Suppose $ \vartriangle ABC$ is similar to $\vartriangle DEF$, with $ A$, $ B$, and $C$ corresponding to $D, E$, and $F$ respectively. If $\overline{AB} = \overline{EF}$, $\overline{BC} = \overline{FD}$, and $\overline{CA} = \overline{DE} = 2$, determine the area of $ \vartriangle ABC$.

How many polynomials of the form $x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$, where $a$, $b$, $c$, and $d$ are real numbers, have the property that whenever $r$ is a root, so is $\frac{-1+i\sqrt{3}}{2} \cdot r$? (Note that $i=\sqrt{-1}$) $\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4$

Find all functions $f: R \rightarrow R$ such that $f(x^{2015} + (f(y))^{2015}) = (f(x))^{2015} + y^{2015}$ holds for all reals $x, y$

Let $n\in N$ $n\geq2$ and the set $X$ with $n+1$ elements. The ordered sequences $(a_{1}, a_{2},\ldots,a_{n})$ and $(b_{1},b_{2},\ldots b_{n})$ of distinct elements of $X$ are said to be $\textit{separated}$ if there exists $i\neq j$ such that $a_{i}=b_{j}$. Determine the maximal number of ordered sequences of $n$ elements from $X$ such that any two of them are $\textit{separated}$. Note: ordered means that, for example $(1,2,3)\neq(2,3,1)$.

Let $ AB$ be a diameter of a circle, $ C$ be any fixed point between $ A$ and $ B$ on this diameter, and $ Q$ be a variable point on the circumference of the circle. Let $ P$ be the point on the line determined by $ Q$ and $ C$ for which $ \frac{AC}{CB}\equal{}\frac{QC}{CP}$. Describe, with proof, the locus of the point $ P$.

Sum of the roots in the range $\left(-\frac{\pi}{2},\frac{\pi}{2} \right)$ of the equation $\sin x\tan x=x^2$ is [list=1] [*] $\frac{\pi}{2}$ [*] 0 [*] 1 [*] None of these [/list]

Isosceles trapezoid $ ABCD$, with $ AB \parallel CD$, is such that there exists a circle $ \Gamma$ tangent to its four sides. Let $ T \equal{} \Gamma \cap BC$, and $ P \equal{} \Gamma \cap AT$ ($ P \neq T$). If $ \frac{AP}{AT} \equal{} \frac{2}{5}$, compute $ \frac{AB}{CD}$.

Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer. [i]Note: A monic polynomial has a leading coefficient equal to 1.[/i] [i](Greece - Panagiotis Lolas and Silouanos Brazitikos)[/i]

How many nonzero coefficients can a polynomial $ P(x)$ have if its coefficients are integers and $ |P(z)| \le 2$ for any complex number $ z$ of unit length?

Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.

The triple bond in carbon monoxide consists of $ \textbf{(A) } \text{3 sigma bonds}\qquad$ $\textbf{(B) } \text{2 sigma bonds and 1 pi bond}\qquad$ $\textbf{(C) } \text{1 sigma bond and 2 pi bonds}\qquad$ $\textbf{(D) } \text{3 pi bonds}\qquad$

Let $p$ be an odd prime number. Prove that $$1^{p-1} + 2^{p-1} +...+ (p-1)^{p-1} \equiv p + (p-1)! \mod p^2$$

Consider a convex $2000$-gon, no three of whose diagonals have a common point. Each of its diagonals is colored in one of $999$ colors. Prove that there exists a triangle all of whose sides lie on diagonals of the same color. (Vertices of the triangle need not be vertices of the original polygon.)

Find all $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $f(x)+f(y) = f(x+y)$ and $f(x^{2013}) = f(x)^{2013}$. [i]Proposed by Calvin Deng[/i]

Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^2+ax+b$ and $x^2+bx+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^2+ax+b$ and $x^2+bx+c$ $x^3-4x^2+x+6$. Find $a+b+c$.

Find the least positive integer $n$ so that both $n$ and $n+1$ have prime factorizations with exactly four (not necessarily distinct) prime factors.

We call an infinite set $S\subseteq\mathbb{N}$ good if for all parwise different integers $a,b,c\in S$, all positive divisors of $\frac{a^c-b^c}{a-b}$ are in $S$. for all positive integers $n>1$, prove that there exists a good set $S$ such that $n \not \in S$. Proposed by Seyed Reza Hosseini Dolatabadi

The incenter of the triangle $ ABC$ is $ K.$ The midpoint of $ AB$ is $ C_1$ and that of $ AC$ is $ B_1.$ The lines $ C_1K$ and $ AC$ meet at $ B_2,$ the lines $ B_1K$ and $ AB$ at $ C_2.$ If the areas of the triangles $ AB_2C_2$ and $ ABC$ are equal, what is the measure of angle $ \angle CAB?$

Arnulfo and Berenice play the following game: One of the two starts by writing a number from $ 1$ to $30$, the other chooses a number from $ 1$ to $30$ and adds it to the initial number, the first player chooses a number from $ 1$ to $30$ and adds it to the previous result, they continue doing the same until someone manages to add $2018$. When Arnulfo was about to start, Berenice told him that it was unfair, because whoever started had a winning strategy, so the numbers had better change. So they asked the following question: Adding chosen numbers from $1 $ to $a$, until reaching the number $ b$, what conditions must meet $a$ and $ b$ so that the first player does not have a winning strategy? Indicate if Arnulfo and Berenice are right and answer the question asked by them.

Let $ABC$ be a triangle with $AB < AC < BC$. Let the incenter and incircle of triangle $ABC$ be $I$ and $\omega$, respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$. Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$. Let $AI$ intersect the circumcircle of triangle $ABC$ at $P \ne A$. Let $K$ and $L$ be the midpoints of $AC$ and $AB$, respectively. Prove that $\angle KIL + \angle YPX = 180^{\circ}$. [i]Proposed by Dominik Burek, Poland[/i]

In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

Let $H\subseteq\mathbb{R}^3$ such that if we reflect any point in $H$ across another point of $H$, the resulting point is also in $H$. Prove that either $H$ is dense in ${R}^3$ or one can find equidistant parallel planes which cover $H$

On the bisector $AL$ of triangle $ABC$ a point $K$ is chosen such that $\angle BKL=\angle KBL=30^{\circ}$. Lines $AB$ and $CK$ intersect at point $M$, lines $AC$ and $BK$ intersect at point $N$. FInd the measure of angle $\angle AMN$ [I]Proposed by D. Shiryaev, S. Berlov[/i]

$z$ is a complex number and $|z|=1$ and $z^2\ne1$. Then $\frac z{1-z^2}$ lies on $\textbf{(A)}~\text{a line not through origin}$ $\textbf{(B)}~\text{|z|=2}$ $\textbf{(C)}~x-\text{axis}$ $\textbf{(D)}~y-\text{axis}$

In a small town, there are $n \times n$ houses indexed by $(i, j)$ for $1 \leq i, j \leq n$ with $(1, 1)$ being the house at the top left corner, where $i$ and $j$ are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by $(1, c)$, where $c \leq \frac{n}{2}$. During each subsequent time interval $[t, t+1]$, the fire fighters defend a house which is not yet on fire while the fire spreads to all undefended [i]neighbors[/i] of each house which was on fire at time t. Once a house is defended, it remains so all the time. The process ends when the fire can no longer spread. At most how many houses can be saved by the fire fighters? A house indexed by $(i, j)$ is a [i]neighbor[/i] of a house indexed by $(k, l)$ if $|i - k| + |j - l|=1$.