George is drawing a Christmas tree; he starts with an isosceles triangle $AB_0C_0$ with $AB_0=AC_0=41$ and $B_0C_0=18$. Then, he draws points $B_i$ and $C_i$ on sides $AB_0$ and $AC_0$, respectively, such that $B_iB_{i+1}=1$ and $C_iC_{i+1}=1$ ($B_{41}=C_{41}=A$). Finally, he uses a green crayon to color in triangles $B_iC_iC_{i+1}$ for $i$ from $0$ to $40$. What is the total area that he colors in?

The circle is inscribed in a triangle, inscribed in a semicircle. Find the marked angle $a$. [img]https://cdn.artofproblemsolving.com/attachments/8/e/334c8662377155086e9211da3589145f460b52.png[/img]

On the sides $AB$ and $AD$ of the square $ABCD$, the points $N$ and $P$ are selected respectively such that $NC=NP$. The point $Q$ is chosen on the segment $AN$ so that $\angle QPN = \angle NCB$. Prove that $2\angle BCQ = \angle AQP$.

Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game? $\textbf{(A) } \text{Bela will always win.}$ $\textbf{(B) } \text{Jenn will always win.} $ $\textbf{(C) } \text{Bela will win if and only if }n \text{ is odd.}$ $\textbf{(D) } \text{Jenn will win if and only if }n \text{ is odd.} $ $\textbf{(E) } \text{Jenn will win if and only if }n > 8.$

Consider all pairs of distinct points on the Cartesian plane $(A, B)$ with integer coordinates. Among these pairs of points, find all for which there exist two distinct points $(X, Y)$ with integer coordinates, such that the quadrilateral $AXBY$ is convex and inscribed. [i]Proposed by Anton Trygub[/i]

Let $x$ and $y$ be positive real numbers with $x + y =1 $. Prove that $$\frac{(3x-1)^2}{x}+ \frac{(3y-1)^2}{y} \ge1.$$ For which $x$ and $y$ equality holds? (K. Czakler, GRG 21, Vienna)

Consider $n$ students with numbers $1, 2, \ldots, n$ standing in the order $1, 2, \ldots, n.$ Upon a command, any of the students either remains on his place or switches his place with another student. (Actually, if student $A$ switches his place with student $B,$ then $B$ cannot switch his place with any other student $C$ any more until the next command comes.) Is it possible to arrange the students in the order $n,1, 2, \ldots, n-1$ after two commands ?

Let $a$ and  $b$ be the roots of the equation $x^2 + x - 3 = 0$. Find the value of the expression $4b^2 -a^3$.

Triangle $ABC$ of area $1$ is given. Point $A'$ lies on the extension of side $BC$ beyond point $C$ with $BC = CA'$. Point $B'$ lies on extension of side $CA$ beyond $A$ and $CA = AB'$. $C'$ lies on extension of $AB$ beyond $B$ with $AB = BC'$. Find the area of triangle $A'B'C'$.

Let $a$ and $b$ be real numbers, where $a \not= 0$ and $a \not= b$ and all the roots of the equation $ax^{3}-x^{2}+bx-1 = 0$ is a real and positive number. Find the smallest possible value of $P = \dfrac{5a^{2}-3ab+2}{a^{2}(b-a)}$. [i](Heir of Ramanujan)[/i]

[list=1] [*] Let $G$ be a $(4, 4)$ unoriented graph, 2-regulate, containing a cycle with the length 3. Find the characteristic polynomial $P_G (\lambda)$ , its spectrum $Spec (G)$ and draw the graph $G$. [*] Let $G'$ be another 2-regulate graph, having its characteristic polynomial $P_{G'} (\lambda) = \lambda^4 - 4\lambda^2 + \alpha, \alpha \in \mathbb{R}$. Find the spectrum $Spec(G')$ and draw the graph $G'$. [*] Are the graphs $G$ and $G'$ cospectral or isomorphic? [/list]

Find all functions $f:\mathbb{R}\to \mathbb R$ satisfying \[f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),\] For all $x,y\in\mathbb R$.

