Snorlax's weight is modeled by the function $w(t)=t2^t$ where $w(t)$ is Snorlax's weight at time $t$ minutes. Find the smallest integer time $t$ such that Snorlax's weight is greater than $10000.$

Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.

Let $ ABC$ be a triangle. Prove that there is a unique point $ U$ in the plane of $ ABC$ such that there exist real numbers $ \alpha, \beta, \gamma, \delta$ not all zero, such that \[ \alpha PL^2 \plus{} \beta PM^2 \plus{} \gamma PN^2 \plus{} \delta UP^2\] is constant for all points $ P$ of the plane, where $ L,M,N$ are the feet of the perpendiculars from $ P$ to $ BC,CA,AB$ respectively. Identify $ U.$

Let $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A)=d\neq 0$ and $\det(A+dA^*)=0$. Prove that $\det(A-dA^*)=4$. [i]Daniel Jinga[/i]

A sequence $x_0=A$ and $x_1=B$ and $x_{n+2}=x_{n+1} +x_n$ is called a Fibonacci type sequence. Call a number $C$ a repeated value if $x_t=x_s=c$ for $t$ different from $s$. Prove one can choose $A$ and $B$ to have as many repeated value as one likes but never infinite.

Let $n \ge 1$ be an integer. Prove that there exists a set $S$ of $n$ positive integers with the following property: if $A$ and $B$ are any two distinct non-empty subsets of $S$, then the averages $\frac{P_{x\in A} x}{|A|}$ and $\frac{P_{x\in B} x}{|B|}$ are two relatively prime composite integers.

Albrecht likes to draw hexagons with all sides having equal length. He calls an angle of such a hexagon [i]nice [/i] if it is exactly $120^o$. He writes the number of its nice angles inside each hexagon. How many different numbers could Albrecht write inside the hexagons? Show examples for as many values as possible and give a reasoning why others cannot appear. [i]Albrecht can also draw concave hexagons[/i]

A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist?

The letter T is formed by placing two $ 2\times 4$ inch rectangles next to each other, as shown. What is the perimeter of the T, in inches? [asy]size(150); draw((0,6)--(4,6)--(4,4)--(3,4)--(3,0)--(1,0)--(1,4)--(0,4)--cycle, linewidth(1));[/asy] $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 24$

A square of side length $1$ centimeter is cut into three convex polygons. Is it possible that the diameter of each of them does not exceed [list][b]a)[/b] $1$ centimeter; [b]b)[/b] $1.01$ centimeters; [b]c)[/b] $1.001$ centimeters?[/list]

Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that \[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]

Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$$ [i]Andrew Carratu[/i]

Let $a, b, c$ and $d$ be four integers such that $7a + 8b = 14c + 28d$. Prove that the product $a\cdot b$ is always divisible by $14$.

An infinite set of rectangles in the Cartesian coordinate plane is given. The vertices of each of these rectangles have coordinates $(0, 0), (p, 0), (p, q), (0, q)$ for some positive integers $p, q$. Show that there must exist two among them one of which is entirely contained in the other.

Let $ ABCD $ be a tetrahedron with $ \angle ABC = \angle ABD = \angle CBD = 90^\circ $ and $ AB = BC $. Let $ E, F, G $ be points on $ \overline{AD} $, $ BD $, and $ \overline{CD} $, respectively, such that each of the quadrilaterals $ AEFB $, $ BFGC $, and $ CGEA $ have an inscribed circle. Let $ r $ be the smallest real number such that $ \dfrac{[\triangle EFG]}{[\triangle ABC]} \leq r $ for all such configurations $ A, B, C, D, E, F, G $. If $ r $ can be expressed as $ \dfrac{\sqrt{a - b\sqrt{c}}}{d} $ where $ a, b, c, d $ are positive integers with $ \gcd(a, b) $ squarefree and $ c $ squarefree, find $ a + b + c + d $. Note: Here, $ [P] $ denotes the area of polygon $ P $. (This wasn't in the original test; instead they used the notation $ \text{area}(P) $, which is clear but frankly cumbersome. :P)

Suppose the numbers $a_0, a_1, a_2, ... , a_n$ satisfy the following conditions: $a_0 =\frac12$, $a_{k+1} = a_k +\frac{1}{n}a_k^2$ for $k = 0, 1, ... , n - 1$. Prove that $1 - \frac{1}{n}< a_n < 1$

Let be two matrices $ A,B\in\mathcal{M}_2\left( \mathbb{C} \right) $ satisfying $ AB-BA=A. $ Show that: [b]a)[/b] $ \text{tr} (A) =\det (A) =0 $ [b]b)[/b] $ AB^nA=0, $ for any natural number $ n $

Let $S$ be a set of positive integers, such that $n \in S$ if and only if $$\sum_{d|n,d<n,d \in S} d \le n$$ Find all positive integers $n=2^k \cdot p$ where $k$ is a non-negative integer and $p$ is an odd prime, such that $$\sum_{d|n,d<n,d \in S} d = n$$

Let $a, b$ be two positive integers such that $b + 1|a^2 + 1$,$ a + 1|b^2 + 1$. Prove that $a, b$ are odd numbers.

Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days? ${ \textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}$

Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.

The sides of the parallelogram serve as the diagonals of the four squares. The vertices of the squares lying in the part of the plane external to the parallelogram (the sides of the squares emerging from these vertices do not have common points with the parallelogram) serve as the vertices of a quadrilateral of area $a$, the four vertices opposite to them form a quadrilateral of area $b$. Find the area of the parallelogram.

The plot of the $y=f(x)$ function, being rotated by the (right) angle around the $(0,0)$ point is not changed. a) Prove that the equation $f(x)=x$ has the unique solution. b) Give an example of such a function.

Consider a checkered board $2m \times 2n$, $m, n \in \mathbb{Z}_{>0}$. A stone is placed on one of the unit squares on the board, this square is different from the upper right square and from the lower left square. A snail goes from the bottom left square and wants to get to the top right square, walking from one square to other adjacent, one square at a time (two squares are adjacent if they share an edge). Determine all the squares the stone can be in so that the snail can complete its path by visiting each square exactly one time, except the square with the stone, which the snail does not visit.

Does there exist a natural number $n$ with exactly 3 different prime divisors $p$, $q$, and $r$, so that $p-1\mid n$, $qr-1\mid n$, $q-1\nmid n$, $r-1\nmid n$, and $3\nmid q+r$?