Prove that there are infinitely many primes $ p$ such that the total number of solutions mod $ p$ to the equation $ 3x^{3}\plus{}4y^{4}\plus{}5z^{3}\minus{}y^{4}z \equiv 0$ is $ p^2$

On a connected graph $G$, one may perform the following operations: [list] [*]choose a vertice $v$, and add a vertice $v'$ such that $v'$ is connected to $v$ and all of its neighbours [*] choose a vertice $v$ with odd degree and delete it [/list] Show that for any connected graph $G$, we may perform a finite number of operations such that the resulting graph is a clique. Proposed by [i]idonthaveanaopsaccount[/i]

Let $ABC$ be a triangle. The line through $A$ tangent to the circumcircle of $ABC$ intersects line $BC$ at point $W$. Points $X,Y \neq A$ lie on lines $AC$ and $AB$, respectively, such that $WA=WX=WY$. Point $X_1$ lies on line $AB$ such that $\angle AXX_1 = 90^{\circ}$, and point $X_2$ lies on line $AC$ such that $\angle AX_1X_2 = 90^{\circ}$. Point $Y_1$ lies on line $AC$ such that $\angle AYY_1 = 90^{\circ}$, and point $Y_2$ lies on line $AB$ such that $\angle AY_1Y_2 = 90^{\circ}$. Let lines $AW$ and $XY$ intersect at point $Z$, and let point $P$ be the foot of the perpendicular from $A$ to line $X_2Y_2$. Let line $ZP$ intersect line $BC$ at $U$ and the perpendicular bisector of segment $BC$ at $V$. Suppose that $C$ lies between $B$ and $U$. Let $x$ be a positive real number. Suppose that $AB=x+1$, $AC=3$, $AV=x$, and $\frac{BC}{CU}=x$. Then $x=\frac{\sqrt{k}-m}{n}$ for positive integers $k$,$m$, and $n$ such that $k$ is not divisible by the square of any integer greater than $1$. Compute $100k+10m+n$. [i]Proposed by Ankit Bisain, Luke Robitaille, and Brandon Wang[/i]

Let $a, b$, and $c$ be the lengths of the sides of an arbitrary triangle, and let $\alpha,\beta$, and $\gamma$ be the radian measures of its corresponding angles. Prove that $$ \frac{\pi}{3}\le \frac{\alpha a +\beta b + \gamma c}{a+b+c} < \frac{\pi}{2}.$$ Suggest spatial analogues of this inequality.

Let $a, b, c, x, y$ and $z$ be complex numbers such that $a=\frac{b+c}{x-2}, b=\frac{c+a}{y-2}, c=\frac{a+b}{z-2}$. If $xy+yz+zx=1000$ and $x+y+z=2016$, find the value of $xyz$.

How many positive integers $n < 500$ exist such that its prime factors are exclusively $2$, $7$, $11$, or a combination of these?

Four unit squares form a $2 \times 2$ grid. Each of the $12$ unit line segments forming the sides of the squares is colored either red or blue in such way that each square has $2$ red sides and blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such colorings. [asy] size(4cm); defaultpen(linewidth(1.2)); draw((0, 0) -- (2, 0) -- (2, 1)); draw((0, 1) -- (1, 1) -- (1, 2) -- (2,2)); draw((0, 0) -- (0, 1), dotted); draw((1, 0) -- (1, 1) -- (2, 1) -- (2, 2), dotted); draw((0, 1) -- (0, 2) -- (1, 2), dotted); [/asy]

Given a triangle $ \triangle{ABC} $ with circumcircle $ \Omega $. Denote its incenter and $ A $-excenter by $ I, J $, respectively. Let $ T $ be the reflection of $ J $ w.r.t $ BC $ and $ P $ is the intersection of $ BC $ and $ AT $. If the circumcircle of $ \triangle{AIP} $ intersects $ BC $ at $ X \neq P $ and there is a point $ Y \neq A $ on $ \Omega $ such that $ IA = IY $. Show that $ \odot\left(IXY\right) $ tangents to the line $ AI $.

On the coordinate plane, draw a set of points $M(x,y)$, the coordinates of which satisfy the inequality $$\log_{x+y} (x^2+y^2) \le 1.$$

Let $n$ be a positive integer and let $a_1, \ldots, a_{n-1} $ be arbitrary real numbers. Define the sequences $u_0, \ldots, u_n $ and $v_0, \ldots, v_n $ inductively by $u_0 = u_1 = v_0 = v_1 = 1$, and $u_{k+1} = u_k + a_k u_{k-1}$, $v_{k+1} = v_k + a_{n-k} v_{k-1}$ for $k=1, \ldots, n-1.$ Prove that $u_n = v_n.$

Let $ABC$ be a triangle with circumcircle $\Omega$. $X$ and $Y$ are points on $\Omega$ such that $XY$ meets $AB$ and $AC$ at $D$ and $E$, respectively. Show that the midpoints of $XY$, $BE$, $CD$, and $DE$ are concyclic. [i]Carl Lian.[/i]

Let $P(x)$ be the unique polynomial of minimal degree with the following properties: $P(x)$ has leading coefficient $1,$ $1$ is a root of $P(x) - 1,$ $2$ is a root of $P(x-2),$ $3$ is a root of $P(3x),$ $4$ is a root of $4P(x)$ The roots of $P(x)$ are integers, with one exception. The root that is not an integer can be written in the form $\frac{m}{n}$, where m and n are relatively prime positive integers. What is $m+n$? $\textbf{(A) }41\qquad\textbf{(B) }43\qquad\textbf{(C) }45\qquad\textbf{(D) }47\qquad\textbf{(E) }49$

In a convex quadrilateral $ABCD$, the lines $AB$ and $CD$ meet at point $E$ and the lines $AD$ and $BC$ meet at point $F$. Let $P$ be the intersection point of diagonals $AC$ and $BD$. Suppose that $\omega_1$ is a circle passing through $D$ and tangent to $AC$ at $P$. Also suppose that $\omega_2$ is a circle passing through $C$ and tangent to $BD$ at $P$. Let $X$ be the intersection point of $\omega_1$ and $AD$, and $Y$ be the intersection point of $\omega_2$ and $BC$. Suppose that the circles $\omega_1$ and $\omega_2$ intersect each other in $Q$ for the second time. Prove that the perpendicular from $P$ to the line $EF$ passes through the circumcenter of triangle $XQY$ . Proposed by Iman Maghsoudi

Let \(n\ge3\) be a fixed integer, and let \(\alpha\) be a fixed positive real number. There are \(n\) numbers written around a circle such that there is exactly one \(1\) and the rest are \(0\)'s. An [i]operation[/i] consists of picking a number \(a\) in the circle, subtracting some positive real \(x\le a\) from it, and adding \(\alpha x\) to each of its neighbors. Find all pairs \((n,\alpha)\) such that all the numbers in the circle can be made equal after a finite number of operations. [i]Proposed by Anthony Wang[/i]

a) Prove that if the sum of the non-zero digits $a_1, a_2, ... , a_n$ is a multiple of $27$, then it is possible to permute these digits in order to obtain an $n$-digit number that is a multiple of $27$. b) Prove that if the non-zero digits $a_1, a_2, ... , a_n$ have the property that every ndigit number obtained by permuting these digits is a multiple of $27$, then the sum of these digits is a multiple of $27$