Sphere $S$ is inscribed in cone $C$. The height of $C$ equals its radius, and both equal $12+12\sqrt2$. Let the vertex of the cone be $A$ and the center of the sphere be $B$. Plane $P$ is tangent to $S$ and intersects $\overline{AB}$. $X$ is the point on the intersection of $P$ and $C$ closest to $A$. Given that $AX=6$, find the area of the region of $P$ enclosed by the intersection of $C$ and $P$.

Let $n$ be a natural number and $P_1, P_2, ... , P_n$ are polynomials with integer coefficients, each of degree at least $2$. Let $S$ be the set of all natural numbers $N$ for which there exists a natural number $a$ and an index $1 \le i \le n$ such that $P_i(a) = N$. Prove, that there are infinitely many primes that do not belong to $S$.

Find all positive integers $n$ so that $$17^n +9^{n^2} = 23^n +3^{n^2} .$$

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

A round pencil has length $ 8$ when unsharpened, and diameter $ \frac {1}{4}$. It is sharpened perfectly so that it remains $ 8$ inches long, with a $ 7$ inch section still cylindrical and the remaining $ 1$ inch giving a conical tip. What is its volume?

How many four digit numbers $\overline{abcd}$ simultaneously satisfy the equalities $a+b=c+d$ and $a^2+b^2=c^2+d^2$?

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.