In the following figure, the largest square is divided into two squares and three rectangles, as shown: The area of each smaller square is equal to $a$ and the area of each small rectangle is equal to $b$. If $a+b=24$ and the root square of $a$ is a natural number, find all possible values for the area of the largest square. [img]https://cdn.artofproblemsolving.com/attachments/f/a/0b424d9c293889b24d9f31b1531bed5081081f.png[/img]

Let $a,b$ be positive integers. Prove that the equation $x^2+(a+b)^2x+4ab=1$ has rational solutions if and only if $a=b$. [i]Mihai Opincariu[/i]

50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine. [i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]

Simplify $ \left(\frac {1 \cdot 2 \cdot 4 \plus{} 2 \cdot 4 \cdot 8 \plus{} \cdots \plus{} n \cdot 2n \cdot 4n}{1 \cdot 3 \cdot 9 \plus{} 2 \cdot 6 \cdot 18 \plus{} \cdots \plus{} n \cdot 3n \cdot 9n}\right)^{\frac {1}{3}}$

Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]

Let $ABC$ be an acute triangle with orthocenter $H$. Let $D$ and $E$ be feet of the altitudes from $B$ and $C$ respectively. Let $M$ be the midpoint of segment $AH$ and $F$ be the intersection point of $AH$ and $DE$. Furthermore, let $P$ and $Q$ be the points inside triangle $ADE$ so that $P$ is an intersection of $CM$ and the circumcircle of $DFH$, and $Q$ is an intersection of $BM$ and the circumcircle of $EFH$. Prove that the intersection of lines $DQ$ and $EP$ lies on segment $AH$.

Every vertex of a polygon has both integer coordinates; the length of each side of this polygon is a natural number. Prove that the perimeter of the polygon is an even number.

Let $P$ be a point inside the acute triangle $ABC$ with $m(\widehat{PAC})=m(\widehat{PCB})$. $D$ is the midpoint of the segment $PC$. $AP$ and $BC$ intersect at $E$, and $BP$ and $DE$ intersect at $Q$. Prove that $\sin\widehat{BCQ}=\sin\widehat{BAP}$.

A finite set $\mathcal S$ of distinct positive real numbers is called [i]radiant[/i] if it satisfies the following property: if $a$ and $b$ are two distinct elements of $\mathcal S$, then $a^2 + b^2$ is also an element of $\mathcal S$. [list=a] [*]Does there exist a radiant set with a size greater than or equal to $4$? [*]Determine all radiant sets of size $2$ or $3$. [/list]

p1. $A$ is a set of numbers. The set $A$ is closed to subtraction, meaning that the result of subtracting two numbers in $A$ will be returns a number in $A$ as well. If it is known that two members of $A$ are $4$ and $9$, show that: a. $0\in A$ b. $13 \in A$ c. $74 \in A$ d. Next, list all the members of the set $A$ . p2. $(2, 0, 4, 1)$ is one of the solutions/answers of $x_1+x_2+x_3+x_4=7$. If all solutions belong on the set of not negative integers , specify as many possible solutions/answers from $x_1+x_2+x_3+x_4=7$ p3. Adi is an employee at a textile company on duty save data. One time Adi was asked by the company leadership to prepare data on production increases over five periods. After searched by Adi only found four data on the increase, namely $4\%$, $9\%$, $7\%$, and $5\%$. One more data, namely the $5$th data, was not found. Investigate increase of 5th data production, if Adi only remembers that the arithmetic mean and median of the five data are the same. p4. Find all pairs of integers $(x,y)$ that satisfy the system of the following equations: $$\left\{\begin{array}{l} x(y+1)=y^2-1 \\ y(x+1)=x^2-1 \end{array} \right. $$ p5. Given the following image. $ABCD$ is square, and $E$ is any point outside the square $ABCD$. Investigate whether the relationship $AE^2 + CE^2 = BE^2 +DE^2$ holds in the picture below. [img]https://cdn.artofproblemsolving.com/attachments/2/5/a339b0e4df8407f97a4df9d7e1aa47283553c1.png[/img]

Circles $C_{1}$ and $C_{2}$ are tangent to each other at $K$ and are tangent to circle $C$ at $M$ and $N$. External tangent of $C_{1}$ and $C_{2}$ intersect $C$ at $A$ and $B$. $AK$ and $BK$ intersect with circle $C$ at $E$ and $F$ respectively. If AB is diameter of $C$, prove that $EF$ and $MN$ and $OK$ are concurrent. ($O$ is center of circle $C$.)

Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

If a, b, and c are non-negative real numbers satisfying $a + b + c = 400$, find the maximum possible value of $\sqrt{2a+b}+\sqrt{2b+c}+\sqrt{2c+a}$.

Let $I = (0, 1]$ be the unit interval of the real line. For a given number $a \in (0, 1)$ we define a map $T : I \to I$ by the formula if \[ T (x, y) = \begin{cases} x + (1 - a),&\mbox{ if } 0< x \leq a,\\ \text{ } \\ x - a, & \mbox{ if } a < x \leq 1.\end{cases} \] Show that for every interval $J \subset I$ there exists an integer $n > 0$ such that $T^n(J) \cap J \neq \emptyset.$

Let $a$ be a positive integer such that $\gcd(an+1, 2n+1) = 1$ for all integer $n$. a) Prove that $\gcd(a-2, 2n+1) = 1$ for all integer $n$. b) Find all possible $a$.

Uri has $99$ empty bags and an unlimited number of balls. The weight of each ball is a number of the form $3^n$ where $n$ is an integer that can vary from ball to ball (negative integer exponents are allowed, such as $3^{-4}=\dfrac{1}{81}$, and the exponent $0$, where $3^0=1$). Uri chose a finite number of balls and distributed them into the bags so that all the bags had the same total weight and there were no balls left over. It is known that Uri chose at most $k$ balls of the same weight. Determine the smallest possible value of $k$.

$p(x)$ is the cubic $x^3 - 3x^2 + 5x$. If $h$ is a real root of $p(x) = 1$ and $k$ is a real root of $p(x) = 5$, find $h + k$.

Jordi stands 20 m from a wall and Diego stands 10 m from the same wall. Jordi throws a ball at an angle of 30 above the horizontal, and it collides elastically with the wall. How fast does Jordi need to throw the ball so that Diego will catch it? Consider Jordi and Diego to be the same height, and both are on the same perpendicular line from the wall. $\textbf{(A) } 11 \text{ m/s}\\ \textbf{(B) } 15 \text{ m/s}\\ \textbf{(C) } 19 \text{ m/s}\\ \textbf{(D) } 30 \text{ m/s}\\ \textbf{(E) } 35 \text{ m/s}$

If $x_1,x_2,...,x_n>0 $ and $x_1^2+x_2^2+...+x_n^2=\dfrac{1}{n}$,prove that $\sum x_i+\sum \dfrac{1}{x_i \cdot x_{i+1}} \ge n^3+1.$

On a straight track are several runners, each running at a different constant speed. They start at one end of the track at the same time. When a runner reaches any end of the track, he immediately turns around and runs back with the same speed (then he reaches the other end and turns back again, and so on). Some time after the start, all runners meet at the same point. Prove that this will happen again.

A zerg player can produce one zergling every minute and a protoss player can produce one zealot every $2.1$ minutes. Both players begin building their respective units immediately from the beginning of the game. In a ght, a zergling army overpowers a zealot army if the ratio of zerglings to zealots is more than $3$. What is the total amount of time (in minutes) during the game such that at that time the zergling army would overpower the zealot army?

$\frac1{1+\sqrt3}+\frac1{\sqrt3+\sqrt5}+\frac1{\sqrt5+\sqrt7}+\ldots+\frac1{\sqrt{2017}+\sqrt{2019}}=?$ $\textbf{(A)}~\frac{\sqrt{2019}-1}2$ $\textbf{(B)}~\frac{\sqrt{2019}+1}2$ $\textbf{(C)}~\frac{\sqrt{2019}-1}4$ $\textbf{(D)}~\text{None of the above}$

In the classroom of at least four students the following holds: no matter which four of them take seats around a round table, there is always someone who either knows both of his neighbours, or does not know either of his neighbours. Prove that it is possible to divide the students into two groups such that in one of them, all students know one another, and in the other, none of the students know each other. (Note: if student A knows student B, then student B knows student A as well.)

prove that if graph $G$ is a tree, then there is a vertex that is common between all of the longest paths. [i]proposed by Sina Rezayi[/i]

The diameter \( AD \) of the circumcircle of triangle \( ABC \) intersects line \( BC \) at point \( K \). Point \( D \) is reflected symmetrically with respect to point \( K \), resulting in point \( L \). On line \( AB \), a point \( F \) is chosen such that \( FL \perp AC \). Prove that \( FK \perp AD \). [i]Proposed by Matthew Kurskyi[/i]