Determine the least real number $a > 1$ such that for any point $P$ in the interior of a square $ABCD$, the ratio of the areas of some two triangle $PAB, PBC, PCD, PDA$ lies in the interval $[1/a, a]$.

For how many three-digit whole numbers does the sum of the digits equal $25$? $\text{(A)}\ 2 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 10$

$(HUN 2)$ Prove that $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{5}{4}$

Let there be $2004$ be bicolor tiles, white on one side and black on the other, placed in a circle. A move consists of choosing a black piece and turning over three pieces: the chosen one, the one on its left and the one on its right. If at the beginning there is only one black piece, will it be possible, repeating the movement described, to make all the pieces have the white face up?

Alex and Jack have $1000$ sheets each. Each of them writes the numbers $1,\ldots,2000$ on his sheets in an arbitrary order, with one number on each side of a sheet. The sheets are to be placed on the floor so that one side of each sheet is visible. Prove that they can do so in such a way that each of the numbers from $1$ to $2000$ is visible.

We are given an arbitrary $\bigtriangleup ABC$. a) Can we dissect $\bigtriangleup ABC$ in $4$ pieces, from which we can make two triangle similar to $\bigtriangleup ABC$ (each piece can be used only once)? Justify your answer! b) Is it possible that for every positive integer $n \ge 2$ , we are able to dissect $\bigtriangleup ABC$ in $2n$ pieces, from which we can make two triangles similar to $\bigtriangleup ABC$ (each piece can be used only once)? Justify your answer!

What is the remainder of the number $1 \binom{2008}{0 }+2\binom{2008}{1}+ ...+2009\binom{2008}{2008}$ when divided by $2008$?

You are traveling in a foreign country whose currency consists of five different-looking kinds of coins. You have several of each coin in your pocket. You remember that the coins are worth $1, 2, 5, 10$, and $20$ florins, but you have no idea which coin is which and you don’t speak the local language. You find a vending machine where a single candy can be bought for $1$ florin: you insert any kind of coin, and receive $1$ candy plus any change owed. You can only buy one candy at a time, but you can buy as many as you want, one after the other. What is the least number of candies that you must buy to ensure that you can determine the values of all the coins? Prove that your answer is correct.

Show that for all positive integers $n$, \[\lfloor \sqrt{n}+\sqrt{n+1}\rfloor =\lfloor \sqrt{4n+1}\rfloor =\lfloor \sqrt{4n+2}\rfloor =\lfloor \sqrt{4n+3}\rfloor.\]

Let $ABC$ be an acute triangle and let $P, Q$ lie on the segment $BC$ such that $BP=PQ=CQ$. The feet of the perpendiculars from $P, Q$ to $AC, AB$ are $X, Y$. Show that the centroid of $ABC$ is equidistant from the lines $QX$ and $PY$.

If $F(n+1)=\frac{2F(n)+1}{2}$ for $n=1,2,\ldots$, and $F(1)=2$, then $F(101)$ equals: $\text{(A)} \ 49 \qquad \text{(B)} \ 50 \qquad \text{(C)} \ 51 \qquad \text{(D)} \ 52 \qquad \text{(E)} \ 53$

Consider the sequence $ (x_n)_{n\ge 0}$ where $ x_n\equal{}2^{n}\minus{}1\ ,\ n\in \mathbb{N}$. Determine all the natural numbers $ p$ for which: \[ s_p\equal{}x_0\plus{}x_1\plus{}x_2\plus{}...\plus{}x_p\] is a power with natural exponent of $ 2$.

A positive integer $ m$ is called [i]good[/i] if there is a positive integer $ n$ such that $ m$ is the quotient of $ n$ by the number of positive integer divisors of $ n$ (including $ 1$ and $ n$ itself). Prove that $ 1, 2, \ldots, 17$ are good numbers and that $ 18$ is not a good number.

Let the medians of the triangle $ABC$ meet at $G$. Let $D$ and $E$ be different points on the line $BC$ such that $DC=CE=AB$, and let $P$ and $Q$ be points on the segments $BD$ and $BE$, respectively, such that $2BP=PD$ and $2BQ=QE$. Determine $\angle PGQ$.

There are nonzero real numbers $a$ and $b$ so that the roots of $x^2 + ax + b$ are $3a$ and $3b$. There are relatively prime positive integers $m$ and $n$ so that $a - b = \tfrac{m}{n}$. Find $m + n$.

If the polynomial $P(x)=ax^2+bx+c$ takes value $0$ for three different values of $x$, then prove the polynomial $P(x)$ takes value $0$ for all $x$.

$x$, $y$, and $z$ are real numbers such that $xyz=10$. What is the maximum possible value of $x^3y^3z^3-3x^4-12y^2-12z^4$?

Suppose $f(x)$ is a polynomial with integer coefficients such that $f(0) = 11$ and $f(x_1) = f(x_2) = ... = f(x_n) = 2002$ for some distinct integers $x_1, x_2, . . . , x_n$. Find the largest possible value of $n$.

Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$. [i]Proposed by Calvin Deng[/i]

Let $ a,b_0,b_1,b_2,...,b_{n\minus{}1}$ be complex numbers, $ A$ a complex square matrix of order $ p$, and $ E$ the unit matrix of order $ p$. Assuming that the eigenvalues of $ A$ are given, determine the eigenvalues of the matrix \[ B\equal{}\begin{pmatrix} b_0E&b_1A&b_2A^2&\cdots&b_{n\minus{}1}A^{n\minus{}1} \\ ab_{n\minus{}1}A^{n\minus{}1}&b_0E&b_1A&\cdots&b_{n\minus{}2}A^{n\minus{}2}\\ ab_{n\minus{}2}A^{n\minus{}2}&ab_{n\minus{}1}A^{n\minus{}1}&b_0E&\cdots&b_{n\minus{}3}A^{n\minus{}3}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\\ ab_1A&ab_2A^2&ab_3A^3&\cdots&b_0E \end{pmatrix}\quad\]

Consider cubes of edge length 5 composed of 125 cubes of edge length 1 where each of the 125 cubes is either coloured black or white. A cube of edge length 5 is called "big", a cube od edge length is called "small". A posititve integer $ n$ is called "representable" if there is a big cube with exactly $ n$ small cubes where each row of five small cubes has an even number of black cubes whose centres lie on a line with distances $ 1,2,3,4$ (zero counts as even number). (a) What is the smallest and biggest representable number? (b) Construct 45 representable numbers.

At a party, there are $100$ cats. Each pair of cats flips a coin, and they shake paws if and only if the coin comes up heads. It is known that exactly $4900$ pairs of cats shook paws. After the party, each cat is independently assigned a ``happiness index" uniformly at random in the interval $[0,1]$. We say a cat is [i]practical[/i] if it has a happiness index that is strictly greater than the index of every cat with which it shook paws. The expected value of the number of practical cats is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m + n$. [i]Proposed by Brandon Wang[/i]

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

Show that it is possible to arrange seven distinct points in the plane so that among any three of these seven points, two of the points are a unit distance apart. (Your solution should include a carefully prepared sketch of the seven points, along with all segments that are of unit length.)

We consider the sum of the digits of a positive integer. For example, the sum of the digits of $2007$ is equal to $9$, since $2 + 0 + 0 + 7 = 9$. (a) How many integers $n$, where $0 < n < 100 000$, have an even sum of digits? (b) How many integers $n$, where $0 < n < 100 000$, have a sum of digits that is less than or equal to $22$?