In space are given $n\ge 2$ points, no four of which are coplanar. Some of these points are connected by segments. Let $K$ be the number of segments $(K>1)$ and $T$ be the number of formed triangles. Prove that $9T^2<2K^3$.

Sara has an ice cream cone with every meal. The cone has a height of $2\sqrt2$ inches and the base of the cone has a diameter of $2$ inches. Ice cream protrudes from the top of the cone in a perfect hempisphere. Find the surface area of the ice cream cone, ice cream included, in square inches.

Compute the remainder when $\underbrace{\hbox{11...1}}_{\hbox{1862}}$ is divided by $2006$

Let $a$, $b$, and $c$ be real numbers satisfying the equations $$a^3+abc=26$$ $$b^3+abc=78$$ $$c^3-abc=104.$$ Find $a^3+b^3+c^3$.

Let $k$ and $d$ be positive integers. Prove that there exists a positive integer $N$ such that for every odd integer $n>N$, the digits in the base-$2n$ representation of $n^k$ are all greater than $d$.

Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively touch externally at $A$. Let $M$ be a point inside $k_2$ and outside the line $O_1O_2$. Find a line $d$ through $M$ which intersects $k_1$ and $k_2$ again at $B$ and $C$ respectively so that the circumcircle of $\Delta ABC$ is tangent to $O_1O_2$.

Let $n$ be a positive integer. Find the greatest common divisor of the numbers $\binom{2n}{1},\binom{2n}{3},\binom{2n}{5},...,\binom{2n}{2n-1}$.

Gabriel plays to draw triangles using the vertices of a regular polygon with $2019$ sides, following these rules: (i) The vertices used by each triangle must not have been previously used. (ii) The sides of the triangle to be drawn must not intersect with the sides of the triangles previously drawn. If Gabriel continues to draw triangles until it is no longer possible, determine the minimum number of triangles that he drew.

Let $ABC$ be a triangle, and let $D$, $A$, $B$, $E$ be points on line $AB$, in that order, such that $AC=AD$ and $BE=BC$. Let $\omega_1, \omega_2$ be the circumcircles of $\triangle ABC$ and $\triangle CDE$, respectively, which meet at a point $F \neq C$. If the tangent to $\omega_2$ at $F$ cuts $\omega_1$ again at $G$, and the foot of the altitude from $G$ to $FC$ is $H$, prove that $\angle AGH=\angle BGH$. [i]Proposed by David Stoner[/i]

For the game show Who Wants To Be A Millionaire?, the dollar values of each question are shown in the following table (where K = 1000). \[ \begin{tabular}{rccccccccccccccc}\text{Question}& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15\\ \text{Value}& 100 & 200 & 300 & 500 & 1\text{K}& 2\text{K}& 4\text{K}& 8\text{K}& 16\text{K}& 32\text{K}& 64\text{K}& 125\text{K}& 250\text{K}& 500\text{K}& 1000\text{K}\end{tabular} \] Between which two questions is the percent increase of the value the smallest? $ \text{(A)}\ \text{From 1 to 2}\qquad\text{(B)}\ \text{From 2 to 3}\qquad\text{(C)}\ \text{From 3 to 4}\qquad\text{(D)}\ \text{From 11 to 12}\qquad\text{(E)}\ \text{From 14 to 15} $

Points$ D, E, F$ are chosen on the sides $AB$, $BC$, $AC$ of a triangle $ABC$, so that $DE = BE$ and $FE = CE$. Prove that the centre of the circle circumscribed around triangle $ADF$ lies on the bisectrix of angle $DEF$.

$ABCD$ is quadrilateral inscribed in a circle $\Gamma$ .Lines $AB$ and $CD$ intersect at $E$ and lines$AD$ and $BC$ intersect at $F$. Prove that the circle with diameter $EF$ and circle $\Gamma$ are orthogonal.

Let $f:\mathbb{N} \longrightarrow \mathbb{N}$ be such that for every positive integer $n$, followings are satisfied. i. $f(n+1) > f(n)$ ii. $f(f(n)) = 2n+2$ Find the value of $f(2013)$. (Here, $\mathbb{N}$ is the set of all positive integers.)

Prove that \[1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots +\frac{1}{2019}-\frac{1}{2020}=\frac{1}{1011}+\frac{1}{1012}+\cdots +\frac{1}{2020}\]

Mildred the cow is tied with a rope to the side of a square shed with side length 10 meters. The rope is attached to the shed at a point two meters from one corner of the shed. The rope is 14 meters long. The area of grass growing around the shed that Mildred can reach is given by $n\pi$ square meters, where $n$ is a positive integer. Find $n$.

Let $ABCDE$ be a right quadrangular pyramid with vertex $E$ and height $EO$. Point $S$ divides this height in the ratio $ES: SO=m$. In which ratio does the plane $(ABC)$ divide the lateral area of the pyramid.

The eight pointed star is a popular quilting pattern. What percent of the entire 4-by-4 grid is covered by the star? $(A)40$ $~~~$ $(B)50$ $~~~$ $(C)60$ $~~~$ $(D)75$ $~~~$ $(E)80$

There are two types of items in Alik's collection: badges and bracelets and there are more badges than bracelets. Alik noticed that if he increases the number of bracelets some (not necessarily integer) number of times without changing the number of icons, then in its collection will be $100$ items. And if, on the contrary, he increases the initial number of badges by the same number of times, leaving the same number of bracelets, then he will have $101$ items. How many badges and how many bracelets could there be in Alik's collection?

$n$ is an odd positive integer and $x,y$ are two rational numbers satisfying $$x^n+2y=y^n+2x.$$Prove that $x=y$.

Prove that for any convex quadrilateral it is always possible to cut out three smaller quadrilaterals similar to the original one with the scale factor equal to 1/2. (The angles of a smaller quadrilateral are equal to the corresponding original angles and the sides are twice smaller then the corresponding sides of the original quadrilateral.)

Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\ge 1,$ find the minimum possible value of $|x_1+x_2+\cdots+x_{2006}|$.

Let $n = 6901$. There are $6732$ positive integers less than or equal to $n$ that are also relatively prime to $n$. Find the sum of the distinct prime factors of $n$.

Let in-circle of $ABC$ touch $AB$, $BC$, $AC$ in $C_1$, $A_1$, $B_1$ respectively. Let $H$- intersection point of altitudes in $A_1B_1C_1$, $I$ and $O$-be in-center and circumcenter of $ABC$ respectively. Prove, that $I, O, H$ lies on one line.

Find $x^2+y^2$ if $x$ and $y$ are positive integers such that \[xy+x+y = 71\qquad\text{and}\qquad x^2y+xy^2 = 880.\]

Gus is an inhabitant on an $11$ by $11$ grid of squares. He can walk from one square to an adjacent square (vertically or horizontally) in $1$ unit of time. There are also two vents on the grid, one at the top left and one at the bottom right. If Gus is at one vent, he can teleport to the other vent in $0.5$ units of time. Let an ordered pair of squares $(a,b)$ on the grid be [i]sus [/i] if the fastest path from $a$ to $b$ requires Gus to teleport between vents. Walking on top of a vent does not count as teleporting between vents. What is the total number of ordered pairs of squares that are [i]sus[/i]? Note that the pairs $(a_1,b_1)$ and $(a_2,b_2)$ are considered distinct if and only if $a_1 \ne a_2$ or $b_1 \ne b_2$.