Rectangle $ABCD$ has an area of 30. Four circles of radius $r_1 = 2$, $r_2 = 3$, $r_3 = 5$, and $r_4 = 4$ are centered on the four vertices $A$, $B$, $C$, and $D$ respectively. Two pairs of external tangents are drawn for the circles at A and $C$ and for the circles at $B$ and $D$. These four tangents intersect to form a quadrilateral $W XY Z$ where $\overline{W X}$ and $\overline{Y Z}$ lie on the tangents through the circles on $A$ and $C$. If $\overline{W X} + \overline{Y Z} = 20$, find the area of quadrilateral $W XY Z$. [img]https://cdn.artofproblemsolving.com/attachments/5/a/cb3b3457f588a15ffb4c875b1646ef2aec8d11.png[/img]

Positive numbers $x,y$ satisfy equality $$x^3+y^3=x-y$$ Prove that $$x^2+y^2<1$$

Let $a_{1}=1$, $a_{2}=2$, $a_{3}$, $a_{4}$, $\cdots$ be the sequence of positive integers of the form $2^{\alpha}3^{\beta}$, where $\alpha$ and $\beta$ are nonnegative integers. Prove that every positive integer is expressible in the form \[a_{i_{1}}+a_{i_{2}}+\cdots+a_{i_{n}},\] where no summand is a multiple of any other.

Let $\mathbb{R}$ denote the set of real numbers. Suppose a function $f: \mathbb{R} \to \mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x\in \mathbb{R}$. Show that [b](i)[/b] $f$ is one-one, [b](ii)[/b] $f$ cannot be strictly decreasing, and [b](iii)[/b] if $f$ is strictly increasing, then $f(x)=x$ for all $x \in \mathbb{R}$.

Points $P_1, P_2, P_3,$ and $P_4$ are $(0,0), (10, 20), (5, 15),$ and $(12, -6)$, respectively. For what point $P \in \mathbb{R}^2$ is the sum of the distances from $P$ to the other $4$ points minimal?

Which is greater: \[\frac{1^{-3}-2^{-3}}{1^{-2}-2^{-2}}-\frac{2^{-3}-3^{-3}}{2^{-2}-3^{-2}}+\frac{3^{-3}-4^{-3}}{3^{-2}-4^{-2}}-\cdots +\frac{2019^{-3}-2020^{-3}}{2019^{-2}-2020^{-2}}\] or \[1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots +\frac{1}{5781}?\]

The sides $PQ$ and $PR$ of triangle $PQR$ are respectively of lengths $4$ inches, and $7$ inches. The median $PM$ is $3\frac{1}{2}$ inches. Then $QR$, in inches, is: $\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad \textbf{(E) }10$

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

Suppose that $a, b,$ and $c$ are positive real numbers such that $$a + b + c \ge \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$ Find the largest possible value of the expression $$\frac{a + b - c}{a^3 + b^3 + abc} + \frac{b + c - a}{b^3 + c^3 + abc} + \frac{c + a - b}{c^3 + a^3 + abc}.$$

Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths. Prove that the lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ are concurrent. [i]Bogdan Enescu, Romania[/i]

In the triangle $ABC$ we have $30^o$ at the vertex $A$, and $50^o$ at the vertex $B$. Let $O$ be the center of inscribed circle. Show that $AC + OC = AB$.

Let $ f: \mathbb{R}\to\mathbb{N}$ be a function which satisfies $ f\left(x \plus{} \dfrac{1}{f(y)}\right) \equal{} f\left(y \plus{} \dfrac{1}{f(x)}\right)$ for all $ x$, $ y\in\mathbb{R}$. Prove that there is a positive integer which is not a value of $ f$. [i]Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithuania[/i]

Prove that one can construct two triangles from six edges of an arbitrary tetrahedron. (VV Proizvolov)

If $P$ is a function such that $P(2x) = 2^{-3}P(x) + 1$, find $P(0)$.

