[b]9.[/b] Let $X_0, X_1, \dots $ be independent, indentically distributed, nondegenerate random variables, and let $0<\alpha <1$ be a real number. Assume that the series $\sum_{k=1}^{\infty} \alpha^{k} X_k$ is convergent with probability one. Prove that the distribution function of the sum is continuous. ([b]P. 23[/b]) [T. F. Móri]

Given a checkered rectangle of size n × m. Is it always possible to mark $3$ or $4$ nodes of a rectangle so that at least one of the marked nodes lay on each straight line containing the side of the rectangle, and the non-self-intersecting polygon with vertices at these nodes has an area equal to $$\dfrac{1}{2}\min \left ( \text{gcd}(n, m), \dfrac{n+m}{\text{gcd}(n, m)} \right)$$?

Inside of a square whose side length is $1$ there are a few circles such that the sum of their circumferences is equal to $10$. Show that there exists a line that meets at least four of these circles.

A flea moves in the positive direction on the real Ox axis, starting from the origin. He can only jump over distances equal with $\sqrt 2$ or $\sqrt{2005}$. Prove that there exists $n_0$ such that the flea can reach any interval $[n,n+1]$ with $n\geq n_0$.

A card deck consists of $1024$ cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of his cards so that each of the ten digits occurs on an even number of his remaining cards. If one player can do this but the other one cannot, the one who can is the winner; otherwise a draw is declared. Determine all possible first moves of the first player after which he has a winning strategy. [i]Proposed by Ilya Bogdanov & Vladimir Bragin, Russia[/i]

Prove that the equation $$\frac x{1+x^2}+\frac y{1+y^2}+\frac z{1+z^2}=\frac1{1996}$$has finitely many solutions in positive integers.

Let $c_1,c_2,\ldots ,c_n,b_1,b_2,\ldots ,b_n$ $(n\geq 2)$ be positive real numbers. Prove that the equation \[ \sum_{i=1}^nc_i\sqrt{x_i-b_i}=\frac{1}{2}\sum_{i=1}^nx_i\] has a unique solution $(x_1,\ldots ,x_n)$ if and only if $\sum_{i=1}^nc_i^2=\sum_{i=1}^nb_i$.

Prove that the product of $8$ consecutive natural numbers can never be a fourth power of natural number.

$A, B$ are points on circle $O$ satisfying $\angle AOB < 120^{\circ} $. $C$ is a point on the tangent line of $O$ passing through $A$ satisfying $AB=AC$ and $\angle BAC < 90^{\circ} $. $D$ is the intersection of $O$ and $BC$ not $B$, and $I$ is the incenter of $ABD$. Prove that $AE=AC$ where $E$ is the intersection of $CI$ and $AD$.

Find the maximum number of $3 \times 5\times 7$ boxes that can be placed inside a $11\times 35\times 39$ box. For the number found, indicate how you would place that number of boxes inside the box.

Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta=\angle PAB=\angle QPC=\angle RQB=\cdots$ will produce in a length that is $120$ meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.) [asy] import olympiad; draw((-50,15)--(50,15)); draw((50,15)--(50,-15)); draw((50,-15)--(-50,-15)); draw((-50,-15)--(-50,15)); draw((-50,-15)--(-22.5,15)); draw((-22.5,15)--(5,-15)); draw((5,-15)--(32.5,15)); draw((32.5,15)--(50,-4.090909090909)); label("$\theta$", (-41.5,-10.5)); label("$\theta$", (-13,10.5)); label("$\theta$", (15.5,-10.5)); label("$\theta$", (43,10.5)); dot((-50,15)); dot((-50,-15)); dot((50,15)); dot((50,-15)); dot((50,-4.09090909090909)); label("$D$",(-58,15)); label("$A$",(-58,-15)); label("$C$",(58,15)); label("$B$",(58,-15)); label("$S$",(58,-4.0909090909)); dot((-22.5,15)); dot((5,-15)); dot((32.5,15)); label("$P$",(-22.5,23)); label("$Q$",(5,-23)); label("$R$",(32.5,23)); [/asy] $\textbf{(A)}~\arccos\frac{5}{6}\qquad\textbf{(B)}~\arccos\frac{4}{5}\qquad\textbf{(C)}~\arccos\frac{3}{10}\qquad\textbf{(D)}~\arcsin\frac{4}{5}\qquad\textbf{(E)}~\arcsin\frac{5}{6}$

Let $m$ and $n$ be positive integers where $m$ has $d$ digits in base ten and $d\leq n$. Find the sum of all the digits (in base ten) of the product $(10^n-1)m$.

