Prove that: $ \sum_{i\equal{}1}^{n^2} \lfloor \frac{i}{3} \rfloor\equal{} \frac{n^2(n^2\minus{}1)}{6}$ For all $ n \in N$.

Find all real values of $x$ for which $$\frac{1}{\sqrt{x} + \sqrt{x - 2}} +\frac{1}{\sqrt{x+2} + \sqrt{x }} =\frac14.$$

Consider polynomials $P$ of degree $2015$, all of whose coefficients are in the set $\{0,1,\dots,2010\}$. Call such a polynomial [i]good[/i] if for every integer $m$, one of the numbers $P(m)-20$, $P(m)-15$, $P(m)-1234$ is divisible by $2011$, and there exist integers $m_{20}, m_{15}, m_{1234}$ such that $P(m_{20})-20, P(m_{15})-15, P(m_{1234})-1234$ are all multiples of $2011$. Let $N$ be the number of good polynomials. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Yang Liu[/i]

Figure $ABCD$ is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded L-shaped region is [asy] pair A,B,C,D; A = (5,5); B = (5,0); C = (0,0); D = (0,5); fill((0,0)--(0,4)--(1,4)--(1,1)--(4,1)--(4,0)--cycle,gray); draw(A--B--C--D--cycle); draw((4,0)--(4,4)--(0,4)); draw((1,5)--(1,1)--(5,1)); label("$A$",A,NE); label("$B$",B,SE); label("$C$",C,SW); label("$D$",D,NW); label("$1$",(1,4.5),E); label("$1$",(0.5,5),N); label("$3$",(1,2.5),E); label("$3$",(2.5,1),N); label("$1$",(4,0.5),E); label("$1$",(4.5,1),N); [/asy] $\text{(A)}\ 7 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12.5 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15$

Five distinct points $A,B,C,D$ and $E$ lie on a line with $|AB|=|BC|=|CD|=|DE|$. The point $F$ lies outside the line. Let $G$ be the circumcentre of the triangle $ADF$ and $H$ the circumcentre of the triangle $BEF$. Show that the lines $GH$ and $FC$ are perpendicular.

Compute \[\displaystyle\int_{0}^{\pi}\frac{2\sin\theta+3\cos\theta-3}{13\cos\theta-5}\mathrm{d}\theta.\]

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

Let the positive integer $n$ have at least for positive divisors and $0<d_1<d_2<d_3<d_4$ be its least positive divisors. Find all positive integers $n$ such that: \[ n=d_1^2+d_2^2+d_3^2+d_4^2. \]

Let convex quadrilateral $ ABCD$ be inscribed in a circle centers at $ O.$ The opposite sides $ BA,CD$ meet at $ H$, the diagonals $ AC,BD$ meet at $ G.$ Let $ O_{1},O_{2}$ be the circumcenters of triangles $ AGD,BGC.$ $ O_{1}O_{2}$ intersects $ OG$ at $ N.$ The line $ HG$ cuts the circumcircles of triangles $ AGD,BGC$ at $ P,Q$, respectively. Denote by $ M$ the midpoint of $ PQ.$ Prove that $ NO \equal{} NM.$

In a circle, chord $AB$ has length $5$ and chord $AC$ has length $7$. Arc $AC$ is twice the length of arc $AB$, and both arcs have degree less than $180$. Compute the area of the circle.

Show that the vertices of a regular pentagon are concyclic. If the length of each side of the pentagon is $x$, show that the radius of the circumcircle is $\frac{x}{2\sin 36^\circ}$.

Sixty-four dice with the numbers ”one” to ”six” are placed on one table and formed into a square with eight horizontal and eight vertical rows of cubes pushed together. By rotating the dice, while maintaining their place, we want to finally have all sixty-four dice the "one" points upwards. Each dice however, may not be turned individually, but only every eight dice in a horizontal or vertical row together by $90^o$ to the longitudinal axis of this row may turn. Prove that it is always possible to solve the dice by repeatedly applying the permitted type of rotation to the required end position.

