Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$

The real function $f$ is defined for $\forall$ $x\in \mathbb{R}$ and $f(0)=0$. Also $f(9+x)=f(9-x)$ and $f(x-10)=f(-x-10)$ for $\forall$ $x\in \mathbb{R}$. What’s the least number of zeros $f$ can have in the interval $[0;2014]$? Does this change, if $f$ is also continuous?

Ryan stands on the bottom-left square of a 2017 by 2017 grid of squares, where each square is colored either black, gray, or white according to the pattern as depicted to the right. Each second he moves either one square up, one square to the right, or both one up and to the right, selecting between these three options uniformly and independently. Noting that he begins on a black square, find the probability that Ryan is still on a black square after 2017 seconds. [center][img]http://i.imgur.com/WNp59XW.png[/img][/center]

Only $A$ and $B$ have $n$ friends in a village of $2010$ people. The other $2008$ people have all different numbers of friends. How many possible values of $n$ are there? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None} $

Prove that if $P(x) = (x-a)^kQ(x)$, where $k$ is a positive integer, $a$ is a nonzero real number, $Q(x)$ is a nonzero polynomial, then $P(x)$ has at least $k + 1$ nonzero coefficients.

Let $ABC$ be a triangle. $D$ and $E$ are points on line segments $BC$ and $AC$, respectively, such that $AD=60$, $BD=189$, $CD=36$, $AE=40$, and $CE=50$. What is $AB+DE$? [i]2017 CCA Math Bonanza Tiebreaker Round #2[/i]

What is greater, $31^{11}$ or $17^{14}$ ?

For what value of $ m $ is the polynomial $ x^3 + y^3 + z^3 + mxyz $ divisible by $ x + y + z $?

Given the linear fractional transformation of $x$ into $f_1(x)=\dfrac{2x-1}{x+1}$. Define $f_{n+1}(x)=f_1(f_n(x))$ for $n=1,2,3,\cdots$. Assuming that $f_{35}(x)=f_5(x)$, it follows that $f_{28}(x)$ is equal to $\textbf{(A) }x\qquad\textbf{(B) }\frac{1}{x}\qquad\textbf{(C) }\frac{x-1}{x}\qquad\textbf{(D) }\frac{1}{1-x}\qquad \textbf{(E) }\text{None of these}$

There is an infinitely tall tetrahedral stack of spheres where each row of the tetrahedron consists of a triangular arrangement of spheres, as shown below. There is $1$ sphere in the top row (which we will call row $0$), $3$ spheres in row $1$, $6$ spheres in row $2$, $10$ spheres in row $3$, etc. The top-most sphere in row $0$ is assigned the number $1$. We then assign each other sphere the sum of the number(s) assigned to the sphere(s) which touch it in the row directly above it. Find a simplified expression in terms of $n$ for the sum of the numbers assigned to each sphere from row $0$ to row $n$. [asy] import three; import solids; size(8cm); //currentprojection = perspective(1, 1, 10); triple backright = (-2, 0, 0), backleft = (-1, -sqrt(3), 0), backup = (-1, -sqrt(3) / 3, 2 * sqrt(6) / 3); draw(shift(2 * backleft) * surface(sphere(1,20)), white); //2 draw(shift(backleft + backright) * surface(sphere(1,20)), white); //2 draw(shift(2 * backright) * surface(sphere(1,20)), white); //3 draw(shift(backup + backleft) * surface(sphere(1,20)), white); draw(shift(backup + backright) * surface(sphere(1,20)), white); draw(shift(2 * backup) * surface(sphere(1,20)), white); draw(shift(backleft) * surface(sphere(1,20)), white); draw(shift(backright) * surface(sphere(1,20)), white); draw(shift(backup) * surface(sphere(1,20)), white); draw(surface(sphere(1,20)), white); label("Row 0", 2 * backup, 15 * dir(20)); label("Row 1", backup, 25 * dir(20)); label("Row 2", O, 35 * dir(20)); dot(-backup); dot(-7 * backup / 8); dot(-6 * backup / 8); dot((backleft - backup) + backleft * 2); dot(5 * (backleft - backup) / 4 + backleft * 2); dot(6 * (backleft - backup) / 4 + backleft * 2); dot((backright - backup) + backright * 2); dot(5 * (backright - backup) / 4 + backright * 2); dot(6 * (backright - backup) / 4 + backright * 2); [/asy]

$ 25$ boys and some girls came to the party and discovered an interesting property of their company. Take an arbitrary group of $ \geq 10$ boys and all the girls which are acquainted with at least one of them. Then in the joint group, the number of girls is by one greater than the number of boys. Prove that there exists a girl who is acquainted with at least $ 16$ boys.