Let $ABC$ be an equilateral triangle of side length $6$ inscribed in a circle $\omega$. Let $A_1,A_2$ be the points (distinct from $A$) where the lines through $A$ passing through the two trisection points of $BC$ meet $\omega$. Define $B_1,B_2,C_1,C_2$ similarly. Given that $A_1,A_2,B_1,B_2,C_1,C_2$ appear on $\omega$ in that order, find the area of hexagon $A_1A_2B_1B_2C_1C_2$.

Find the mean value of the function $\ln r$ on the circle $(x - a)^2 + (y-b)^2 = R^2$ (of the function $1/r$ on the sphere).

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. [asy]unitsize(.3cm); defaultpen(linewidth(.8pt)); path c=Circle((0,2),1); filldraw(Circle((0,0),3),grey,black); filldraw(Circle((0,0),1),white,black); filldraw(c,white,black); filldraw(rotate(60)*c,white,black); filldraw(rotate(120)*c,white,black); filldraw(rotate(180)*c,white,black); filldraw(rotate(240)*c,white,black); filldraw(rotate(300)*c,white,black);[/asy]$ \textbf{(A)}\ \pi \qquad \textbf{(B)}\ 1.5\pi \qquad \textbf{(C)}\ 2\pi \qquad \textbf{(D)}\ 3\pi \qquad \textbf{(E)}\ 3.5\pi$

Prove that a right triangle has an angle equal to $30^o$ if and only if the center of the circle inscribed in this triangle is located on the perpendicular bisector of the median taken from the vertex of the right angle of the triangle.

Place three discs with radius $ r$ in a square with sides of length 1 so that the discs do not intersect: as on the figure. What is the greatest possible value of $ r$? [img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1997Number8.jpg[/img] A. $ \frac {1}{3}$ B. $ \frac {1}{4}$ C. $ \frac {\sqrt {2}}{6}$ D. $ 2 \sqrt {2} \minus{} \sqrt {6}$ E. $ \frac {\sqrt {2}}{1 \plus{} 2 \sqrt {2} \plus{} \sqrt {3}}$

A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether? $ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 15 $

Points $A,B,C$ with $AB<BC$ lie in this order on a line. Let $ABDE$ be a square. The circle with diameter $AC$ intersects the line $DE$ at points $P$ and $Q$ with $P$ between $D$ and $E$. The lines $AQ$ and $BD$ intersect at $R$. Prove that $DP=DR$.

For any two points $A (x_1 , y_1)$ and $B (x_2, y_2)$, the distance $r (A, B)$ between them is determined by the equality $r(A, B) = | x_1- x_2 | + | y_1 - y_2 |$. Prove that the triangle inequality $r(A, C) + r(C, B) \ge r(A, B)$. holds for the distance introduced in this way . Let $A$ and $B$ be two points of the plane (you can take $A(1, 3)$, $B(3, 7)$). Find the locus of points $C$ for which a) $r(A, C) + r(C, B) = r(A, B)$ b) $r(A, C) = r(C, B).$

Is it possible in a plane mark $10$ red, $10$ blue and $10$ green points (all distinct) such that three conditions hold: $i)$ For every red point $A$ there exists a blue point closer to point $A$ than any other green point $ii)$ For every blue point $B$ there exists a green point closer to point $B$ than any other red point $iii)$ For every green point $C$ there exists a red point closer to point $C$ than any other blue point

Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$

Let $A$ consist of $16$ elements of the set $\{1,2,3,\ldots, 106\}$, so that no two elements of $A$ differ by $6, 9, 12, 15, 18,$ or $21$. Prove that two elements of $A$ must differ by $3$.

Let ABC be an acute-angled triangle and AD one of its altitudes (D on BC). The line through D parallel to AB is denoted by $l$, and t is the tangent to the circumcircle of ABC at A. Finally, let E be the intersection of $l$ and t. Show that CE and t are perpendicular to each other.

One hundred points are marked inside a circle, with no three in a line. Prove that it is possible to connect the points in pairs such that all fifty lines intersect one another inside the circle.