A pentagon $ABCDE$ inscribed in a circle for which $BC<CD$ and $AB<DE$ is the base of a pyramid with vertex $S$. If $AS$ is the longest edge starting from $S$, prove that $BS>CS$.

You have $\text{5}$ different strings with weights tied at various point, all hanging from the ceiling, and reaching down to the floor. The string is released at the top, allowing the weights to fall. Which one will create a regular, uniform beating sound as the weights hit the floor? [asy] size(300); // (A) bar picture bar; draw(bar,(0,0)--(0,42)); for (int i=0;i<43;i+=2) { draw(bar,(-2,i)--(-3,i)); } add(bar); picture ball; filldraw(ball,circle((0,0),0.5),gray); add(ball); add(shift(12*up)*ball); add(shift(22*up)*ball); add(shift(30*up)*ball); add(shift(36*up)*ball); add(shift(40*up)*ball); add(shift(42*up)*ball); label(scale(0.75)*"(A)",(-1,0),2*S); // (B) bar add(shift(15*right)*bar); add(shift(15*right)*ball); add(shift(6*up)*shift(15*right)*ball); add(shift(12*up)*shift(15*right)*ball); add(shift(18*up)*shift(15*right)*ball); add(shift(24*up)*shift(15*right)*ball); add(shift(30*up)*shift(15*right)*ball); add(shift(36*up)*shift(15*right)*ball); add(shift(42*up)*shift(15*right)*ball); label(scale(0.75)*"(B)",(14,0),2*S); // (C) bar add(shift(30*right)*bar); add(shift(30*right)*ball); add(shift(2*up)*shift(30*right)*ball); add(shift(6*up)*shift(30*right)*ball); add(shift(12*up)*shift(30*right)*ball); add(shift(20*up)*shift(30*right)*ball); add(shift(30*up)*shift(30*right)*ball); add(shift(42*up)*shift(30*right)*ball); label(scale(0.75)*"(C)",(29,0),2*S); // (D) bar add(shift(45*right)*bar); add(shift(45*right)*ball); add(shift(2*up)*shift(45*right)*ball); add(shift(8*up)*shift(45*right)*ball); add(shift(18*up)*shift(45*right)*ball); add(shift(32*up)*shift(45*right)*ball); label(scale(0.75)*"(D)",(44,0),2*S); // (E) bar add(shift(60*right)*bar); add(shift(60*right)*ball); add(shift(2*up)*shift(60*right)*ball); add(shift(10*up)*shift(60*right)*ball); add(shift(28*up)*shift(60*right)*ball); label(scale(0.75)*"(E)",(59,0),2*S); [/asy]

Let there be $n$ students and $m$ clubs. The students joined the clubs so that the following is true: - For all students $x$, you can choose some clubs such that $x$ is the only student who joined all of the chosen clubs. Let the number of clubs each student joined be $a_1,a_2,...,a_m$. Prove that $$a_1!(m - a_1)! + a_2!(m - a_2)! + ... + a_n!(m -a_n)! \le m!$$

Consider two triangles, $ABC$ and $DEF$, and any point $O$. We take any point $X$ in $\vartriangle ABC$ and any point $Y$ in $\vartriangle DEF$ and draw a parallelogram $OXY Z$. Prove that the locus of all possible points $Z$ form a polygon. How many sides can it have? Prove that its perimeter is equal to the sum of perimeters of the original triangles.

Fernanda decided to decorate a square blanket with a ribbon and buttons, placing a button in the center of each square where the ribbon passes and forming the design indicated in the figure. If Fernanda sews the first button in the shaded square on line $0$, on which line does she sew the $2007$th button? [img]https://cdn.artofproblemsolving.com/attachments/2/9/0c9c85ec6448ee3f6f363c8f4bcdd5209f53f6.png[/img]

An equilateral triangle $ABC$ is inscribed in a circle $\Omega$ and has incircle $\omega$. Points $P$ and $Q$ are in segments $AC$ and $AB$, respectively, such that $PQ$ is tangent to $\omega$. The circle $\Omega_B$ has center $P$ and radius $PB$ and the circle $\Omega_C$ is defined similarly. Prove that $\Omega$, $\Omega_B$ and $\Omega_C$ have a common point.

Solve the system $\begin{cases} 10x_1 + 3x_2 + 4x_3 + x_4 + x_5 = 0 \\ 11x_2 + 2x_3 + 2x_4 + 3x_5 + x_6 = 0 \\ 15x_3 + 4x_4 + 5x_5 + 4x_6 + x_7 = 0 \\ 2x_1 + x_2 - 3x_3 + 12x_4 - 3x_5 + x_6 + x_7 = 0 \\ 6x_1 - 5x_2 + 3x_3 - x_4 + 17x_5 + x_6 = 0 \\ 3x_1 + 2x_2 - 3x_3 + 4x_4 + x_5 - 16x_6 + 2x_7 = 0\\ 4x_1 - 8x_2 + x_3 + x_4 + 3x_5 + 19x_7 = 0 \end{cases}$

In triangle $ABC$ the bisector of angle $A$, the perpendicular to side $AB$ from its midpoint, and the altitude from vertex $B$, intersect in the same point. Prove that the bisector of angle $A$, the perpendicular to side $AC$ from its midpoint, and the altitude from vertex $C$ also intersect in the same point.

