Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$. Find $\lim_{n\rightarrow\infty}a_{n}$.

A $tetromino$ is a figure made up of four unit squares connected by common edges. [List=i] [*] If we do not distinguish between the possible rotations of a tetromino within its plane, prove that there are seven distinct tetrominos. [*]Prove or disprove the statement: It is possible to pack all seven distinct tetrominos into $4\times 7$ rectangle without overlapping. [/list]

We call a positive integer [i]heinersch[/i] if it can be written as the sum of a positive square and positive cube. Prove: There are infinitely many heinersch numbers $h$, such that $h-1$ and $h+1$ are also heinersch.

We want to partition the integers $1, 2, 3, . . . , 100$ into several groups such that within each group either any two numbers are coprime or any two are not coprime. At least how many groups are needed for such a partition? [i]We call two integers coprime if they have no common divisor greater than $1$.[/i]

Determine all non-constant polynomials $X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0$ with integer coefficients for which the roots are exactly the numbers $a_0,a_1,\ldots ,a_{n-1}$ (with multiplicity).

Consider all segments dividing the area of a triangle $ABC$ in two equal parts. Find the length of the shortest segment among them, if the side lengths $a,$ $b,$ $c$ of triangle $ABC$ are given. How many of these shortest segments exist ?

Four $L$s are equivalent to three $M$s. Nine $M$s are equivalent to fourteen $T$ s. Seven $T$ s are equivalent to two $W$ s. If Kevin has thirty-six $L$s, how many $W$ s would that be equivalent to?

Mr. and Mrs. Zeta want to name their baby Zeta so that its monogram (first, middle, and last initials) will be in alphabetical order with no letter repeated. How many such monograms are possible? $\text{(A)} \ 276 \qquad \text{(B)} \ 300 \qquad \text{(C)} \ 552 \qquad \text{(D)} \ 600 \qquad \text{(E)} \ 15600$

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Find all positive integers $a, b, c$ such that $3abc+11(a+b+c)=6(ab+bc+ac)+18.$

This graph depicts the torque output of a hypothetical gasoline engine as a function of rotation frequency. The engine is incapable of running outside of the graphed range. [asy] size(200); draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); for (int i=1;i<10;++i) { draw((i,0)--(i,7),dashed+linewidth(0.5)); } for (int j=1;j<7;++j) { draw((0,j)--(10,j),dashed+linewidth(0.5)); } draw((0,0)--(0,-0.3)); draw((4,0)--(4,-0.3)); draw((8,0)--(8,-0.3)); draw((0,0)--(-0.3,0)); draw((0,2)--(-0.3,2)); draw((0,4)--(-0.3,4)); draw((0,6)--(-0.3,6)); label("0",(0,-0.5),S); label("1000",(4,-0.5),S); label("2000",(8,-0.5),S); label("0",(-0.5,0),W); label("10",(-0.5,2),W); label("20",(-0.5,4),W); label("30",(-0.5,6),W); label("I",(1,-1.5),S); label("II",(6,-1.5),S); label("III",(9,-1.5),S); label(scale(0.95)*"Engine Revolutions per Minute",(5,-3.5),N); label(scale(0.95)*rotate(90)*"Output Torque (Nm)",(-1.5,3),W); path A=(0.9,2.7)--(1.213, 2.713)-- (1.650, 2.853)-- (2.087, 3)-- (2.525, 3.183)-- (2.963, 3.471)-- (3.403, 3.888)-- (3.823, 4.346)-- (4.204, 4.808)-- (4.565, 5.277)-- (4.945, 5.719)-- (5.365, 6.101)-- (5.802, 6.298)-- (6.237, 6.275)-- (6.670, 6.007)-- (7.101, 5.600)-- (7.473, 5.229)-- (7.766, 4.808)-- (8.019, 4.374)-- (8.271, 3.894)-- (8.476, 3.445)-- (8.568, 2.874)-- (8.668, 2.325)-- (8.765, 1.897)-- (8.794, 1.479)--(8.9,1.2); draw(shift(0.1*right)*shift(0.2*down)*A,linewidth(3)); [/asy] At what engine RPM (revolutions per minute) does the engine produce maximum power? (A) $\text{I}$ (B) At some point between $\text{I}$ and $\text{II}$ (C) $\text{II}$ (D) At some point between $\text{II}$ and $\text{III}$ (E) $\text{III}$

[asy] for(int a=0; a<4; ++a) { draw((a,0)--(a,3)); } for(int b=0; b<4; ++b) { draw((0,b)--(3,b)); }[/asy] Placing no more than one $x$ in each small square, what is the greatest number of $x$'s that can be put on the grid shown without getting three $x$'s in a row vertically, horizontally, or diagonally? $ \text{(A)}\ 2\qquad\text{(B)}\ 3\qquad\text{(C)}\ 4\qquad\text{(D)}\ 5\qquad\text{(E)}\ 6 $

([b]8[/b]) Evaluate the integral $ \int_0^1\ln x \ln(1\minus{}x)\ dx$.

