For $n \geq 3$ and $a_{1} \leq a_{2} \leq \ldots \leq a_{n}$ given real numbers we have the following instructions: - place out the numbers in some order in a ring; - delete one of the numbers from the ring; - if just two numbers are remaining in the ring: let $S$ be the sum of these two numbers. Otherwise, if there are more the two numbers in the ring, replace Afterwards start again with the step (2). Show that the largest sum $S$ which can result in this way is given by the formula \[S_{max}= \sum^n_{k=2} \begin{pmatrix} n -2 \\ [\frac{k}{2}] - 1\end{pmatrix}a_{k}.\]

Let $f\in C^1[a,b]$, $f(a)=0$ and suppose that $\lambda\in\mathbb R$, $\lambda >0$, is such that $$|f'(x)|\leq \lambda |f(x)|$$ for all $x\in [a,b]$. Is it true that $f(x)=0$ for all $x\in [a,b]$?

For an integer $n\ge 3$ and a permutation $\sigma=(p_{1},p_{2},\cdots ,p_{n})$ of $\{1,2,\cdots , n\}$, we say $p_{l}$ is a $landmark$ point if $2\le l\le n-1$ and $(p_{l-1}-p_{l})(p_{l+1}-p_{l})>0$. For example , for $n=7$,\\ the permutation $(2,7,6,4,5,1,3)$ has four landmark points: $p_{2}=7$, $p_{4}=4$, $p_{5}=5$ and $p_{6}=1$. For a given $n\ge 3$ , let $L(n)$ denote the number of permutations of $\{1,2,\cdots ,n\}$ with exactly one landmark point. Find the maximum $n\ge 3$ for which $L(n)$ is a perfect square.

Determine the largest positive integer that is a divisor of all three of $A=2^{2010}\times3^{2010}, B=3^{2010}\times5^{2010},$ and $C=5^{2010}\times2^{2010}.$

Find the last eight digits of the binary development of $27^{1986}.$

Let $\alpha$ be an irrational number, and denote $F = \{ (x,y) \in R^2 : y \geq \alpha x \}$ as a closed half-plane bounded by a line. Let $P(\alpha,n) = P(X_1,...,X_n \in F)$, where $X_n$ is a simple, symmetric random walk that starts at the origin and moves with probability 1/4 in each direction. Prove that $P(\alpha,n)$ does not depend on $\alpha$.

Let $ABC$ be a triangle with $AC = 18$ and $D$ is the point on the segment $AC$ such that $AD = 5$. Draw perpendiculars from $D$ on $AB$ and $BC$ which have lengths $4$ and $5$ respectively. Find the area of the triangle $ABC$.

Let $ABCD$ be a rectangle consisting of unit squares. All vertices of these unit squares inside the rectangle and on its sides have been colored in four colors. Additionally, it is known that: [list] [*] every vertex that lies on the side $AB$ has been colored in either the $1.$ or $2.$ color; [*] every vertex that lies on the side $BC$ has been colored in either the $2.$ or $3.$ color; [*] every vertex that lies on the side $CD$ has been colored in either the $3.$ or $4.$ color; [*] every vertex that lies on the side $DA$ has been colored in either the $4.$ or $1.$ color; [*] no two neighboring vertices have been colored in $1.$ and $3.$ color; [*] no two neighboring vertices have been colored in $2.$ and $4.$ color. [/list] Notice that the constraints imply that vertex $A$ has been colored in $1.$ color etc. Prove that there exists a unit square that has all vertices in different colors (in other words it has one vertex of each color).

Let $P$ be a regular $2n$-sided polygon. A [b]rhombus-ulation[/b] of $P$ is dividing $P$ into rhombuses such that no two intersect and no vertex of any rhombus is on the edge of other rhombuses or $P$. (a) Prove that number of rhombuses is a function of $n$. Find the value of this function. Also find the number of vertices and edges of the rhombuses as a function of $n$. (b) Prove or disprove that there always exists an edge $e$ of $P$ such that by erasing all the segments parallel to $e$ the remaining rhombuses are connected. (c) Is it true that each two rhombus-ulations can turn into each other using the following algorithm multiple times? Algorithm: Take a hexagon -not necessarily regular- consisting of 3 rhombuses and re-rhombus-ulate the hexagon. (d) Let $f(n)$ be the number of ways to rhombus-ulate $P$. Prove that:\[\Pi_{k=1}^{n-1} ( \binom{k}{2} +1) \leq f(n) \leq \Pi_{k=1}^{n-1} k^{n-k} \]

The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is: \[ \begin{tabular}[t]{|c|c|c|c|c|c|}\hline x&0&1&2&3&4\\\hline y&100&90&70&40&0\\\hline \end{tabular}\] $\textbf{(A)}\ y=100-10x \qquad \textbf{(B)}\ y=100-5x^2 \qquad \textbf{(C)}\ y=100-5x-5x^2 \qquad\\ \textbf{(D)}\ y=20-x-x^2 \qquad \textbf{(E)}\ \text{None of these}$

Horizontal line $m$ passes the center of circle $\odot O$. Line $l\perp m$, $l$ and $m$ intersect at $M$, and $M$ is on the right side of $O$. Three points $A,B,C$ ($B$ is in the middle) lie on line $l$, which are outside the circle, above line $m$. $AP,BQ,CR$ are tangent to $\odot O$ at $P,Q,R$. Prove: [b](a)[/b] If $l$ is tangent to $\odot O$, then $AB\cdot CR+BC\cdot AP=AC\cdot BQ$. [b](b)[/b] If $l$ and $\odot O$ intersect, then $AB\cdot CR+BC\cdot AP<AC\cdot BQ$. [b](c)[/b] If $l$ and $\odot O$ are apart, then $AB\cdot CR+BC\cdot AP>AC\cdot BQ$.

Frankie the Frog starts his morning at the origin in $\mathbb{R}^2$. He decides to go on a leisurely stroll, consisting of $3^1+3^{10}+3^{11}+3^{100}+3^{111}+3^{1000}$ moves, starting with the first move. On the $n$th move, he hops a distance of $$\max\{k\in\mathbb{Z}:3^k|n\}+1,$$ then turns $90^{\circ}$ counterclockwise. What is the square of the distance from his final position to the origin?

There are $n$ guests at a party. Any two guests are either friends or not friends. Every guest is friends with exactly four of the other guests. Whenever a guest is not friends with two other guests, those two other guests cannot be friends with each other either. What are the possible values of $n$?

For a positive integer $m$, prove that the number of pairs of positive integers $(x,y)$ which satisfies the following two conditions is even or $0$. (i): $x^2-3y^2+2=16m$ (ii): $2y \le x-1$

Solve in the set of rational numbers the equation $$2\sqrt{3(x + 1)^2} -3 \sqrt{2(y - 2)^2}= 4\sqrt2 + 5|\sqrt2 - \sqrt3|$$

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

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.