There are $n$ sticks which have distinct integer length. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What's the largest possible value of $n$ under such conditions? [i](Proposed by Bogdan Rublov)[/i]

Let $ABC$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $BC$ such that $\angle{CAX} = \angle{YAB}$. Suppose that: $1)$ $K$ and $S$ are the feet of the perpendiculars from from $B$ to the lines $AX$ and $AY$ respectively. $2)$ $T$ and $L$ are the feet of the perpendiculars from $C$ to the lines $AX$ and $AY$ respectively. Prove that $KL$ and $ST$ intersect on the line $BC$.

For a positive integer $n$, let $\Omega(n)$ denote the number of prime factors of $n$, counting multiplicity. Let $f_1(n)$ and $f_3(n)$ denote the sum of positive divisors $d|n$ where $\Omega(d)\equiv 1\pmod 4$ and $\Omega(d)\equiv 3\pmod 4$, respectively. For example, $f_1(72) = 72 + 2 + 3 = 77$ and $f_3(72) = 8+12+18 = 38$. Determine $f_3(6^{2020}) - f_1(6^{2020})$.

For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.

$a,b,c \geq-3/4$ and $a+b+c=1$. Show that: $\frac{a}{1+a^{2}}+\frac{b}{1+b^{2}}+\frac{c}{1+c^{2}}\leq \frac{9}{10}$

Find all positive integers $N$ such that an $N\times N$ board can be tiled using tiles of size $5\times 5$ or $1\times 3$. Note: The tiles must completely cover all the board, with no overlappings.

Two $n$-sided polygons are said to be of the same type if we can label their vertices in clockwise order as $A_1$, $A_2$, ..., $A_n$ and $B_1$, $B_2$, ..., $B_n$ respectively such that each pair of interior angles $A_i$ and $B_i$ are either both reflex angles or both non-reflex angles. How many different types of $11$-sided polygons are there?

$a,b,c$ are three non-zero-complex numbers, and $\frac{a}{b}=\frac{b}{c}=\frac{c}{a}$, then the value of $\frac{a+b-c}{a-b+c}$ is ($\omega=-\frac{1}{2}+\frac{\sqrt3}{2}\text{i}$) $\text{(A)}1\qquad\text{(B)}\pm\omega\qquad\text{(C)}1,\omega,\omega^2\qquad\text{(D)}1,-\omega,-\omega^2$

$M$ is the midpoint of the side $BC$ of a triangle $ABC$. Construct a line $\ell$ intersecting the triangle and parallel to $BC$ such that the segment of $\ell$ between the sides $AB$ and $AC$ is the hypotenuse of a right-angled triangle with $M$ being its third vertex. (Folklore)

Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.

An integer sequence $\{a_{n}\}_{n \ge 1}$ is given such that \[2^{n}=\sum^{}_{d \vert n}a_{d}\] for all $n \in \mathbb{N}$. Show that $a_{n}$ is divisible by $n$ for all $n \in \mathbb{N}$.

Is it true that the center of the inscribed circle of the triangle lies inside the triangle formed by the lines of connecting it's midpoints?

$x_1,x_2,...,x_n>1$ are natural numbers and $n \geq 3$ Prove that : $(x_1x_2...x_n)^2 \ne x_1^3 + x_2^3 +...+x_n^3$

In acute triangle $ABC,$ $AB<AC,$ $O$ is the circumcenter of the triangle. $M$ is the midpoint of segment $BC,$ $(AOM)$ intersects the line $AB$ again at $D$ and intersects the segment $AC$ at $E.$ Prove that $DM=EC.$