Let $ABCDEF$ be a regular hexagon with side length $2$. A circle with radius $3$ and center at $A$ is drawn. Find the area inside quadrilateral $BCDE$ but outside the circle. [i]Proposed by Carl Joshua Quines.[/i]

Let $m$ and $n$ be arbitrary non-negative integers. Prove that \[\frac{(2m)!(2n)!}{m! n!(m+n)!}\] is an integer.

$101$ people, sitting at a round table in any order, had $1,2,... , 101$ cards, respectively. A transfer is someone give one card to one of the two people adjacent to him. Find the smallest positive integer $k$ such that there always can through no more than $ k $ times transfer, each person hold cards of the same number, regardless of the sitting order.

$ABC$ is a triangle with a right angle at $C$. $M_1$ and $M_2$ are two arbitrary points inside $ABC$, and $M$ is the midpoint of $M_1M_2$. The extensions of $BM_1,BM$ and $BM_2$ intersect $AC$ at $N_1,N$ and $N_2$ respectively. Prove that $\frac{M_1N_1}{BM_1}+\frac{M_2N_2}{BM_2}\geq 2\frac{MN}{BM}$

Let $ABC$ be a triangle and $G$ its centroid. Let $G_a, G_b$ and $G_c$ be the orthogonal projections of $G$ on sides $BC, CA$, respectively $AB$. If $S_a, S_b$ and $S_c$ are the symmetrical points of $G_a, G_b$, respectively $G_c$ with respect to $G$, prove that $AS_a, BS_b$ and $CS_c$ are concurrent. Liana Topan

In a city of gnomes there are $1000$ identical towers, each of which has $1000$ stories, with exactly one gnome living on each story. Every gnome in the city wears a hat colored in one of $1000$ possible colors and any two gnomes in the same tower have different hats. A pair of gnomes are friends if they wear hats of the same color, one of them lives in the $k$-th story of his tower and the other one in the $(k+1)$-st story of his tower. Determine the maximal possible number of pairs of gnomes which are friends. [i]Authored by Nikola Velov[/i]

The numbers $1, 2, \ldots 49$ are placed in a $7\times 7$ array, and the sum of the numbers in each row and in each column is computed. Some of these $14$ sums are odd while others are even. Let $A$ denote the sum of all the odd sums and $B$ the sum of all even sums. Is it possible that the numbers were placed in the array in such a way that $A = B$?

Simplify $\frac{0.\overline{3}}{1.\overline{3}}$. $\text{(A) }\frac{3}{13}\qquad\text{(B) }\frac{1}{4}\qquad\text{(C) }\frac{3}{11}\qquad\text{(D) }\frac{1}{3}\qquad\text{(E) }\frac{3}{4}$

The square $A_1B_1C_1D_1$ is inscribed in the right triangle $ABC$ (with $C=90$) so that points $A_1$, $B_1$ lie on the legs $CB$ and $CA$ respectively ,and points $C_1$, $D_1$ lie on the hypotenuse $AB$. The circumcircle of triangles $B_1A_1C$ an $AC_1B_1$ intersect at $B_1$ and $Y$. Prove that the lines $A_1X$ and $B_1Y$ meet on the hypotenuse $AB$.

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

Which is the best value for the mass of the block? (a) $\text{3 kg}$ (b) $\text{5 kg}$ (c) $\text{10 kg}$ (d) $\text{20 kg}$ (e) $\text{30 kg}$

$a,b,c $ are positive real numbers . Prove that $\sqrt[7]{\frac{a}{b+c}+\frac{b}{c+a}} +\sqrt[7]{\frac{b}{c+a}+\frac{c}{b+a}}+\sqrt[7]{\frac{c}{a+b}+\frac{a}{b+c}}\geq 3$

Let $ABCDEFGHI$ be a regular polygon with $9$ sides and the vertices are written in the counterclockwise and let $ABJKLM$ be a regular polygon with $6$ sides and the vertices are written in the clockwise. Prove that $\angle HMG=\angle KEL$. Note: The polygon $ABJKLM$ is inside of $ABCDEFGHI$.