[b]p1.[/b] $ABCD$ is a convex quadrilateral such that $AB = 20$, $BC = 24$, $CD = 7$, $DA = 15$, and $\angle DAB$ is a right angle. What is the area of $ABCD$? [b]p2.[/b] A triangular number is one that can be written in the form $1 + 2 +...·+n$ for some positive number $n$. $ 1$ is clearly both triangular and square. What is the next largest number that is both triangular and square? [b]p3.[/b] Find the last (i.e. rightmost) three digits of $9^{2008}$. [b]p4.[/b] When expressing numbers in a base $b \ge 11$, you use letters to represent digits greater than $9$. For example, $A$ represents $10$ and $B$ represents $11$, so that the number $110$ in base $10$ is $A0$ in base $11$. What is the smallest positive integer that has four digits when written in base $10$, has at least one letter in its base $12$ representation, and no letters in its base $16$ representation? [b]p5.[/b] A fly starts from the point $(0, 16)$, then flies straight to the point $(8, 0)$, then straight to the point $(0, -4)$, then straight to the point $(-2, 0)$, and so on, spiraling to the origin, each time intersecting the coordinate axes at a point half as far from the origin as its previous intercept. If the fly flies at a constant speed of $2$ units per second, how many seconds will it take the fly to reach the origin? [b]p6.[/b] A line segment is divided into two unequal lengths so that the ratio of the length of the short part to the length of the long part is the same as the ratio of the length of the long part to the length of the whole line segment. Let $D$ be this ratio. Compute $$D^{-1} + D^{[D^{-1}+D^{(D^{-1}+D^2)}]}.$$ [b]p7.[/b] Let $f(x) = 4x + 2$. Find the ordered pair of integers $(P, Q)$ such that their greatest common divisor is $1, P$ is positive, and for any two real numbers $a$ and $b$, the sentence: “$P a + Qb \ge 0$” is true if and only if the following sentence is true: “For all real numbers x, if $|f(x) - 6| < b$, then $|x - 1| < a$.” [b]p8.[/b] Call a rectangle “simple” if all four of its vertices have integers as both of their coordinates and has one vertex at the origin. How many simple rectangles are there whose area is less than or equal to $6$? [b]p9.[/b] A square is divided into eight congruent triangles by the diagonals and the perpendicular bisectors of its sides. How many ways are there to color the triangles red and blue if two ways that are reflections or rotations of each other are considered the same? [b]p10.[/b] In chess, a knight can move by jumping to any square whose center is $\sqrt5$ units away from the center of the square that it is currently on. For example, a knight on the square marked by the horse in the diagram below can move to any of the squares marked with an “X” and to no other squares. How many ways can a knight on the square marked by the horse in the diagram move to the square with a circle in exactly four moves? [img]https://cdn.artofproblemsolving.com/attachments/d/9/2ef9939642362182af12089f95836d4e294725.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Initially, on the blackboard are written all natural numbers from $1$ to $20$. A move consists of selecting $2$ numbers $a<b$ written on the blackboard such that their difference is at least $2$, erasing these numbers and writting $a+1$ and $b-1$ instead. What is the maximum numbers of moves one can perform?

A $45 \times 45$ grid has had the central unit square removed. For which positive integers $n$ is it possible to cut the remaining area into $1 \times n$ and $n\times 1$ rectangles?

For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied: (a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$; (b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$. Determine $N(n)$ for all $n\ge 2$.

Find maximal possible number of points lying on or inside a circle with radius $R$ in such a way that the distance between every two points is greater than $R\sqrt2$. [i]H. Lesov[/i]

2. There are three hands on a clock. Each of them rotates in a normal direction at some non-zero speed, which can be wrong. In the morning the long and the short hands coincided. Just in three hours after that moment the long and the mid-length hands coincided. After next four hours the short and the mid-length hands coincided. Will it necessarily occur that all three hands will coincide? Alexandr Yuran

Supppose that there are roads $AB$ and $CD$ but there are no roads $BC$ and $AD$ between four cities $A$, $B$, $C$, and $D$. Define [i]restructing[/i] to be the changing a pair of roads $AB$ and $CD$ to the pair of roads $BC$ and $AD$. Initially there were some cities in a country, some of which were connected by roads and for every city there were exactly $100$ roads starting in it. The minister drew a new scheme of roads, where for every city there were also exactly $100$ roads starting in it. It's known also that in both schemes there were no cities connected by more than one road. Prove that it's possible to obtain the new scheme from the initial after making a finite number of restructings. [i] (Т. Зубов)[/i] [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

There are $n$ points on a circle ($n>1$). Define an "interval" as an arc of a circle such that it's start and finish are from those points. Consider a family of intervals $F$ such that for every element of $F$ like $A$ there is almost one other element of $F$ like $B$ such that $A \subseteq B$ (in this case we call $A$ is sub-interval of $B$). We call an interval maximal if it is not a sub-interval of any other interval. If $m$ is the number of maximal elements of $F$ and $a$ is number of non-maximal elements of $F,$ prove that $n\geq m+\frac a2.$

The squares of an n*n chessboard (n >1) are filled with 1s and -1s. A series of steps are performed. For each step, the number in each square is replaced with the product of the numbers that were in the squares adjacent. Find all values of n for which, starting from an arbitrary collection of numbers, after finitely many steps one obtains a board filled with 1s.

Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex). Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.

