The sides of the triangle $ABC$ lie on the sides of the angle $MAN$. Construct a triangle $ABC$ if the point $O$ of the intersection of its medians is given.

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

Peter Rabbit is hopping along the number line, always jumping in the positive $x$ direction. For his first jump, he starts at $0$ and jumps $1$ unit to get to the number $1.$ For his second jump, he jumps $4$ units to get to the number $5.$ He continues jumping by jumping $1$ unit whenever he is on a multiple of $3$ and by jumping $4$ units whenever he is on a number that is not a multiple of $3.$ What number does he land on at the end of his $100$th jump? $$ \mathrm a. ~ 297\qquad \mathrm b.~298\qquad \mathrm c. ~299 \qquad \mathrm d. ~300 \qquad \mathrm e. ~301 $$

Each face of a cube is divided into $n^2$ equal squares. The vertices of the squares are called [i]nodes[/i], so each face has $(n+1)^2$ nodes. $(a)$ If $n=2$,does there exist a closed polygonal line whose links are sids of the squares and which passes through each node exactly once? $(b)$ Prove that, for each $n$, such a polygonal line divides the surface area of the cube into two equal parts

Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$. [i]Amol Aggarwal.[/i]

Find the number of positive integer divisors of \[\sum_{k=0}^{50}(-3)^k\dbinom{100}{2k}.\] [i]Proposed by Serena Xu[/i]

Let $N$ be a natural number. Find (with prove) the number of solutions in the segment $[1,N]$ of the equation $x^2-[x^2]=(x-[x])^2$, where $[x]$ means the floor function of $x$.

The sequence $a_i$ is defined as $a_1 = 1$ and \[a_n = a_{\left\lfloor \dfrac{n}{2} \right\rfloor} + a_{\left\lfloor \dfrac{n}{3} \right\rfloor} + a_{\left\lfloor \dfrac{n}{4} \right\rfloor} + \cdots + a_{\left\lfloor \dfrac{n}{n} \right\rfloor} + 1\] for every positive integer $n > 1$. Prove that there are infinitely many values of $n$ such that $a_n \equiv n \mod 2012$.

Bobo the clown was juggling his spherical cows again when he realized that when he drops a cow is related to how many cows he started off juggling. If he juggles $1$, he drops it after $64$ seconds. When juggling $2$, he drops one after $55$ seconds, and the other $55$ seconds later. In fact, he was able to create the following table: \begin{tabular}{c|c ccccccccccc} cows started juggling & 1 & 2 & 3 & 4 & 5 & 6 &7 &8 &9 &10 & 11\\ seconds he drops after & 64 & 55 & 47 & 40 & 33 & 27 & 22& 18& 14 &13 &12\\ \hline cows started juggling & 12 &13 &14& 15 &16& 17& 18& 19 &20& 21 &22\\ seconds he drops after & 11 &10 &9 &8 &7& 6 &5 &4& 3& 2& 1 \end{tabular} He can only juggle up to $22$ cows. To juggle the cows the longest, what number of cows should he start off juggling? How long (in minutes) can he juggle for?

For each integer $n \ge 2$ we consider the last digit different from zero in the decimal expansion of $n!$. The infinite sequence of these digits starts with $2,6,4,2,2$. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.

Find at least one real number $A{}$ such that for any positive integer $n{}$ the distance between $\lceil A^n\rceil$ and the nearest square of an integer is equal to two. [i]Dmitry Krekov[/i]

Call an $n$-gon to be [i]lattice[/i] if its vertices are lattice points. Prove that inside every lattice convex pentagon there exists a lattice point.

Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.

Two players play alternatively on an infinite square grid. The first player puts an $X$ in an empty cell and the second player puts an $O$ in an empty cell. The first player wins if he gets $11$ adjacent $X$'s in a line - horizontally, vertically or diagonally. Show that the second player can always prevent the first player from winning.

The lines $y=x$, $y=2x$, and $y=3x$ are the three medians of a triangle with perimeter $1$. Find the length of the longest side of the triangle.

We know that the function $ f: \left[ 0,\frac{\pi }{2}\right]\longrightarrow [a,b], f(x)=\sqrt[n]{\cos x } +\sqrt[n]{\sin x} , $ is surjective for a given natural number $ n\ge 2. $ Determine the numbers $ a,b, $ and the monotony of $ f. $

Let $P_1(x)=\frac{1}{x}$ and $P_n(x)=P_{n-1}(x)+P_{n-1}(x-1)$ for every natural $ n$ greater than $1$. Find the value of $P_{2008}(2008)$. [i](Mathophile)[/i]

It is possible to choose $ x > \frac {2}{3}$ in such a way that the value of $ \log_{10}(x^2 \plus{} 3) \minus{} 2 \log_{10}x$ is $ \textbf{(A)}\ \text{negative} \qquad \textbf{(B)}\ \text{zero} \qquad \textbf{(C)}\ \text{one}$ $ \textbf{(D)}\ \text{smaller than any positive number that might be specified}$ $ \textbf{(E)}\ \text{greater than any positive number that might be specified}$

$ 2000(2000^{2000}) \equal{}$ $ \textbf{(A)}\ 2000^{2001} \qquad \textbf{(B)}\ 4000^{2000} \qquad \textbf{(C)}\ 2000^{4000}\qquad \textbf{(D)}\ 4,000,000^{2000} \qquad \textbf{(E)}\ 2000^{4,000,000}$

There are $n$ students in a circle, one behind the other, all facing clockwise. The students have heights $h_1 <h_2 < h_3 < \cdots < h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or lesss, the two students are permitted to switch places Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.

John, Paul, George, and Ringo baked a circular pie. Each cut a piece that was a sector of the circle. John took one-third of the whole pie. Paul took one-fourth of the whole pie. George took one-fifth of the whole pie. Ringo took one-sixth of the whole pie. At the end the pie had one sector remaining. Find the measure in degrees of the angle formed by this remaining sector.

An arbitrary triangle $ABC$ is given and a point $P$ lies on the side $AB$. It is requested to draw through $P$ a line that divides the triangle into two figures of the same area.

Evaluate $\int_e^{e^2} \frac{4(\ln x)^2+1}{(\ln x)^{\frac 32}}\ dx.$

The Feuerbach point of a scalene triangle lies on one of its bisectors. Prove that it bisects the segment between the corresponding vertex and the incenter. Proposed by: A.Zaslavsky

Let $ A_0 \equal{} (a_1,\dots,a_n)$ be a finite sequence of real numbers. For each $ k\geq 0$, from the sequence $ A_k \equal{} (x_1,\dots,x_k)$ we construct a new sequence $ A_{k \plus{} 1}$ in the following way. 1. We choose a partition $ \{1,\dots,n\} \equal{} I\cup J$, where $ I$ and $ J$ are two disjoint sets, such that the expression \[ \left|\sum_{i\in I}x_i \minus{} \sum_{j\in J}x_j\right| \] attains the smallest value. (We allow $ I$ or $ J$ to be empty; in this case the corresponding sum is 0.) If there are several such partitions, one is chosen arbitrarily. 2. We set $ A_{k \plus{} 1} \equal{} (y_1,\dots,y_n)$ where $ y_i \equal{} x_i \plus{} 1$ if $ i\in I$, and $ y_i \equal{} x_i \minus{} 1$ if $ i\in J$. Prove that for some $ k$, the sequence $ A_k$ contains an element $ x$ such that $ |x|\geq\frac n2$. [i]Author: Omid Hatami, Iran[/i]