Prove that for every pair of positive integers $k$ and $n$, there exists integer $x_1$, $x_2$,$...$, $x_k$ with $0 \le x_j \le 2^{k-1}\cdot \sqrt[k]{n}$ for $j = 1$, $2$, $...$, $k$, and such that $$x_1 + x^2_2+ x^3_3+...+ x^k_k= n.$$

Two circles $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ intersects $\omega_1$ again at $C$ and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ intersects the circle $\omega$ through $AO_1O_2$ at $F$. Prove that the length of segment $EF$ is equal to the diameter of $\omega$.

A finite sequence of decimal digits from $\{0,1,\cdots, 9\}$ is said to be [i]common[/i] if for each sufficiently large positive integer $n$, there exists a positive integer $m$ such that the expansion of $n$ in base $m$ ends with this sequence of digits. For example, $0$ is common because for any large $n$, the expansion of $n$ in base $n$ is $10$, whereas $00$ is not common because for any squarefree $n$, the expansion of $n$ in any base cannot end with $00$. Determine all common sequences. [i]Proposed by Wong Jer Ren[/i]

Piglet added together three consecutive whole numbers, then the next three numbers, and multiplied one sum by the other. Could the product be equal to $111,111,111$?

Let $H$ be a regular hexagon with side length one. Peter picks a point $P$ uniformly and at random within $H$, then draws the largest circle with center $P$ that is contained in $H$. What is this probability that the radius of this circle is less than $1/2$ ?

An $n \times n$ table is called good if one can paint its cells with three colors so that, for any two different rows and two different columns, the four cells at their intersections are not all of the same color. a)Show, that exists good $9 \times 9$ good table. b)Prove, that fif $n \times n$ table is good, then $n<11$

Triangle $T_1$ has sides of length $a_1$, $b_1$, and $c_1$; its area is $K_1$. Triangle $T_2$ has sides of length $a_2$, $b_2$, and $c_2$; its area is $K_2$. Triangle $T_3$ has sides of length $a_1 + a_2$, $b_1 + b_2$, and $c_1 + c_2$; its area is $K_3$. (a) Prove that $K_1^2 + K_2^2 < K_3^2$. (b) Prove that $\sqrt{K_1} + \sqrt{K_2} \le \sqrt{K_3} \,$.

The distance between the city and the house of Borya is 2km. Once Borya went from the city to his house with speed 4km/h. Simultaneously with that a dog Sharik started running out of house in the direction to city, and whenever Sharik meets Borya or the house, it starts running back (so the dog runs between Borya and the house), and when the dog runs to the house, its speed is 8km/h, and when it runs from the house, its speed is 12km/h. What distance will Sharik run until Borya comes to the house? [i]Yauheni Barabanau[/i]

A parallelogram $ABCD$ is given. The perpendicular from $C$ to $CD$ meets the perpendicular from $A$ to $BD$ at point $F$, and the perpendicular from $B$ to $AB$ meets the perpendicular bisector to $AC$ at point $E$. Find the ratio in which side $BC$ divides segment $EF$.

The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is $ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{2}{3} \qquad\textbf{(E)}\ 2 $

A chord $CD$ of a circle with center $O$ is perpendicular to a diameter $AB$. A chord $AE$ bisects the radius $OC$. Show that the line $DE$ bisects the chord $BC$ [i]V. Gordon[/i]

Find the remainder when \[9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}\] is divided by $ 1000$.

Let $ a\equal{}\sqrt[1992]{1992}$. Which number is greater \[ \underbrace{a^{a^{a^{\ldots^{a}}}}}_{1992}\quad\text{or}\quad 1992? \]

Determine the minimum value of $\dfrac{m^m}{1\cdot 3\cdot 5\cdot \ldots \cdot(2m-1)}$ for positive integers $m$.

$\xi, \xi'$ are iid random variables. let F have the distribution function $\xi+\xi'$, and G have the uniform distribution over the interval [-1,1]. Prove that $\max | F ( x ) - G ( x ) | \geq 10^{-1994}$ .

Brownian motion (for example, pollen grains in water randomly pushed by collisions from water molecules) simplified to one dimension and beginning at the origin has several interesting properties. If $B(t)$ denotes the position of the particle at time $t$, the average of $B(t)$ is $x=0$, but the averate of $B(t)^{2}$ is $t$, and these properties of course still hold if we move the space and time origins ($x=0$ and $t=0$) to a later position and time of the particle (past and future are independent). What is the average of the product $B(t)B(s)$?

Let $n \ge 3$ be an integer. On a circle, there are $n$ points. Each of them is labelled with a real number at most $1$ such that each number is the absolute value of the difference of the two numbers immediately preceding it in clockwise order. Determine the maximal possible value of the sum of all numbers as a function of $n$. (Walther Janous)

Along consecutive seven days, from Sunday to Saturday, let us call the days belonging to the same month a MB. For example, if the last day of a month is Sunday, the last MB of that month consists of the last day of that month. If a year is from January first to December $31$, find the maximum and minimum values of MB in one year.

Point $ P$ is inside equilateral $ \triangle ABC$. Points $ Q, R$ and $ S$ are the feet of the perpendiculars from $ P$ to $ \overline{AB}, \overline{BC}$, and $ \overline{CA}$, respectively. Given that $ PQ \equal{} 1, PR \equal{} 2$, and $ PS \equal{} 3$, what is $ AB$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 3 \sqrt {3}\qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 4 \sqrt {3}\qquad \textbf{(E)}\ 9$

Let $ N $ be a positive integer. A set $ S \subset \{ 1, 2, \cdots, N \} $ is called [i]allowed[/i] if it does not contain three distinct elements $ a, b, c $ such that $ a $ divides $ b $ and $ b $ divides $c$. Determine the largest possible number of elements in an allowed set $ S $.

Reduce the number $\sqrt[3]{2 +\sqrt5} + \sqrt[3]{2 -\sqrt5}$.

Let $C: y=x^2+ax+b$ be a parabola passing through the point $(1,\ -1)$. Find the minimum volume of the figure enclosed by $C$ and the $x$ axis by a rotation about the $x$ axis. Proposed by kunny

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

Consider the expression $M(n, m)=|n\sqrt{n^2+a}-bm|$, where $n$ and $m$ are arbitrary positive integers and the numbers $a$ and $b$ are fixed, moreover $a$ is an odd positive integer and $b$ is a rational number with an odd denominator of its representation as an irreducible fraction. Prove that there is [b]a)[/b] no more than a finite number of pairs $(n, m)$ for which $M(n, m)=0$; [b]b)[/b] a positive constant $C$ such that the inequality $M(n, m)\geqslant0$ holds for all pairs $(n, m)$ with $M(n, m)\ne 0$.

[asy]draw((0,0)--(1,0)--(1,1)--(2,1)--(2,2)--(3,2)--(3,3)--(2,3)--(2,4)--(1,4)--(1,5)--(0,5)--(0,4)--(-1,4)--(-1,1)--(0,1)--cycle); draw((0,1)--(1,1)); draw((-1,2)--(2,2)); draw((-1,3)--(2,3)); draw((0,4)--(1,4)); draw((0,1)--(0,4)); draw((1,1)--(1,4)); draw((2,2)--(2,3)); draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle); draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle); draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle); draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle); draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle); draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle); label("H",(0.5,0.2),N); label("G",(1.5,1.2),N); label("F",(-0.5,1.2),N); label("E",(2.5,2.2),N); label("D",(-0.5,2.2),N); label("C",(1.5,3.2),N); label("B",(-0.5,3.2),N); label("A",(0.5,4.2),N);[/asy] Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box? \[ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6 \]