Let $S$ be a circle with radius $2$, let $S_1$ be a circle,with radius $1$ and tangent, internally to $S$ in $B$ and let $S_2$ be a circle, with radius $1$ and tangent to $S_1$ in $A$, but $S_2$ isn't tangent to $S$. If $K$ is the point of intersection of the line $AB$ and the circle $S$, prove that $K$ is in the circle $S_2$.

$ ABC$ is a triangle which is not isosceles. Let the circumcenter and orthocenter of $ ABC$ be $ O$, $ H$, respectively, and the altitudes of $ ABC$ be $ AD$, $ BC$, $ CF$. Let $ K\neq A$ be the intersection of $ AD$ and circumcircle of triangle $ ABC$, $ L$ be the intersection of $ OK$ and $ BC$, $ M$ be the midpoint of $ BC$, $ P$ be the intersection of $ AM$ and the line that passes $ L$ and perpendicular to $ BC$, $ Q$ be the intersection of $ AD$ and the line that passes $ P$ and parallel to $ MH$, $ R$ be the intersection of line $ EQ$ and $ AB$, $ S$ be the intersection of $ FD$ and $ BE$. If $ OL \equal{} KL$, then prove that two lines $ OH$ and $ RS$ are orthogonal.

Let $A\subset \{1,2,\ldots,4014\}$, $|A|=2007$, such that $a$ does not divide $b$ for all distinct elements $a,b\in A$. For a set $X$ as above let us denote with $m_{X}$ the smallest element in $X$. Find $\min m_{A}$ (for all $A$ with the above properties).

Prove that for each prime number $p$ and positive integer $n$, $p^n$ divides \[\binom{p^n}{p}-p^{n-1}. \]

The sum of the (decimal) digits of a natural number $n$ equals $100$, and the sum of digits of $44n$ equals $800$. Determine the sum of digits of $3n$.

Let $A=\frac{2^2+3\cdot 2 + 1}{3! \cdot 4!} + \frac{3^2+3\cdot 3 + 1}{4! \cdot 5!} + \frac{4^2+3\cdot 4 + 1}{5! \cdot 6!} + \dots + \frac{10^2+3\cdot 10 + 1}{11! \cdot 12!}$. What is the remainder when $11!\cdot 12! \cdot A$ is divided by $11$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10 $

The number of real roots of the equation $\lg^2x-[\lg x]-2=0$ is________.

A postman wants to divide $n$ packages with weights $1, 2, 3, 4, n$ into three groups of exactly the same weight. Can he do this if (a) $n = 2011$ ? (b) $n = 2012$ ?

The numbers from $1$ to $49$ are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number $7.$ How many of these four numbers are prime? [asy] size(6cm); fill((4,0)--(5,0)--(5,1)--(4,1)--cycle,mediumgray); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,mediumgray); fill((1,3)--(1,4)--(2,4)--(2,3)--cycle,mediumgray); fill((0,4)--(0,5)--(1,5)--(1,4)--cycle,mediumgray); label(scale(.9)*"$1$", (3.5,3.5)); label(scale(.9)*"$2$", (4.5,3.5)); label(scale(.9)*"$3$", (4.5,4.5)); label(scale(.9)*"$4$", (3.5,4.5)); label(scale(.9)*"$5$", (2.5,4.5)); label(scale(.9)*"$6$", (2.5,3.5)); label(scale(.9)*"$7$", (2.5,2.5)); draw((1,0)--(1,7)--(2,7)--(2,0)--(3,0)--(3,7)--(4,7)--(4,0)--(5,0)--(5,7)--(6,7)--(6,0)--(7,0)--(7,7),gray); draw((0,1)--(7,1)--(7,2)--(0,2)--(0,3)--(7,3)--(7,4)--(0,4)--(0,5)--(7,5)--(7,6)--(0,6)--(0,7)--(7,7),gray); draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(1.25)); [/asy] $\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$

Let $ABCD$ be a trapezoid with $AD \parallel BC$ and $\angle BCD < \angle ABC < 90^\circ$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$. The circumcircle $\omega$ of $\triangle BEC$ intersects the segment $CD$ at $X$. The lines $AX$ and $BC$ intersect at $Y$, while the lines $BX$ and $AD$ intersect at $Z$. Prove that the line $EZ$ is tangent to $\omega$ iff the line $BE$ is tangent to the circumcircle of $\triangle BXY$.

Determine all the solutions of the equation $x^3 + y^3 + z^3 = wx^2y^2z^2$ in natural numbers $x, y, z, w$. Justify your answer

Let the curve $ C: y\equal{}x\sqrt{9\minus{}x^2}\ (x\geq 0)$. (1) Find the maximum value of $ y$. (2) Find the area of the figure bounded by the curve $ C$ and the $ x$ axis. (3) Find the volume of the solid generated by rotation of the figure about the $ y$ axis.

Let $n$ be a positive integer. Show that there are positive real numbers $a_0, a_1, \dots, a_n$ such that for each choice of signs the polynomial $$\pm a_nx^n\pm a_{n-1}x^{n-1} \pm \dots \pm a_1x \pm a_0$$ has $n$ distinct real roots. (Proposed by Stephan Neupert, TUM, München)

We call a pair of distinct numbers $(a, b)$ a [i]binary pair[/i] if $ab+1$ is a power of two. Given a set $S$ of $n$ positive integers, what is the maximum possible numbers of binary pairs in S?

circle is inscribed in an equilateral triangle of side length $1$. Tangents to the circle are drawn that cut off equilateral triangles at each corner. Circles are inscribed in each of these equilateral triangles. If this process is repeated infinitely many times, what is the sum of the areas of all the circles? [img]https://cdn.artofproblemsolving.com/attachments/c/e/ef4000989155708db8cfa674dd00857afb9919.png[/img]

One hundred and one beetles are crawling in the plane. Some of the beetles are friends. Every one hundred beetles can position themselves so that two of them are friends if and only if they are at unit distance from each other. Is it always true that all one hundred and one beetles can do the same?

We say that $H$ permeates $G$ if $G$ and $H$ are finite groups and for all subgroup $F$ of $G$ there is $H'\cong H$ with $H'\le F$ or $F\le H'\le G$. Suppose that a non-abelian group $H$ permeates $G$ and let $S=\langle H'\le G | H'\cong H\rangle$. Show that $$|\bigcap_{H'\in S} H'|>1$$