Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are $60$, $20$, and $15$ respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area.

Let $x, y$, and $z$ be positive integers satisfying the following system of equations: $$x^2 +\frac{2023}{x}= 2y^2$$ $$y +\frac{2028}{y^2} = z^2$$ $$2z +\frac{2025}{z^2} = xy$$ Find $x + y + z$.

The extension of the bisector of angle $A$ of triangle $ABC$ intersects with the circumscribed circle of this triangle at point $W$. A straight line is drawn through $W$, which is parallel to side $AB$ and intersects sides $BC$ and $AC$ , at points $N$ and $K$, respectively. Prove that the line $AW$ is tangent to the circumscribed circle of $\vartriangle CNW$. (Sergey Yakovlev)

Which bond is strongest? ${ \textbf{(A)}\ \text{C=C}\qquad\textbf{(B)}\ \text{C=N}\qquad\textbf{(C)}\ \text{C=O}\qquad\textbf{(D)}}\ \text{C=S}\qquad $

Find the largest positive integer $n$ such that no two adjacent digits are the same, and for any two distinct digits $0 \leq a,b \leq 9 $, you can't get the string $abab$ just by removing digits from $n$.

Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.

Is there an injective function $f : Z^+ \to Q$ satisfying the equation $f(xy) = f(x) + f(y)$ for all positive integers $x$ and $y$?

For a positive integer $n$, let $p(n)$ denote the product of the digits of $n$, and let $s(n)$ denote the sum of the digits of $n$. Find the sum of all positive integers $n$ satisfying $p(n)s(n)=8$.

Let be a natural number $ n\ge 2, $ and a matrix $ A\in\mathcal{M}_n\left( \mathbb{C} \right) $ whose determinant vanishes. Show that $$ \left( A^* \right)^2 =A^*\cdot\text{tr} A^*, $$ where $ A^* $ is the adjugate of $ A. $

It is possible to partition the set $\{100, 101,\cdots , 1000\}$ into two subsets so that for any two distinct elements $x$ and $y$ belonging to the same subset $ \sqrt[3]{x + y}$ is irrational?

Prove that any triangle can be divided into $22$ triangles, each of which has an angle of $22^o$, and another $23$ triangles, each of which has an angle of $23^o$.

Recall that in any row of Pascal's Triangle, the first and last elements of the row are $1$ and each other element in the row is the sum of the two elements above it from the previous row. With this in mind, define the $\textit{Pascal Squared Triangle}$ as follows: [list] [*] In the $n^{\text{th}}$ row, where $n\geq 1$, the first and last elements of the row equal $n^2$; [*] Each other element is the sum of the two elements directly above it. [/list] The first few rows of the Pascal Squared Triangle are shown below. \[\begin{array}{c@{\hspace{7em}} c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{4pt}}c@{\hspace{2pt}} c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{3pt}}c@{\hspace{2pt}} c@{\hspace{2pt}}c} \vspace{4pt} \text{Row 1: } & & & & & & 1 & & & & & \\\vspace{4pt} \text{Row 2: } & & & & & 4 & & 4 & & & & \\\vspace{4pt} \text{Row 3: } & & & & 9 & & 8 & & 9 & & & \\\vspace{4pt} \text{Row 4: } & & &16& &17& &17& & 16& & \\\vspace{4pt} \text{Row 5: } & &25 & &33& &34 & &33 & &25 & \end{array}\] Let $S_n$ denote the sum of the entries in the $n^{\text{th}}$ row. For how many integers $1\leq n\leq 10^6$ is $S_n$ divisible by $13$?

Every point in the plane is coloured one of seven distinct colours. Is there an inscribed trapezoid whose vertices are all of the same colour?

Solve the system of equations: $$\frac{x-1}{xy-3}=\frac{y-2}{xy-4}=\frac{3-x-y}{7-x^2-y^2}$$

Given is an acute angled triangle $ABC$. A circle going through $B$ and the triangle's circumcenter, $O$, intersects $BC$ and $BA$ at points $P$ and $Q$ respectively. Prove that the intersection of the heights of the triangle $POQ$ lies on line $AC$.

In a convex cyclic hexagon $ABCDEF$, $BC=EF$ and $CD=AF$. Diagonals $AC$ and $BF$ intersect at point $Q,$ and diagonals $EC$ and $DF$ intersect at point $P.$ Points $R$ and $S$ are marked on the segments $DF$ and $BF$ respectively so that $FR=PD$ and $BQ=FS.$ [b]The segments[/b] $RQ$ and $PS$ intersect at point $T.$ Prove that the line $TC$ bisects the diagonal $DB$.

There are $ 2016 $ positions marked around a circle, with a token on one of them. A legitimate move is to move the token either 1 position or 4 positions from its location, clockwise. The restriction is that the token can not occupy the same position more than once. Players $ A $ and $ B $ take turns making moves. Player $ A $ has the first move. The first player who cannot make a legitimate move loses. Determine which of the two players has a winning strategy.

The number $2^{1997}$ has $m$ decimal digits, while the number $5^{1997}$ has $n$ digits. Evaluate $m+n$.

How many positive three-digit integers $\underline a\, \underline b\,\underline c$ can represent a valid date in $2013$, where either $a$ corresponds to a month and $\underline b\,\underline c$ corresponds to the day in that month, or $\underline a\, \underline b$ corresponds to a month and $c$ corresponds to the day? For example, 202 is a valid representation for February 2nd, and 121 could represent either January 21st or December 1st.

$n$ balls are placed independently uniformly at random into $n$ boxes. One box is selected at random, and is found to contain $b$ balls. Let $e_n$ be the expected value of $b^4$. Find $$\lim_{n \to \infty}e_n.$$

$A, B, C$ are the angles of a triangle. Show that $2\frac{\sin A}{A} + 2\frac{\sin B}{B} + 2\frac{\sin C}{C} \le \left(\frac{1}{B} + \frac{1}{C}\right) \sin A + \left(\frac{1}{C} + \frac{1}{A}\right) \sin B + \left(\frac{1}{A} + \frac{1}{B}\right) \sin C$

Call a set of some cells in infinite chess field as board. Set of rooks on the board call as awesome if no one rook can beat another, but every empty cell is under rook attack. There are awesome set with $2008$ rooks and with $2010$ rooks. Prove, that there are awesome set with $2009$ rooks.

Let $A, B, C, D$ be points, in order, on a straight line such that $AB=BC=CD$. Let $E$ be a point closer to $B$ than $D$ such that $BE=EC=CD$ and let $F$ be the midpoint of $DE$. Let $AF$ intersect $EC$ at $G$ and let $BF$ intersect $EC$ at $H$. If $[BHC]+[GHF]=1$, then $AD^2 = \frac{a\sqrt{b}}{c}$ where $a,b,$ and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a+b+c$. [i]Proposed by [b]AOPS12142015[/b][/i]

Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admits bounded primitives. Prove that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\left\{ \begin{matrix} x, & \quad x\le 0 \\ f(1/x)\cdot\ln x ,& \quad x>0 \end{matrix}\right. $$ admits primitives. [i]Florian Dumitrel[/i]

Two sequences of nonzero reals $a_1, a_2, a_3, \dots$ and $b_2, b_3, \dots$ are such that $b_n=\prod_{i=1}^{n} a_i$ and $a_n=\frac{b_n^2}{3b_n-3}$ for all integers $n > 1$. Given that $a_1=\frac{1}{2}$, find $\lvert b_{60}\rvert$. [i]Proposed by Andrew Zhao[/i]