A freight train leaves the town of Jenkinsville at $1:00$ PM traveling due east at constant speed. Jim, a hobo, sneaks onto the train and falls asleep. At the same time, Julie leaves Jenkinsville on her bicycle, traveling along a straight road in a northeasterly direction (but not due northeast) at $10$ miles per hour. At $1:12$ PM, Jim rolls over in his sleep and falls from the train onto the side of the tracks. He wakes up and immediately begins walking at $3:5$ miles per hour directly towards the road on which Julie is riding. Jim reaches the road at $2:12$ PM, just as Julie is riding by. What is the speed of the train in miles per hour?

Find the value of $a+b$ given that $(a,b)$ is a solution to the system \begin{align*}3a+7b&=1977,\\5a+b&=2007.\end{align*} $\begin{array}{c@{\hspace{14em}}c@{\hspace{14em}}c} \textbf{(A) }488&\textbf{(B) }498&\end{array}$

A permutation of $\{1, 2, 3, \dots, 16\}$ is called \emph{blocksum-simple} if there exists an integer $n$ such that the sum of any $4$ consecutive numbers in the permutation is either $n$ or $n+1$. How many blocksum-simple permutations are there? [i]Proposed by Yannick Yao[/i]

Let $T=\text{TNFTPP}$. How many positive integers are within $T$ of exactly $\lfloor \sqrt T\rfloor$ perfect squares? (Note: $0^2=0$ is considered a perfect square.)

$m>1$ is an integer such that $[2m-\sqrt{m}+1, 2m]$ contains a prime. Prove that for any pairwise distinct positive integers $a_1$, $a_2$, $\dots$, $a_m$, there is always $1\leq i,j\leq m$ such that $\frac{a_i}{(a_i, a_j)}\geq m$.

Given that six-digit positive integer $\overline{ABCDEF}$ has distinct digits $A,$ $B,$ $C,$ $D,$ $E,$ $F$ between $1$ and $8$, inclusive, and that it is divisible by $99$, find the maximum possible value of $\overline{ABCDEF}$. [i]Proposed by Andrew Milas[/i]

Are there integers $m, n \geq 2$ such that the following property is always true? $$``\text{For any real numbers } x, y, \text{ if } x^m + y^m \text{ and } x^n + y^n \text{ are integers, then } x + y \text{ is an integer}".$$

Let $f$ be a real-valued function such that \[f(x)+2f\left(\dfrac{2002}x\right)=3x\] for all $x>0$. Find $f(2)$. $\textbf{(A) }1000\qquad\textbf{(B) }2000\qquad\textbf{(C) }3000\qquad\textbf{(D) }4000\qquad\textbf{(E) }6000$

Prove that $\sqrt{a_1}+\sqrt{a_2}+\cdots+\sqrt{a_{119}}$ is an integer, where \[a_n=2-\frac{1}{n^2+\sqrt{n^4+1/4}}.\]

Alice and Bob play a game. Initially, they write the pair $(1012,1012)$ on the board. They alternate their turns with Alice going first. In each turn the player can turn the pair $(a,b)$ to either $(a-2, b+1), (a+1, b-2)$ or $(a-1, b)$ as long as the resulting pair has only nonnegative values. The game terminates, when there is no legal move possible. Alice wins if the game terminates at $(0,0)$ and Bob wins if the game terminates at $(0,1)$. Determine who has the winning strategy? [i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]

In rectangle $ ABCD$, we have $ A \equal{} (6, \minus{} 22)$, $ B \equal{} (2006,178)$, and $ D \equal{} (8,y)$, for some integer $ y$. What is the area of rectangle $ ABCD$? $ \textbf{(A) } 4000 \qquad \textbf{(B) } 4040 \qquad \textbf{(C) } 4400 \qquad \textbf{(D) } 40,000 \qquad \textbf{(E) } 40,400$

Let us define the sequences $a_1, a_2, a_3,...$ and $b_1, b_2, b_3,...$. with the following conditions $a_1 = 3, b_1 = 1$ and $a_{n +1} =\frac{a_n^2+b_n^2}{2}$ and $b_{n + 1}= a_n \cdot b_n$ for each $n = 1, 2,...$. Find all different prime factors οf the number $a_{2000} + b_{2000}$.

Find all continuous function $f:\mathbb{R}^{\geq 0}\rightarrow \mathbb{R}^{\geq 0}$ such that : \[f(xf(y))+f(f(y)) = f(x)f(y)+2 \: \: \forall x,y\in \mathbb{R}^{\geq 0}\] [i]Proposed by Mohammad Ahmadi[/i]