Call a positive integer $x$ $n$-[i]cube-invariant[/i] if the last $n$ digits of $x$ are equal to the last $n$ digits of $x^3$. For example, $1$ is $n$-cube invariant for any integer $n$. How many $2015$-cube-invariant numbers $x$ are there such that $x < 10^{2015}$?

For which acute-angled triangles is the ratio of the smallest side to the inradius the maximum?

Rațiu and Horațiu are playing a game on a $100\times 100$ grid. They make moves alternatively, starting with Rațiu. At a move, a player places a token on an empty cell of the grid. If a player places a token on a cell which is adjacent to another cell with a token, he loses. Determine who has a winning strategy.

Let $ ABC$ be an equilateral triangle. Let $ \Omega$ be its incircle (circle inscribed in the triangle) and let $ \omega$ be a circle tangent externally to $ \Omega$ as well as to sides $ AB$ and $ AC$. Determine the ratio of the radius of $ \Omega$ to the radius of $ \omega$.

Evaluate $\int_0^{\pi} \sqrt[4]{1+|\cos x|}\ dx.$ created by kunny

Right triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. What is $OP$? $\textbf{(A)} ~\frac{23}{8}\qquad\textbf{(B)} ~\frac{29}{10}\qquad\textbf{(C)} ~\frac{35}{12}\qquad\textbf{(D)} ~\frac{73}{25}\qquad\textbf{(E)} ~3$

Ana and Celia sell various objects and obtain for each object as many euros as objects they sold. The money obtained is made up of some $10$ euro bills and less than $10$ coins of $1$ euro . They decide to distribute the money as follows: Ana takes a $10$ euro bill and then Celia, and so on successively until Ana takes the last $10$ euro note, and Celia takes all the $1$ euro coins . How many euros more than Celia did Ana take? Give all the possibilities. [hide=original wording]Ana y Celia venden varios objetos y obtienen por cada objeto tantos euros como objetos vendieron. El dinero obtenido está constituido por algunos billetes de 10 euros y menos de 10 monedas de 1 euro. Deciden repartir el dinero del siguiente modo: Ana toma un billete de 10 euros y después Celia, y así sucesivamente hasta que Ana toma el último billete de 10 euros, y Celia se lleva todas las monedas de 1 euro. ¿Cuántos euros más que Celia se llevó Ana? Dar todas las posibilidades.[/hide]

Let $a$, $b$, $c$, $d$ be positive real numbers such that $abcd=1$. Prove that $1/[(1/2 +a+ab+abc)^{1/2}]+ 1/[(1/2+b+bc+bcd)^{1/2}] + 1/[(1/2+c+cd+cda)^{1/2}] + 1/[1(1/2+d+da+dab)^{1/2}]$ is greater than or equal to $2^{1/2}$.

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? $(A)40$ $~~~$ $(B)50$ $~~~$ $(C)60$ $~~~$ $(D)75$ $~~~$ $(E)80$