On a board with $3$ columns and $4$ rows, each of the $12$ squares will be painted green or white. In the first and last row, the number of squares painted green must be the same. Furthermore, in the first and last column, the number of squares painted green must also be unequal. How many different ways can you paint the board?

Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$. Circle $\omega_1$ with center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\omega_2$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\omega_1$ and $\omega_2$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.

The integer $n$ is the smallest positive multiple of 15 such that every digit of $n$ is either 8 or 0. Compute $\frac{n}{15}$.

Find the number of integer polynomials $P$ such that $P(x)^2 = P(P(x)) \forall x$.

Planet Tarator is a planet in the Yoghurty way galaxy. This planet has a shape of convex $1392$-hedron. On earth we don't have any other information about sides of planet tarator. We have discovered that each side of the planet is a country, and has it's own currency. Each two neighbour countries have their own constant exchange rate, regardless of other exchange rates. Anybody who travels on land and crosses the border must change all his money to the currency of the destination country, and there's no other way to change the money. Incredibly, a person's money may change after crossing some borders and getting back to the point he started, but it's guaranteed that crossing a border and then coming back doesn't change the money. On a research project a group of tourists were chosen and given same amount of money to travel around the Tarator planet and come back to the point they started. They always travel on land and their path is a nonplanar polygon which doesn't intersect itself. What is the maximum number of tourists that may have a pairwise different final amount of money? [b]Note 1:[/b] Tourists spend no money during travel! [b]Note 2:[/b] The only constant of the problem is 1392, the number of the sides. The exchange rates and the way the sides are arranged are unknown. Answer must be a constant number, regardless of the variables. [b]Note 3:[/b] The maximum must be among all possible polyhedras. Time allowed for this problem was 90 minutes.

Let $a_1<a_2$ be two given integers. For any integer $n\ge 3$, let $a_n$ be the smallest integer which is larger than $a_{n-1}$ and can be uniquely represented as $a_i+a_j$, where $1\le i<j\le n-1$. Given that there are only a finite number of even numbers in $\{a_n\}$, prove that the sequence $\{a_{n+1}-a_{n}\}$ is eventually periodic, i.e. that there exist positive integers $T,N$ such that for all integers $n>N$, we have \[a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.\]

There are $n$ cards. Max and Lewis play, alternately, the following game Max starts the game, he removes exactly $1$ card, in each round the current player can remove any quantity of cards, from $1$ card to $t+1$ cards, which $t$ is the number of removed cards by the previous player, and the winner is the player who remove the last card. Determine all the possible values of $n$ such that Max has the winning strategy.

The midpoints of the sides $BC$, $CA$ and $AB$ of triangle $ABC$ are $M$, $N$ and $P$ respectively . $G$ is the intersection point of the medians. The circumscribed circle around $BGP$ intersects the line $MP$ at the point $K$ (different than $P$).The circle circumscribed around $CGN$ intersects the line $MN$ at point $L$ (different than $N$). Prove that $\angle BAK = \angle CAL$.

Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride? $\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }15$

The non-decreasing functions $f,g: \mathbb{R}\rightarrow \mathbb{R}$ are such that $f(r)\leq g(r)$ for $\forall$ rational numbers $r$. Is it true that $f(x)\leq g(x)$ for $\forall$ real numbers $x$?

