Kayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region, that the paper can be divided into by the sides of the triangles? [i]Proposed by Michael Tang[/i]

Given a convex quadrilateral $ABCD$, find the locus of the points $P$ inside the quadrilateral such that \[S_{PAB}\cdot S_{PCD} = S_{PBC}\cdot S_{PDA}\] (where $S_X$ denotes the area of triangle $X$).

1. The numbers $1,2, \ldots$ are placed in a triangle as following: \[ \begin{matrix} 1 & & & \\ 2 & 3 & & \\ 4 & 5 & 6 & \\ 7 & 8 & 9 & 10 \\ \ldots \end{matrix} \] What is the sum of the numbers on the $n$-th row?

George has $150$ cups of flour and $200$ eggs. He can make a cupcake with $3$ cups of flour and $2$ eggs, or he can make an omelet with $4$ eggs. What is the maximum number of treats (both omelets and cupcakes) he canmake?

Find the value of $n$ such that $\frac{2019 + n}{2019 - n}= 5$

In the triangle $ABC$, the side $BC$ is equal to $a$. Point $F$ is midpoint of $AB$, $I$ is the point of intersection of the bisectors of triangle $ABC$. It turned out that $\angle AIF = \angle ACB$. Find the perimeter of the triangle $ABC$. (Grigory Filippovsky)

What is the minimum number of squares that is necessary to draw on a white sheet to obtain a square grid of side $n$?

Find all non constant polynomials $P(x),Q(x)$ with real coefficients such that: $P((Q(x))^3)=xP(x)(Q(x))^3$

If $x$ and $y$ are positive real numbers such that $\sqrt(2x)+\sqrt(y)=13$ and $\sqrt(8x)+\sqrt(9y)=35$, calculate $20x+23y$.

Define the function $f: \mathbb N \cup \{0\} \to \mathbb{Q}$ as follows: $f(0) = 0$ and \[ f(3n+k) = -\frac{3f(n)}{2} + k , \] for $k = 0, 1, 2$. Show that $f$ is one-to-one and determine the range of $f$.

Let $\triangle ABC$ be a triangle such that $AB=40$ and $AC=30.$ Points $X$ and $Y$ are on the segment $AB$ and $BC,$ respectively such that $AX:BX=3:2$ and $BY:CY=1:4.$ Given that $XY=12,$ the area of $\triangle ABC$ can be written as $a\sqrt{b}$ where $a$ and $b$ are positive integers and $b$ is squarefree. Compute $a+b.$

Let $S$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S$ so that the union of the two subsets is $S$? The order of selection does not matter; for example, the pair of subsets $\{a, c\}$, $\{b, c, d, e, f\}$ represents the same selection as the pair $\{b, c, d, e, f\}$, $\{a, c\}$.

Suppose that $x$ and $y$ are different real numbers such that $\frac{x^n-y^n}{x-y}$ is an integer for some four consecutive positive integers $n$. Prove that $\frac{x^n-y^n}{x-y}$ is an integer for all positive integers n.

Let $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^2-PQ^2$.

Evaluate $\prod_{k=1}^{254}\log_{k+1}(k + 2)^{u_k}$, where $u_k = \begin{cases}- k & \text{if} \,\, k \,\, \text{is odd}\\ \frac{1}{k-1} & \text{if} \,\, k \,\, \text{is even} \end{cases}$

Let $\{a_k\}^{2011}_{k=1}$ be the sequence of real numbers defined by $$a_1=0.201, \quad a_2=(0.2011)^{a_1},\quad a_3=(0.20101)^{a_2},\quad a_4=(0.201011)^{a_3},$$ and more generally \[ a_k = \begin{cases}(0.\underbrace{20101\cdots0101}_{k+2 \ \text{digits}})^{a_{k-1}}, &\text {if } k \text { is odd,} \\ (0.\underbrace{20101\cdots01011}_{k+2 \ \text{digits}})^{a_{k-1}}, &\text {if } k \text { is even.}\end{cases} \] Rearranging the numbers in the sequence $\{a_k\}^{2011}_{k=1}$ in decreasing order produces a new sequence $\{b_k\}^{2011}_{k=1}$. What is the sum of all the integers $k$, $1\le k \le 2011$, such that $a_k = b_k$? $ \textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\ 2012 $

Given positive numbers $a_1,a_2,...,a_{99},a_{100}$. It is known, that $$a_1>a_0, a_2=3a_1-2a_0, a_3=3a_2-2a_1, ..., a_{100}=3a_{99}-2a_{98}$$ Prove that $$a_{100}>2^{99}.$$