Find the highest power of 2 that is a factor of the number $$ L_n = (n+1)(n+2)... 2n,$$ where $n$is a natural number.

Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to him in the circle. Then each person computes and announces the average of the numbers of his two neighbors. The figure shows the average announced by each person ([u]not[/u] the original number the person picked). The number picked by the person who announced the average $6$ was [asy] label("(1)", (0,.9)); label("(2)", (.4,.65)); label("(3)", (.8,.25)); label("(4)", (.8,-.2)); label("(5)", (.4,-.65)); label("(6)", (0,-.9)); label("(7)", (-.4,-.65)); label("(8)", (-.8,-.2)); label("(9)", (-.8,.25)); label("(10)", (-.4,.65)); [/asy] $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{not uniquely determined from the given information} $

Let $a,b,c$ be positive real numbers such that $ab+bc+ca=1$. Prove that \[\sqrt[4]{\frac{\sqrt{3}}{a}+6\sqrt{3}b}+\sqrt[4]{\frac{\sqrt{3}}{b}+6\sqrt{3}c}+\sqrt[4]{\frac{\sqrt{3}}{c}+6\sqrt{3}a}\le\frac{1}{abc}\] When does inequality hold?

Triangle $ ABC$ is inscribed in circle $ \omega$. The tangent lines to $ \omega$ at $ B$ and $ C$ meet at $ T$. Point $ S$ lies on ray $ BC$ such that $ AS \perp AT$. Points $ B_1$ and $ C_1$ lie on ray $ ST$ (with $ C_1$ in between $ B_1$ and $ S$) such that $ B_1T \equal{} BT \equal{} C_1T$. Prove that triangles $ ABC$ and $ AB_1C_1$ are similar to each other.

Let $ ABC$ be a triangle. The $ A$-excircle of triangle $ ABC$ has center $ O_{a}$ and touches the side $ BC$ at the point $ A_{a}$. The $ B$-excircle of triangle $ ABC$ touches its sidelines $ AB$ and $ BC$ at the points $ C_{b}$ and $ A_{b}$. The $ C$-excircle of triangle $ ABC$ touches its sidelines $ BC$ and $ CA$ at the points $ A_{c}$ and $ B_{c}$. The lines $ C_{b}A_{b}$ and $ A_{c}B_{c}$ intersect each other at some point $ X$. Prove that the quadrilateral $ AO_{a}A_{a}X$ is a parallelogram. [i]Remark.[/i] The $ A$[i]-excircle[/i] of a triangle $ ABC$ is defined as the circle which touches the segment $ BC$ and the extensions of the segments $ CA$ and $ AB$ beyound the points $ C$ and $ B$, respectively. The center of this circle is the point of intersection of the interior angle bisector of the angle $ CAB$ and the exterior angle bisectors of the angles $ ABC$ and $ BCA$. Similarly, the $ B$-excircle and the $ C$-excircle of triangle $ ABC$ are defined. [hide="Source of the problem"][i]Source of the problem:[/i] Theorem (88) in: John Sturgeon Mackay, [i]The Triangle and its Six Scribed Circles[/i], Proceedings of the Edinburgh Mathematical Society 1 (1883), pages 4-128 and drawings at the end of the volume.[/hide]

$ x^2 \plus{} y^2 \plus{} z^2 \equal{} 1993$ then prove $ x \plus{} y \plus{} z$ can't be a perfect square:

Given a natural number $a_0$, we construct the sequence $\{a_n\}$ as follows $a_{n+1} = a^2_n-5$ if $a_n$ is odd, and $\frac{a_n}{2}$ if $a_n$ is even. Prove that for any odd $a_0 > 5$ in the sequence $\{a_n\}$ arbitrarily large numbers will occur.

Three coins lie on integer points on the number line. A move consists of choosing and moving two coins, the first one $ 1$ unit to the right and the second one $ 1$ unit to the left. Under which initial conditions is it possible to move all coins to one single point?

Find all $g:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in R$, \[(4x+g(x)^2)g(y)=4g(\frac{y}{2}g(x))+4xyg(x)\]

Let $A,B \in M_{n}(\mathbb{R}).$ Show that $rank(A) = rank(B)$ if and only if there exist nonsingular matrices $X,Y,Z \in M_{n}(\mathbb{R})$ such that \[ AX + YB = AZB. \]

A random selector can only select one of the nine integers $ 1,2,\ldots,9$, and it makes these selections with equal probability. Determine the probability that after $ n$ selections ($ n>1$), the product of the $ n$ numbers selected will be divisible by 10.