[b]p1.[/b] Consider a parallelogram $ABCD$ with sides of length $a$ and $b$, where $a \ne b$. The four points of intersection of the bisectors of the interior angles of the parallelogram form a rectangle $EFGH$. A possible configuration is given below. Show that $$\frac{Area(ABCD)}{Area(EFGH)}=\frac{2ab}{(a - b)^2}$$ [img]https://cdn.artofproblemsolving.com/attachments/e/a/afaf345f2ef7c8ecf4388918756f0b56ff20ef.png[/img] [b]p2.[/b] A metal wire of length $4\ell$ inches (where $\ell$ is a positive integer) is used as edges to make a cardboard rectangular box with surface area $32$ square inches and volume $8$ cubic inches. Suppose that the whole wire is used. (i) Find the dimension of the box if $\ell= 9$, i.e., find the length, the width, and the height of the box without distinguishing the different orders of the numbers. Justify your answer. (ii) Show that it is impossible to construct such a box if $\ell = 10$. [b]p3.[/b] A Pythagorean n-tuple is an ordered collection of counting numbers $(x_1, x_2,..., x_{n-1}, x_n)$ satisfying the equation $$x^2_1+ x^2_2+ ...+ x^2_{n-1} = x^2_{n}.$$ For example, $(3, 4, 5)$ is an ordinary Pythagorean $3$-tuple (triple) and $(1, 2, 2, 3)$ is a Pythagorean $4$-tuple. (a) Given a Pythagorean triple $(a, b, c)$ show that the $4$-tuple $(a^2, ab, bc, c^2)$ is Pythagorean. (b) Extending part (a) or using any other method, come up with a procedure that generates Pythagorean $5$-tuples from Pythagorean $3$- and/or $4$-tuples. Few numerical examples will not suffice. You have to find a method that will generate infinitely many such $5$-tuples. (c) Find a procedure to generate Pythagorean $6$-tuples from Pythagorean $3$- and/or $4$- and/or $5$-tuples. Note. You can assume without proof that there are infinitely many Pythagorean triples. [b]p4.[/b] Consider the recursive sequence defined by $x_1 = a$, $x_2 = b$ and $$x_{n+2} =\frac{x_{n+1} + x_n - 1}{x_n - 1}, n \ge 1 .$$ We call the pair $(a, b)$ the seed for this sequence. If both $a$ and $b$ are integers, we will call it an integer seed. (a) Start with the integer seed $(2, 2019)$ and find $x_7$. (b) Show that there are infinitely many integer seeds for which $x_{2020} = 2020$. (c) Show that there are no integer seeds for which $x_{2019} = 2019$. [b]p5.[/b] Suppose there are eight people at a party. Each person has a certain amount of money. The eight people decide to play a game. Let $A_i$, for $i = 1$ to $8$, be the amount of money person $i$ has in his/her pocket at the beginning of the game. A computer picks a person at random. The chosen person is eliminated from the game and their money is put into a pot. Also magically the amount of money in the pockets of the remaining players goes up by the dollar amount in the chosen person's pocket. We continue this process and at the end of the seventh stage emerges a single person and a pot containing $M$ dollars. What is the expected value of $M$? The remaining player gets the pot and the money in his/her pocket. What is the expected value of what he/she takes home? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and the line $ED$ is perpendicular to the line $AD$. Segment $\overline{AE}$ intersects the circle at point $C$ between $A$ and $E$. What is the area of $\triangle ABC$? $\textbf{(A) \ } \frac{120}{37}\qquad \textbf{(B) \ } \frac{140}{39}\qquad \textbf{(C) \ } \frac{145}{39}\qquad \textbf{(D) \ } \frac{140}{37}\qquad \textbf{(E) \ } \frac{120}{31}$

In the cabinet 2004 telephones are located; each two of these telephones are connected by a cable, which is colored in one of four colors. From each color there is one cable at least. Can one always select several telephones in such a way that among their pairwise cable connections exactly 3 different colors occur?

What are the signs of $\triangle{H}$ and $\triangle{S}$ for a reaction that is spontaneous only at low temperatures? $ \textbf{(A)}\ \triangle{H} \text{ is positive}, \triangle{S} \text{ is positive} \qquad\textbf{(B)}\ \triangle{H}\text{ is positive}, \triangle{S} \text{ is negative} \qquad$ $\textbf{(C)}\ \triangle{H} \text{ is negative}, \triangle{S} \text{ is negative} \qquad\textbf{(D)}\ \triangle{H} \text{ is negative}, \triangle{S} \text{ is positive} \qquad $

A right rectangular prism of silly powder has dimensions $20 \times 24 \times 25.$ Jerry the wizard applies $10$ bouts of highdroxylation to the box, each of which increases one dimension of the silly powder by $1$ and decreases a different dimension of the silly powder by $1,$ with every possible choice of dimensions equally likely to be chosen and independent of all previous choices. Compute the expected volume of the silly powder after Jerry’s routine.

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\odot O.$ The lines tangent to $\odot O$ at $A,B$ intersect at $L.$ $M$ is the midpoint of the segment $AB.$ The line passing through $D$ and parallel to $CM$ intersects $ \odot (CDL) $ at $F.$ Line $CF$ intersects $DM$ at $K,$ and intersects $\odot O$ at $E$ (different from point $C$). Prove that $EK=DK.$