$3m$ balls numbered $1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m$ are distributed into $8$ boxes so that any two boxes contain identical balls. Find the minimal possible value of $m$.

An ant lies on each corner of a $20 \times 23$ rectangle. Each second, each ant independently and randomly chooses to move one unit vertically or horizontally away from its corner. After $10$ seconds, find the expected area of the convex quadrilateral whose vertices are the positions of the ants.

Let $ABC$ be a triangle with $\angle ABC = 60^{\circ}$. Find the angles of the triangle if $\angle BHI = 60^{\circ}$, where $H$ and $I$ are the orthocenter and incenter of $ABC$

Let $I$ be the incenter of the acute triangle $\triangle ABC$, and $BI$, $CI$ intersect the altitude of $\triangle ABC$ through $A$ at $U$, $V$, respectively. The circle with $AI$ as a diameter intersects $\odot(ABC)$ again at $T$, and $\odot(TUV)$ intersects the segment $BC$ and $\odot(ABC)$ at $P$, $Q$, respectively. Let $R$ be another intersection of $PQ$ and $\odot(ABC)$. Show that $AR\parallel BC$.

Let $ABCD$ be a square with the side of $20$ units. Amir divides this square into $400$ unit squares. Reza then picks $4$ of the vertices of these unit squares. These vertices lie inside the square $ABCD$ and define a rectangle with the sides parallel to the sides of the square $ABCD.$ There are exactly $24$ unit squares which have at least one point in common with the sides of this rectangle. Find all possible values for the area of a rectangle with these properties. [hide="Note"][i]Note:[/i] Vid changed to Amir, and Eva change to Reza![/hide]

Prove that any identity that holds for every finite $ n$-distributive lattice also holds for the lattice of all convex subsets of the $ (n\minus{}1)$-dimensional Euclidean space. (For convex subsets, the lattice operations are the set-theoretic intersection and the convex hull of the set-theoretic union. We call a lattice $ n$-$ \textit{distributive}$ if \[ x \wedge (\bigvee_{i\equal{}0}^n y_i)\equal{}\bigvee_{j\equal{}0}^n(x \wedge (\bigvee_{0\leq i \leq n, \;i \not\equal{} j\ }y_i))\] holds for all elements of the lattice.) [i]A. Huhn[/i]

In some galaxy there are $1000000$ planets and on each of them there are at least $101$ portals. Each portal allows to teleport between some two planets, no two planets are connected by more than one portal. It is known that starting from any planet using portals you can get to any other planet, while it is impossible to return to that planet using once at most 5 different portals. Prove that starting from any planet you can get to any other planet within a year, using at most one portal daily (a year consists of 365 days). [i]M. Zorka[/i]

Let $ABC$ be an acute triangle with $AB<AC$ and incenter $I$. Let $D$ be the projection of $I$ onto $BC$. Let $H$ be the orthocenter of $ABC$ and suppose that $\angle IDH=\angle CBA-\angle ACB$. Prove that $AH=2ID$.

All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of$ k$ for which the same $k$-digit number can be found in at least two different places in this string?

The best parts of grandma’s $30$ cm $ \times 30$ cm square shaped pie are the edges. For this reason grandma’s three grandchildren would like to split the pie between each other so that everyone gets the same amount (of the area) of the pie, but also of the edges. Can they cut the pie into three connected pieces like that?

Determine, whether exists function $f$, which assigns each integer $k$, nonnegative integer $f(k)$ and meets the conditions: $f(0) > 0$, for each integer $k$ minimal number of the form $f(k - l) + f(l)$, where $l \in \mathbb{Z}$, equals $f(k)$.

Find all integers $a$ such that there are infinitely many positive integers $n$ such that $n$ divides $\phi(n)!+a$.

There is a sequence of integers $ a_1, a_2, \ldots, a_{2n+1} $ with the following property: after eliminating any term, the remaining ones can be divided into two groups of $ n $ terms such that the sum of the terms in the first group is equal to the sum words in the second. Prove that all terms of the sequence are equal.

There are three squares in the picture. Find the sum of angles $ADC$ and $BDC$. [img]https://cdn.artofproblemsolving.com/attachments/c/9/885a6c6253fca17e24528f8ba8a5d31a18c845.png[/img]

Find \[\lim_{n\to\infty}\int _0^{2n} e^{-2x}\left|x-2\lfloor\frac{x+1}{2}\rfloor\right|\ dx.\] [i]1985 Tohoku University entrance exam/Mathematics, Physics, Chemistry, Biology[/i]

Let $ABC$ be a triangle with side lengths $AB=13$, $BC=15$, $CA=14$. Let $\ell$ be the line passing through $A$ parallel to $BC$. Define $H$ as the orthocenter of $\triangle ABC$, and extend $BH$ to intersect $AC$ at $E$ and $\ell$ at $G$. Similarly, extend $CH$ to intersect $AB$ at $F$ and $\ell$ at $D$. Let $M$ be the midpoint of $BC$, and let $AM$ intersect the circumcircle of $AEF$ again at $P$. The ratio $\frac{PD}{PG}$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Individual #15[/i]

Let $a$ and $b$ be positive integers such that $\text{gcd}(a, b) + \text{lcm}(a, b)$ is a multiple of $a+1$. If $b \le a$, show that $b$ is a perfect square.