b] or [b]OK[/b]. For the tournament of mathematical battles, the teacher wants to partition all the students into $4$ teams of $6$ students each. May it happen that every such partition contains either $3$ [b]GOOD[/b] teams or exactly one [b]GOOD[/b] team and both options are present?

An ant sits at vertex $A$ of unit square $ABCD$. He needs to get to point $C$, where the entrance to the anthill is located. Points $A$ and $C$ are separated by a vertical wall in the form of an isosceles right triangle with hypotenuse $BD$. Find the length of the shortest path that an ant must overcome in order to get into the anthill.

Can the number \( \overline{abc} + \overline{bca} + \overline{cab} \) be a perfect square?

What is the largest integer $n < 2018$ such that for all integers $b > 1$, $n$ has at least as many $1$'s in its base-$4$ representation as it has in its base-$b$ representation?

Triangle $ABC$ has incenter $I$. Let $D$ be the foot of the perpendicular from $A$ to side $BC$. Let $X$ be a point such that segment $AX$ is a diameter of the circumcircle of triangle $ABC$. Given that $ID = 2$, $IA = 3$, and $IX = 4$, compute the inradius of triangle $ABC$.

Let $S$ be a randomly selected four-element subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Let $m$ and $n$ be relatively prime positive integers so that the expected value of the maximum element in $S$ is $\dfrac{m}{n}$. Find $m + n$.

a) A certain locker room contains $n$ lockers numbered $1,2,\ldots,n$ and all are originally locked. An attendant performs a sequence of operations $T_1, T_2 ,\ldots, T_n$, whereby with the operation $T_k$ the state of those lockers whose number is divisible by $k$ is swapped. After all $n$ operations have been performed, it is observed that all lockers whose number is a perfect square (and only those lockers) are open. Prove this. b) Investigate in a meaningful mathematical way a procedure or set of operations similar to those above which will produce the set of cubes, or the set of numbers of the form $2 m^2 $, or the set of numbers of the form $m^2 +1$, or some nontrivial similar set of your own selection.

a) Let $A$ be a $n\times n$, $n\geq 2$, symmetric, invertible matrix with real positive elements. Show that $z_n\leq n^2-2n$, where $z_n$ is the number of zero elements in $A^{-1}$. b) How many zero elements are there in the inverse of the $n\times n$ matrix $$A=\begin{pmatrix} 1&1&1&1&\ldots&1\\ 1&2&2&2&\ldots&2\\ 1&2&1&1&\ldots&1\\ 1&2&1&2&\ldots&2\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&2&1&2&\ldots&\ddots \end{pmatrix}$$

Let $A(n)$ denote the number of sequences $a_1\ge a_2\ge\cdots{}\ge a_k$ of positive integers for which $a_1+\cdots{}+a_k = n$ and each $a_i +1$ is a power of two $(i = 1,2,\cdots{},k)$. Let $B(n)$ denote the number of sequences $b_1\ge b_2\ge \cdots{}\ge b_m$ of positive integers for which $b_1+\cdots{}+b_m =n$ and each inequality $b_j\ge 2b_{j+1}$ holds $(j=1,2,\cdots{}, m-1)$. Prove that $A(n) = B(n)$ for every positive integer $n$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]