Georg investigates which integers are expressible in the form $$\pm 1^2 \pm 2^2 \pm 3^2 \pm \dots \pm n^2.$$ For example, the number $3$ can be expressed as $ -1^2 + 2^2$, and the number $-13$ can be expressed as $+1^2 + 2^2 + 3^2 - 4^2 + 5^2 - 6^2$. Are all integers expressible in this form?

Let $ABC$ be an acute scalene triangle. Its $C$-excircle tangent to the segment $AB$ meets $AB$ at point $M$ and the extension of $BC$ beyond $B$ at point $N$. Analogously, its $B$-excircle tangent to the segment $AC$ meets $AC$ at point $P$ and the extension of $BC$ beyond $C$ at point $Q$. Denote by $A_1$ the intersection point of the lines $MN$ and $PQ$, and let $A_2$ be defined as the point, symmetric to $A$ with respect to $A_1$. Define the points $B_2$ and $C_2$, analogously. Prove that $\triangle ABC$ is similar to $\triangle A_2B_2C_2$.

How many sets of two or more consecutive positive integers have a sum of 15? $ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

Let ${p}$ be a prime. $a,b,c\in\mathbb Z,\gcd(a,p)=\gcd(b,p)=\gcd(c,p)=1.$ Prove that: $\exists x_1,x_2,x_3,x_4\in\mathbb Z,| x_1|,|x_2|,|x_3|,|x_4|<\sqrt p,$ satisfying $$ax_1x_2+bx_3x_4\equiv c\pmod p.$$ [i]Proposed by Wang Guangting[/i]

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

Determine all real solutions to the following equation: \[2^{(2^x)}-3\cdot2^{(2^{x-1}+1)}+8=0.\]

Let be an holomorphic function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ having the property that $ |f(z)|\le e^{|\text{Im}(z)|} , $ for all complex numbers $ z. $ Prove that the restriction of any of its derivatives (of any order) to the real numbers is everywhere dominated by $ 1. $

A random walk is a process in which something moves from point to point, and where the direction of movement at each step is randomly chosen. Suppose that a person conducts a random walk on a line: he starts at $0$ and each minute randomly moves either $1$ unit in the positive direction or $1$ unit in the negative direction. What is his expected distance from the origin after $6$ moves? [i]2017 CCA Math Bonanza Lightning Round #3.4[/i]

Two points are selected in a convex pentagon. Prove that you can choose a quadrilateral with vertices at the vertices of a pentagon so that both selected points fall into it.

Let each sequence of capital Bulgarian letters be a word (Note that the Bulgarian alphabet consists of 30 letters). We say that a word is [i]calm[/i], if there is no sequence of the letter $P$ in it (two letters $P$ next to each other). Find a clear expression for $f(n)$ (closed formula), which represents the number of calm words with length $n$. Example: [i]РНТ[/i] and [i]ТТРВ[/i] are calm, but [i]ЖГРР[/i] and [i]ЖРРРП[/i] aren’t.

Players $A$ and $B$ play the following game. Each of them throws a dice, and if the outcomes are $x$ and $y$ respectively, a list of all two digit numbers $10a + b$ with $a,b\in \{1,..,6\}$ and $10a + b \le 10x + y$ is created. Then the players alternately reduce the list by replacing a pair of numbers in the list by their absolute difference, until only one number remains. If the remaining number is of the same parity as the outcome of $A$’s throw, then $A$ is proclaimed the winner. What is the probability that $A$ wins the game?

Compute the number of digits is $2015!$. Your score will be given by $\max\{\lfloor125(\min\{\tfrac{A}{C},\tfrac{C}{A}\}-\tfrac{1}{5})\rfloor,0\}$, where $A$ is your answer and $C$ is the actual answer.

Prove that the product of the radii of three circles exscribed to a given triangle does not exceed $A=\frac{3\sqrt 3}{8}$ times the product of the side lengths of the triangle. When does equality hold?