Let $n>k>1$ be positive integers and let $G$ be a graph with $n$ vertices such that among any $k$ vertices, there is a vertex connected to the rest $k-1$ vertices. Find the minimal possible number of edges of $G$. Proposed by V. Dolnikov

Alice and Bob play the following game on a square grid with $2024 \times 2024$ unit squares. They take turns covering unit squares with stickers including their names. Alice plays the odd-numbered turns, and Bob plays the even-numbered turns. \\ On the $k$-th turn, let $n_k$ be the least integer such that $n_k\geqslant\tfrac{k}{2024}$. If there is at least one square without a sticker, then the player taking the turn: [list = i] [*] selects at most $n_k$ unit squares on the grid such that at least one of the chosen unit squares does not have a sticker. [*] covers each of the selected unit squares with a sticker that has their name on it. If a selected square already has a sticker on it, then that sticker is removed first. [/list] At the end of their turn, a player wins if there exist $123$ unit squares containing stickers with that player's name that are placed on horizontally, vertically, or diagonally consecutive unit squares. We consider the game to be a draw if all of the unit squares are covered but no player has won yet. \\ Does Alice have a winning strategy? [i]Proposed by Erik Paemurru, Estonia[/i]

There are n cards on a table numbered from $1$ to $n$, where $n$ is an even number. Two people take turns taking away the cards. The first player will always take the card with the largest number on it, but the second player will take a random card. Prove: the probability that the first player takes the card with the number $i$ is $ \frac{i-1}{n-1} $

Each element of set $\mathcal{S}$ is colored with multiple colors. A $\textit{rainbow}$ is a subset of $\mathcal{S}$ which has amongst its elements at least $1$ color from each element of $\mathcal{S}$. A $\textit{minimal rainbow}$ is a rainbow where removing any single element gives a non-rainbow. Prove that the union of all minimal rainbows is $\mathcal{S}$. [i]Grisham Paimagam[/i]

There were $10$ boys and $10$ girls at the party. Every boy likes a different 'positive' number of girls. Every girl likes a different positive number of boys. Define the largest non-negative integer $n$ such that it is always possible to form at least $n$ disjoint pairs in which both like the other.

For which integers $N$ it is possible to write real numbers into the cells of a square of size $N \times N$ so that among the sums of each pair of adjacent cells there are all integers from $1$ to $2(N-1)N$ (each integer once)? Maxim Didin

The game Prongle is played with a special deck of cards: on each card is a nonempty set of distinct colors. No two cards in the deck contain the exact same set of colors. In this game, a “Prongle” is a set of at least $2$ cards such that each color is on an even number of cards in the set. Let k be the maximum possible number of prongles in a set of $2019$ cards. Find $\lfloor \log 2 (k) \rfloor$.

Two players $A$ and $B$ play a game with a ball and $n$ boxes placed onto the vertices of a regular $n$-gon where $n$ is a positive integer. Initially, the ball is hidden in a box by player $A$. At each step, $B$ chooses a box, then player $A$ says the distance of the ball to the selected box to player $B$ and moves the ball to an adjacent box. If $B$ finds the ball, then $B$ wins. Find the least number of steps for which $B$ can guarantee to win.

Ten volleyball teams played a tournament; every two teams met exactly once. The winner of the play gets 1 point, the loser gets 0 (there are no draws in volleyball). If the team that scored $n$-th has $x_{n}$ points ($n=1, \dots, 10$), prove that $x_{1}+2x_{2}+\dots+10x_{10}\geq 165$. [i]Proposed by D. Teryoshin[/i]

There are two-way flights between some of the $2017$ cities in a country, such that given two cities, it is possible to reach one from the other. No matter how the flights are appointed, one can define $k$ cities as "special city", so that there is a direct flight from each city to at least one "special city". Find the minimum value of $k$.

The doctor instructed a person to take $48$ pills for next $30$ days. Every day he take at least $1$ pill and at most $6$ pills. Show that exist the numbers of conscutive days such that the total numbers of pills he take is equal with $11$.

A [u]positive sequence[/u] is a finite sequence of positive integers. [u]Sum of a sequence[/u] is the sum of all the elements in the sequence. We say that a sequence $A$ can be [u]embedded[/u] into another sequence $B$, if there exists a strictly increasing function $$\phi : \{1,2, \ldots, |A|\} \rightarrow \{1,2, \ldots, |B|\},$$ such that $\forall i \in \{1, 2, \ldots ,|A|\}$, $$A[i] \leq B[\phi(i)],$$ where $|S|$ denotes the length of a sequence $S$. For example, $(1,1,2)$ can be embedded in $(1,2,3)$, but $(3,2,1)$ can not be in $(1,2,3)$\\ Given a positive integer $n$, construct a positive sequence $U$ with sum $O(n \, \log \, n)$, such that all the positive sequences with sum $n$, can be embedded into $U$.\\

Adam has a secret natural number $x$ which Eve is trying to discover. At each stage Eve may only ask questions of the form "is $x+n$ a prime number?" for some natural number $n$ of her choice. Prove that Eve may discover $x$ using finitely many questions.

There are $ k$ rooks on a $ 10 \times 10$ chessboard. We mark all the squares that at least one rook can capture (we consider the square where the rook stands as captured by the rook). What is the maximum value of $ k$ so that the following holds for some arrangement of $ k$ rooks: after removing any rook from the chessboard, there is at least one marked square not captured by any of the remaining rooks.

Given right hexagon. Each side is divided onto $1000$ equal segments. All the points of division are connected with the segments, parallel to sides. Let us paint in turn the triples of unpainted nodes of obtained net, if they are vertices of the unilateral triangle, doesn't matter of what size an orientation. Suppose, we have managed to paint all the vertices except one. Prove that the unpainted node is not a hexagon vertex.