$x_1,x_2,\cdots,x_n$ are positive real numbers. Prove that $$\frac{x_1^2}{x_2}+\frac{x_2^2}{x_3}+\cdots+\frac{x_n^2}{x_1}\geq x_1+x_2+\cdots x_n.$$

Let $ABC$ be a triangle, and $O$ the center of its circumcircle. Let a line through the point $O$ intersect the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Denote by $S$ and $R$ the midpoints of the segments $BN$ and $CM$, respectively. Prove that $\measuredangle ROS=\measuredangle BAC$.

Beatrix is going to place six rooks on a $6\times6$ chessboard where both the rows and columns are labelled $1$ to $6$; the rooks are placed so that no two rooks are in the same row or the same column. The [i]value[/i] of a square is the sum of its row number and column number. The [i]score[/i] of an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Let $n$ be a given positive integer. The sequence of real numbers $a_1, a_2, a_3, \cdots, a_n$ satisfy for each $m \leq n$, $$\left|\sum_{k=1}^m\frac{a_k}k\right| \leq 1.$$ Given this information, find the greatest possible value of $\left|\sum_{k=1}^n a_k\right|$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 13)[/i]

Let $n \ge 1$ be an integer. Prove that there exists a set $S$ of $n$ positive integers with the following property: if $A$ and $B$ are any two distinct non-empty subsets of $S$, then the averages $\frac{P_{x\in A} x}{|A|}$ and $\frac{P_{x\in B} x}{|B|}$ are two relatively prime composite integers.

[b]p1.[/b] A right triangle has hypotenuse of length $12$ cm. The height corresponding to the right angle has length $7$ cm. Is this possible? [img]https://cdn.artofproblemsolving.com/attachments/0/e/3a0c82dc59097b814a68e1063a8570358222a6.png[/img] [b]p2.[/b] Prove that from any $5$ integers one can choose $3$ such that their sum is divisible by $3$. [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a knight on an arbitrary square. Then the second player can put another knight on a free square that is not controlled by the first knight. Then the first player can put a new knight on a free square that is not controlled by the knights on the board. Then the second player can do the same, etc. A player who cannot put a new knight on the board loses the game. Who has a winning strategy? [b]p4.[/b] Consider a regular octagon $ABCDEGH$ (i.e., all sides of the octagon are equal and all angles of the octagon are equal). Show that the area of the rectangle $ABEF$ is one half of the area of the octagon. [img]https://cdn.artofproblemsolving.com/attachments/d/1/674034f0b045c0bcde3d03172b01aae337fba7.png[/img] [b]p5.[/b] Can you find a positive whole number such that after deleting the first digit and the zeros following it (if they are) the number becomes $24$ times smaller? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $b$ be the probability that the cards are from different suits. Compute $\lfloor1000b\rfloor$.

Represent the polynomial $P(X) = X^{100} + X^{20} + 1$ as the product of 4 polynomials with integer coefficients.

There are $300$ apples in a table and the heaviest apple is [b]not[/b] heavier than three times the weight of the lightest apple. Prove that the apples can be splitted in sets of $4$ elements such that [b]no[/b] set is heavier than $\frac{3}{2}$ times the weight of any other set.

Five squirrels together have a supply of 2022 nuts. On the first day 2 nuts are added, on the second day 4 nuts, on the third day 6 nuts and so on, i.e. on each further day 2 nuts more are added than on the day before. At the end of any day the squirrels divide the stock among themselves. Is it possible that they all receive the same number of nuts and that no nut is left over?

Triangle $ABC$ is not isosceles. The incircle of $\triangle ABC$ touches the sides $BC$, $CA$, $AB$ in the points $K$, $L$, $M$. The parallel with $LM$ through $B$ meets $KL$ at $D$, the parallel with $LM$ through $C$ meets $KM$ at $E$. Prove that $DE$ passes through the midpoint of $\overline{LM}$.