Let $k>1$ be a positive integer. On a $\text{k} \times \text{k}$ square grid, Tom and Jerry are on opposite corners, with Tom at the top right corner. Both can move to an adjacent square every move, where two squares are adjacent if they share a side. Tom and Jerry alternate moves, with Jerry going first. Tom [i]catches[/i] Jerry if they are on the same square. We aim to answer to the following question: What is the smallest number of moves that Tom needs to guarantee catching Jerry? (a) Without proof, find the answer in the cases of $k=2,3,4$, and (correctly) guess what the answer is in terms of $k$. We'll refer to this answer as $A(k)$. (b) Find a strategy that Jerry can use to guarantee that Tom takes at least $A(k)$ moves to catch Jerry. Now, you will find a strategy for Tom to catch Jerry in at most $A(k)$ moves, no matter what Jerry does. (c) Find, with proof, a working strategy for $k=5$. (d) Find, with proof, a working strategy for all $k \geq 2$.

The function $f$ is given by the table \[\begin{array}{|c||c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 4 & 1 & 3 & 5 & 2 \\ \hline \end{array}\] If $u_0=4$ and $u_{n+1}=f(u_n)$ for $n\geq 0$, find $u_{2002}$. $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

The sum of $n$ consecutive integers is divisible by $n$ for some $n > 1$. For which $n$ is this always true? $\textbf{(A) }\text{even }n\qquad\textbf{(B) }\text{odd }n\text{ divisible by }3\qquad\textbf{(C) }\text{odd }n\qquad\textbf{(D) }\text{prime }n\qquad\textbf{(E) }\text{no such }n\text{ exists}$

In the acute-angled triangle $ABC$, let $CD, AE$ be the altitudes. Points $F$ and $G$ are the projections of $A$ and $C$ on the line $DE$, respectively, $H$ and $K$ are the projections of $D$ and $E$ on the line $AC$, respectively. The lines $HF$ and $KG$ intersect at point $P$. Prove that line $BP$ bisects the segment $DE$.

The integers from $1$ to $n$ are arranged along a circle such that each number is a multiple of difference of its adjacents. For which $n$ below such an arrangement is possible? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 13 $

The sum of the measures of all the face angles of a given complex polyhedral angle is equal to the sum of all its dihedral angles. Prove that the polyhedral angle is a trihedral angle. $\mathbf{Note:}$ A convex polyhedral angle may be formed by drawing rays from an exterior point to all points of a convex polygon.

Let $a,\ b$ and $c$ be positive real numbers so that $a+b+c=1$. Show that $$a\sqrt{a^2+6bc}+b\sqrt{b^2+6ac}+c\sqrt{c^2+6ab}\leq\frac{3\sqrt{2}}{4}$$

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.

Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions: (i) $f(x)$ is an integer if and only if $x$ is an integer; (ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.

Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$. Find the least number $k$ such that $s(M,N)\le k$, for all points $M,N$. [i]Dinu Șerbănescu[/i]

[b] Problem Section #3 NOTE: Neglect that HF and CD.

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

Aidan rolls a pair of fair, six sided dice. Let$ n$ be the probability that the product of the two numbers at the top is prime. Given that $n$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers, find $a +b$. [i]Proposed by Aidan Duncan[/i]

We have a hexagon such that all its edges touch a circle. If five of the edges have lengths 1,2,3,4, and 5 as on the figure, how long is the last edge? [img]http://i250.photobucket.com/albums/gg265/geometry101/HexagonImage.jpg[/img] A. 1 B. 3 C. 15/8 D. $ \sqrt{15}$ E. Not uniquely determined, more than one possibility

Prove that there exist infinitely many pairs $\left(x;\;y\right)$ of different positive rational numbers, such that the numbers $\sqrt{x^2+y^3}$ and $\sqrt{x^3+y^2}$ are both rational.

Let $b$ be the length of the largest diagonal and $c$ be the length of the smallest diagonal of a regular nonagon with side length $a$. Which one of the followings is true? $ \textbf{(A)}\ b=\dfrac{a+c}2 \qquad\textbf{(B)}\ b=\sqrt {ac} \qquad\textbf{(C)}\ b^2=\dfrac{a^2+c^2}2 \\ \textbf{(D)}\ c=a+b \qquad\textbf{(E)}\ c^2=a^2+b^2 $

On a table there is a pile with $ T$ tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process? a.) $ T \equal{} 1000$ (Cono Sur) b.) $ T \equal{} 2001$ (BWM)

After simple interest for two months at $5\%$ per annum was credited, a Boy Scout Troop had a total of $\textdollar 255.31$ in the Council Treasury. The interest credited was a number of dollars plus the following number of cents $\textbf{(A) }11\qquad\textbf{(B) }12\qquad\textbf{(C) }13\qquad\textbf{(D) }21\qquad \textbf{(E) }31$

[b]Q9.[/b] Evaluate the integer part of the number \[H= \sqrt{1+2011^2+ \frac{2011^2}{2012^2}}+ \frac{2011}{2012}.\]

Find all triples $ (x,y,z) $ of natural numbers that are in geometric progression and verify the inequalities $$ 4016016\le x<y<z\le 4020025. $$

At a two-day team competition in chess, three schools with $15$ pupils each attend. Each student plays one game against each player on the other two teams, ie a total of $30$ chess games per student. a) Is it possible for each student to play exactly $15$ games after the first day? b) Show that it is possible for each student to play exactly $16$ games after the first day. c) Assume that each student has played exactly $16$ games after the first day. Show that there are three students, one from each school, who have played their three parties

Let $a$, $b$ and $c$ be positive real numbers such that $abc=2015$. Prove that $$\frac{a+b}{a^2+b^2}+\frac{b+c}{b^2+c^2}+\frac{c+a}{c^2+a^2} \leq \frac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{\sqrt{2015}}$$

We have $2012$ sticks with integer length, and sum of length is $n$. We need to have sticks with lengths $1,2,....,2012$. For it we can break some sticks ( for example from stick with length $6$ we can get $1$ and $4$). For what minimal $n$ it is always possible?

Prove that if the diagonals of a convex quadrilateral are perpendicular, then the feet of perpendiculars dropped from the intersection point of diagonals on the sides of this quadrilateral lie on one circle. Is the converse true?