Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac{1}{p}, $ where $p$ is a real parameter. Investigate for which values of $p$ solutions exist and how many solutions exist. (Of course, the last question ''how many solutions exist'' should be understood as ''how many solutions exists modulo $2\pi $''.)

Let $H\subseteq\mathbb{R}^3$ such that if we reflect any point in $H$ across another point of $H$, the resulting point is also in $H$. Prove that either $H$ is dense in ${R}^3$ or one can find equidistant parallel planes which cover $H$

Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon.

Let $a, b$ be positive reals such that $a^3 + b^3 = ab + 1$. Prove that \[(a-b)^2 + a + b \geq 2\]

What is the sum of the digits of the smallest prime that can be written as a sum of $5$ distinct primes? $\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$

[i][b]a)[/b][/i] Determine if there are matrices $A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $A^2+B^2=C^2$. [b][i]b)[/i][/b] Determine if there are matrices $A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $A^4+B^4=C^4$. [b]Note[/b]: The notation $A\in \mathrm{SL}_{2}(\mathbb{Z})$ means that $A$ is a $2\times 2$ matrix with integer entries and $\det A=1$.

Suppose $\{x_n\}$ is a decreasing sequence that $\displaystyle\lim_{n \rightarrow\infty}x_n=0$. Prove that $\sum(-1)^nx_n$ is convergent

For $j \in \{ 1,2,3,4\}$, let $z_j$ be a complex number with $| z_j | = 1$ and $z_j \neq 1$. Prove that $$3 - z_1 - z_2 - z_3 - z_4 + z_1z_2z_3z_4 \neq 0.$$

Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard? [hide=Solution] With probability $1$ the number of corners covered is $0$, $1$, or $2$ for example by the diameter of a square being $\sqrt{2}$ so it suffices to compute the probability that the square covers $2$ corners. This is due to the fact that density implies the mean number of captured corners is $1$. For the lattice with offset angle $\theta \in \left[0,\frac{\pi}{2}\right]$ consider placing a lattice uniformly randomly on to it and in particular say without loss of generality consider the square which covers the horizontal lattice midpoint $\left(\frac{1}{2},0 \right)$. The locus of such midpoint locations so that the square captures the $2$ points $(0,0),(1,0)$, is a rectangle. As capturing horizontally adjacent points does not occur when capturing vertically adjacent points one computes twice that probability as $\frac{4}{\pi} \int_0^{\frac{\pi}{2}} (1-\sin(\theta))(1-\cos(\theta)) d\theta=\boxed{\frac{2(\pi-3)}{\pi}}$ \\ [asy] draw((0,0)--(80,40)); draw((0,0)--(-40,80)); draw((80,40)--(40,120)); draw((-40,80)--(40,120)); draw((80,40)--(-20,40)); draw((-40,80)--(60,80)); draw((32*sqrt(5),16*sqrt(5))--(-8*sqrt(5),16*sqrt(5))); draw((40+8*sqrt(5),120-16*sqrt(5))--(40-32*sqrt(5),120-16*sqrt(5))); draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)+2*(40-16*sqrt(5)),16*sqrt(5)+(40-16*sqrt(5)))); draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)-(80-16*sqrt(5))/2,16*sqrt(5)+(80-16*sqrt(5)))); draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)+(120-16*sqrt(5)-40)/2,120-16*sqrt(5)-(120-16*sqrt(5)-40))); draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)-2*(120-16*sqrt(5)-80),120-16*sqrt(5)-(120-16*sqrt(5)-80))); [/asy] [/hide]

Let $O$ be the circumcircle of $\triangle ABC$, with $AC=BC$ end let $D=AO\cap BC$. If $BD$ and $CD$ are integer numbers and $AO-CD$ is prime, determine such three numbers.

Let $a_n$ be a sequence defined as $a_1=1$, $a_2=2$, and $a_n=a_{n-1}-a_{n-2}$. Compute $a_{2015}$.

Let $\mathbb{R}[x,y]$ denote the set of two-variable polynomials with real coefficients. We say that the pair $(a,b)$ is a [i]zero[/i] of the polynomial $f \in \mathbb{R}[x,y]$ if $f(a,b)=0$. If polynomials $p,q \in \mathbb{R}[x,y]$ have infinitely many common zeros, does it follow that there exists a non-constant polynomial $r \in \mathbb{R}[x,y]$ which is a factor of both $p$ and $q$?

Define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. Define $\{x \} = x - \lfloor x \rfloor$. For example, $\{ 3 \} = 3-3 = 0$, $\{ \pi \} = \pi - 3$, and $\{ - \pi \} = 4-\pi$. If $\{n\} + \{ 3n\} = 1.4$, then find the sum of all possible values of $100\{n\}$. [i]Proposed by Isabella Grabski [/i]

Consider a pyramid with a regular $n$-gon as its base. We colour all the segments connecting two of the vertices of the pyramid except for the sides of the base either red or blue. Show that if $n=9$ then for each such colouring there are three vertices of the pyramid connecting by three segments of the same colour, and that this is not necessarily the case if $n=8$.

Determine all finite nonempty sets $S$ of positive integers satisfying \[ {i+j\over (i,j)}\qquad\mbox{is an element of S for all i,j in S}, \] where $(i,j)$ is the greatest common divisor of $i$ and $j$.

The lengths in inches of the three sides of each of four triangles I, II, III, and IV are as follows: \[ \begin{array}{rlrl} \hbox {I}& 3,\ 4,\ \hbox{and}\ 5 \qquad & \hbox{III}& 7,\ 24,\ \hbox{and}\ 25 \\ \hbox{II}& 4,\ 7\frac{1}{2},\ \hbox{and}\ 8\frac{1}{2} \qquad & \hbox{IV}& 3\frac{1}{2},\ 4\frac{1}{2},\ \hbox{and}\ 5\frac{1}{2}. \end{array} \] Of these four given triangles, the only right triangles are \[ \begin{tabular}{rlrlrl} (A) & I and II \qquad & (B) & I and III \qquad & (C) & I and IV \\ (D) & I, II, and III \qquad & (E) & I, II, and IV & \end{tabular} \]