A $9\times 9$ table is filled with zeroes.In every step we can either take a row add $1$ to every cell and shift it one unit to right or take a column reduce every cell by $1$ and shift it one cell down. Can the table with the top right $-1$ and bottom left $+1$ and all other cells zero be reached?

For a sequence $(a_{n})_{n\geq 1}$ of real numbers it is known that $a_{n}=a_{n-1}+a_{n+2}$ for $n\geq 2$. What is the largest number of its consecutive elements that can all be positive?

Given positive integers $m$ and $n \ge m$, determine the largest number of dominoes ($1\times2$ or $2 \times 1$ rectangles) that can be placed on a rectangular board with $m$ rows and $2n$ columns consisting of cells ($1 \times 1$ squares) so that: (i) each domino covers exactly two adjacent cells of the board; (ii) no two dominoes overlap; (iii) no two form a $2 \times 2$ square; and (iv) the bottom row of the board is completely covered by $n$ dominoes.

A binary string is a string consisting of only 0’s and 1’s (for instance, 001010, 101, etc.). What is the probability that a randomly chosen binary string of length 10 has 2 consecutive 0’s? Express your answer as a fraction.

Let $(a_n)_{n=1}^\infty$ be a real sequence such that \[a_n = (n-1)a_1 + (n-2)a_2 + \dots + 2a_{n-2} + a_{n-1}\] for every $n\geq 3$. If $a_{2011} = 2011$ and $a_{2012} = 2012$, what is $a_{2013}$? $ \textbf{(A)}\ 6025 \qquad\textbf{(B)}\ 5555 \qquad\textbf{(C)}\ 4025 \qquad\textbf{(D)}\ 3456 \qquad\textbf{(E)}\ 2013 $

Altitudes $BE$ and $CF$ of acute triangle $ABC$ intersect at $H$. Suppose that the altitudes of triangle $EHF$ concur on line $BC$. If $AB=3$ and $AC=4$, then $BC^2=\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a+b$.

There are real numbers $x,y,h$ and $k$ that satisfy the system of equations $$x^2 + y^2 - 6x - 8y = h$$ $$x^2 + y^2 - 10x + 4y = k$$ What is the minimum possible value of $h+k$? $ \textbf{(A) }-54 \qquad \textbf{(B) }-46 \qquad \textbf{(C) }-34 \qquad \textbf{(D) }-16 \qquad \textbf{(E) }16 \qquad $

Suppose that $X$ and $Y$ are metric spaces and $f:X \longrightarrow Y$ is a continious function. Also $f_1: X\times \mathbb R \longrightarrow Y\times \mathbb R$ with equation $f_1(x,t)=(f(x),t)$ for all $x\in X$ and $t\in \mathbb R$ is a closed function. Prove that for every compact set $K\subseteq Y$, its pre-image $f^{pre}(K)$ is a compact set in $X$.

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

$x_{1},...,x_{n}$ are positive reals such that their sum and their squares' sum are equal to $t$. Prove that $\sum_{i\neq{j}}\frac{x_{i}}{x_{j}}\ge\frac{(n-1)^{2}\cdot{t}}{t-1}$

For any two rational numbers $ p$ and $ q$ in the interval $ (0,1)$ and function $ f$, there is always $ \displaystyle f \left( \frac{p\plus{}q}{2} \right) \leq \frac{f(p) \plus{} f(q)}{2}$. Then prove that for any rational numbers $ \lambda, x_1, x_2 \in (0,1)$, there is always: \[ f( \lambda x_1 \plus{} (1\minus{}\lambda) x_2 ) \leq \lambda f(x_i) \plus{} (1\minus{}\lambda) f(x_2)\]

Draw the smaller number of line segments connecting points of the figure such that the new figure obtained to have exactly: [img]https://cdn.artofproblemsolving.com/attachments/d/1/098e03714904573a1eacd2d3dc28b4e8c42c7c.png[/img] i) one axis of symmetry ii) two axes of symmetry iii) four axes of symmetry Draw a new figure, at each case.

If ${a, b}$ and $c$ are positive real numbers, prove that \begin{align*} a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 &\ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}. \end{align*} [i](Montenegro).[/i]

A palindrome is a number that remains the same when its digits are reversed. Let $n$ be a product of distinct primes not divisible by $10$. Prove that infinitely many multiples of $n$ are palindromes.