Which one divides $2^{2^{2010}}+2^{2^{2009}}+1$? $ \textbf{(A)}\ 19 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ \text{None} $

The positive real numbers $a, b, c$ with $abc = 1$ Show that: $\sqrt{a + \frac{1}{a}} + \sqrt{b + \frac{1}{b}} + \sqrt{c + \frac{1}{c}}\geq 2(\sqrt{a} + \sqrt{b} + \sqrt{c})$

In a regular hexagon, segments with lengths from $1$ to $6$ were drawn as shown in the right figure (the segments go sequentially in increasing length, all the angles between them are right). Find the side length of this hexagon. [img]https://cdn.artofproblemsolving.com/attachments/3/1/82e4225b56d984e897a43ba1f403d89e5f4736.png[/img]

Let $ABC$ be a triangle such that $AC\not= BC,AB<AC$ and let $K$ be it's circumcircle. The tangent to $K$ at the point $A$ intersects the line $BC$ at the point $D$. Let $K_1$ be the circle tangent to $K$ and to the segments $(AD),(BD)$. We denote by $M$ the point where $K_1$ touches $(BD)$. Show that $AC=MC$ if and only if $AM$ is the bisector of the $\angle DAB$. [i]Neculai Roman[/i]

There exist some number of ordered triples of real numbers $(x,y,z)$ that satisfy the following system of equations: \begin{align*} x+y+2z &= 6\\ x^2+y^2+2z^2 &= 18\\ x^3+y^3+2z^3&=54 \end{align*} Given that the sum of all possible positive values of $x$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ where $a$,$b$,$c$, and $d$ are positive integers, $c$ is squarefree, and $\gcd(a,b,d)=1$, find the value of $a+b+c+d$.

Let $ABC$ be an acute triangle (with $AB \ne AC$) and $M$ be the midpoint of $BC$. The circle of diameter $AM$ cuts $AC$ at $N$ and $BC$ again at $H$. A point $K$ is taken on $AC$ (between $A$ and $N$) such that $CN = NK$. Segments $AH$ and $BK$ intersect at $L$. The circle that passes through $A$,$K$ and $L$ cuts $AB$ at $P$. Prove that $C$,$L$ and $P$ are collinear.

Given are six reals $a, b, c, x, y, z$ such that $(a + b + c)(x + y + z) = 3$ and $(a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = 4$. Prove that $ax + by + cz \ge 0$.

For each value of parameter $a$, solve the the equation $$ x - \sqrt{x^2-a^2} = \frac{(x-a)^2}{2(x+a)}$$

Consider a $25 \times 25$ grid of unit squares. Draw with a red pen contours of squares of any size on the grid. What is the minimal number of squares we must draw in order to colour all the lines of the grid?

In the convex quadrilateral $ABCD$, the bisectors of angles $A$ and $C$ intersect in $I$. Prove that $ABCD$ is circumscriptible if and only if $$S[AIB] + S[CID] =S[AID]+S[BIC]$$ ( $S[XYZ]$ denotes the area of the triangle $XYZ$)

Find all triples of nonnegative integers $(m,n,k)$ satisfying $5^m+7^n=k^3$.

Prove that the parallelogram $ABCD$ with relation $\angle ABD + \angle DAC = 90^o$, is either a rectangle or a rhombus.

A sequence $P=\left \{ a_{n} \right \}$ is called a $ \text{Permutation}$ of natural numbers (positive integers) if for any natural number $m,$ there exists a unique natural number $n$ such that $a_n=m.$ We also define $S_k(P)$ as: $S_k(P)=a_{1}+a_{2}+\cdots +a_{k}$ (the sum of the first $k$ elements of the sequence). Prove that there exists infinitely many distinct $ \text{Permutations}$ of natural numbers like $P_1,P_2, \cdots$ such that$:$ $$\forall k, \forall i<j: S_k(P_i)|S_k(P_j)$$

The Fibonacci numbers are defined recursively by $F_0=0, F_1=1,$ and $F_i=F_{i-1}+F_{i-2}$ for $i \ge 2.$ Given $15$ wooden blocks of weights $F_2, F_3, \ldots, F_{16},$ compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks.

Is it possible to arrange integers in the cells of the infinite chechered sheet so that every integer appears at least in one cell, and the sum of any $10$ numbers in a row vertically or horizontal, would be divisible by $101$?

It is given circle with center in center of coordinate center with radius of $2016$. On circle and inside it are $540$ points with integer coordinates such that no three of them are collinear. Prove that there exist two triangles with vertices in given points such that they have same area

Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://imgur.com/1mfBNNL.png[/img]

Consider the $5$ by $5$ by $5$ equilateral triangular grid as shown: [asy] size(5cm); real n = 5; for (int i = 0; i < n; ++i) { draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i)); } for (int i = 0; i < n; ++i) { draw((n-i,0)--((n-i)/2,(n-i)*0.866)); } for (int i = 0; i < n; ++i) { draw((i,0)--((n+i)/2,(n-i)*0.866)); } [/asy] Ethan chooses two distinct upward-oriented equilateral triangles bounded by the gridlines. The probability that Ethan chooses two triangles that share exactly one vertex can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Andrew Wen[/i]

Find the largest real constant $a$ such that for all $n \geq 1$ and for all real numbers $x_0, x_1, ... , x_n$ satisfying $0 = x_0 < x_1 < x_2 < \cdots < x_n$ we have \[\frac{1}{x_1-x_0} + \frac{1}{x_2-x_1} + \dots + \frac{1}{x_n-x_{n-1}} \geq a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right)\]

Let $ABC$ be an equilateral triangle and a point M inside the triangle such that $MA^2 = MB^2 +MC^2$. Draw an equilateral triangle $ACD$ where $D \ne B$. Let the point $N$ inside $\vartriangle ACD$ such that $AMN$ is an equilateral triangle. Determine $\angle BMC$.

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

Two towns, $A$ and $B$, are $100$ miles apart. Every $20$ minutes, (starting at midnight), a bus traveling at $60$ mph leaves town $A$ for town $B$, and every $30$ minutes (starting at midnight) a bus traveling at $20$ mph leaves town $B$ for town $A$. Dirk starts in Town $A$ and gets on a bus leaving for town $B$ at noon. However, Dirk is always afraid he has boarded a bus going in the wrong direction, so each time the bus he is in passes another bus, he gets out and transfers to that other bus. How many hours pass before Dirk finally reaches Town $B$?

Let $a,b$ and $c$ be positive integers that $$\frac{a\sqrt{3}+b}{b\sqrt3+c}$$ is a rational number, show that $$\frac{a^2+b^2+c^2}{a+b+ c}$$ is an integer. [i](Anonymous314)[/i]

Let $ABCD$ be a trapezoid in which $AB \parallel CD$ and $AB = 2CD$. A line $\ell$ perpendicular to $CD$ was drawn through point $C$. A circle with center at point $D$ and radius $DA$ intersects line $\ell$ at points $P$ and $Q$. Prove that $AP \perp BQ$.