A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive $x$ and $y$ axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly $1989$ minutes? [asy] draw((0,0)--(20,0), EndArrow); draw((0,0)--(0,25), EndArrow); draw((0,0)--(5,0)--(5,5)--(0,5)--(0,10)--(10,10)--(10,0)--(15,0)--(15,15)--(0,15)--(0,20)--(10,20),linewidth(2)); draw((0,20)--(10,20), EndArrow); draw((3.5,.5)--(4,.5)--(4,2), EndArrow); draw((4,3.5)--(4,4)--(2.5,4), EndArrow); draw((2,5.5)--(1,5.5)--(1,7), EndArrow); draw((1,8)--(1,9)--(2.5,9), EndArrow); draw((8,9.5)--(9,9.5)--(9,8), EndArrow); draw((10.5,2)--(10.5,1)--(12,1), EndArrow); draw((13,.5)--(14,.5)--(14,2), EndArrow); draw((14.5,13)--(14.5,14)--(13,14), EndArrow); draw((2,15.5)--(1,15.5)--(1,17), EndArrow); draw((.5,18)--(.5,19)--(2,19), EndArrow); label("x", (21,0), E); label("y", (0,26), N); label("4", (0,20), W); label("3", (0,15), W); label("2", (0,10), W); label("1", (0,5), W); label("0", (0,0), SW); label("1", (5,0), S); label("2", (10,0), S); label("3", (15,0), S); [/asy] $\textbf{(A)}\ (35,44) \qquad\textbf{(B)}\ (36,45) \qquad\textbf{(C)}\ (37,45) \qquad\textbf{(D)}\ (44,35) \qquad\textbf{(E)}\ (45,36)$

What is the greatest positive integer $x$ for which $2^{2^x+1}+2$ is divisible by $17$?

Given two distinct numbers $ b_1$ and $ b_2$, their product can be formed in two ways: $ b_1 \times b_2$ and $ b_2 \times b_1.$ Given three distinct numbers, $ b_1, b_2, b_3,$ their product can be formed in twelve ways: $ b_1\times(b_2 \times b_3);$ $ (b_1 \times b_2) \times b_3;$ $ b_1 \times (b_3 \times b_2);$ $ (b_1 \times b_3) \times b_2;$ $ b_2 \times (b_1 \times b_3);$ $ (b_2 \times b_1) \times b_3;$ $ b_2 \times(b_3 \times b_1);$ $ (b_2 \times b_3)\times b_1;$ $ b_3 \times(b_1 \times b_2);$ $ (b_3 \times b_1)\times b_2;$ $ b_3 \times(b_2 \times b_1);$ $ (b_3 \times b_2) \times b_1.$ In how many ways can the product of $ n$ distinct letters be formed?

Rectangle $ ABCD$ and a semicircle with diameter $ AB$ are coplanar and have nonoverlapping interiors. Let $ \mathcal{R}$ denote the region enclosed by the semicircle and the rectangle. Line $ \ell$ meets the semicircle, segment $ AB$, and segment $ CD$ at distinct points $ N$, $ U$, and $ T$, respectively. Line $ \ell$ divides region $ \mathcal{R}$ into two regions with areas in the ratio $ 1: 2$. Suppose that $ AU \equal{} 84$, $ AN \equal{} 126$, and $ UB \equal{} 168$. Then $ DA$ can be represented as $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.

A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square$?$ $\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{5}{8} \qquad \textbf{(C) } \frac{2}{3} \qquad \textbf{(D) } \frac{3}{4} \qquad \textbf{(E) } \frac{7}{8}$

The point $M$ is inside the tetrahedron $ABCD$ and the intersection points of the lines $AM,BM,CM$ and $DM$ with the opposite walls are denoted with $A_1,B_1,C_1,D_1$ respectively. It is given also that the ratios $\frac{MA}{MA_1}$, $\frac{MB}{MB_1}$, $\frac{MC}{MC_1}$, and $\frac{MD}{MD_1}$ are equal to the same number $k$. Find all possible values of $k$. [i]K. Petrov[/i]

We have $31$ pieces where $1$ is written on two of them, $2$ is written on eight of them, $3$ is written on twelve of them, $4$ is written on four of them, and $5$ is written on five of them. We place $30$ of them into a $5\times 6$ chessboard such that the sum of numbers on any row is equal to a fixed number and the sum of numbers on any column is equal to a fixed number. What is the number written on the piece which is not placed? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

[u]Round 1[/u] [b]p1.[/b] Goldilocks enters the home of the three bears – Papa Bear, Mama Bear, and Baby Bear. Each bear is wearing a different-colored shirt – red, green, or blue. All the bears look the same to Goldilocks, so she cannot otherwise tell them apart. The bears in the red and blue shirts each make one true statement and one false statement. The bear in the red shirt says: “I'm Blue's dad. I'm Green's daughter.” The bear in the blue shirt says: “Red and Green are of opposite gender. Red and Green are my parents.” Help Goldilocks find out which bear is wearing which shirt. [b]p2.[/b] The University of Washington is holding a talent competition. The competition has five contests: math, physics, chemistry, biology, and ballroom dancing. Any student can enter into any number of the contests but only once for each one. For example, a student may participate in math, biology, and ballroom. It turned out that each student participated in an odd number of contests. Also, each contest had an odd number of participants. Was the total number of contestants odd or even? [b]p3.[/b] The $99$ greatest scientists of Mars and Venus are seated evenly around a circular table. If any scientist sees two colleagues from her own planet sitting an equal number of seats to her left and right, she waves to them. For example, if you are from Mars and the scientists sitting two seats to your left and right are also from Mars, you will wave to them. Prove that at least one of the $99$ scientists will be waving, no matter how they are seated around the table. [b]p4.[/b] One hundred boys participated in a tennis tournament in which every player played each other player exactly once and there were no ties. Prove that after the tournament, it is possible for the boys to line up for pizza so that each boy defeated the boy standing right behind him in line. [b]p5.[/b] To celebrate space exploration, the Science Fiction Museum is going to read Star Wars and Star Trek stories for $24$ hours straight. A different story will be read each hour for a total of $12$ Star Wars stories and $12$ Star Trek stories. George and Gene want to listen to exactly $6$ Star Wars and $6$ Star Trek stories. Show that no matter how the readings are scheduled, the friends can find a block of $12$ consecutive hours to listen to the stories together. [u]Round 2[/u] [b]p6.[/b] $2013$ people attended Cinderella's ball. Some of the guests were friends with each other. At midnight, the guests started turning into mice. After the first minute, everyone who had no friends at the ball turned into a mouse. After the second minute, everyone who had exactly one friend among the remaining people turned into a mouse. After the third minute, everyone who had two human friends left in the room turned into a mouse, and so on. What is the maximal number of people that could have been left at the ball after $2013$ minutes? [b]p7.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If we have a number $x$ at a certain step, then at the next step we have $x+1$ or $-\frac 1x$. If we start with the number $1$, which of the following cannot be got after a finite number of steps? $ \textbf{(A)}\ -2 \qquad\textbf{(B)}\ \dfrac 12 \qquad\textbf{(C)}\ \dfrac 53 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None of above} $

Let $ABCD$ be a unit square. Draw a quadrant of the a circle with $A$ as centre and $B,D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as end points of the arc. Inscribe a circle $ \Gamma$ touching arcs $AC$ and $BD$ both externally and also touching the side $CD$. Find the radius of $ \Gamma$.

We define the [i]Fibonacci sequence[/i] $\{F_n\}_{n\ge0}$ by $F_0=0$, $F_1=1$, and for $n\ge2$, $F_n=F_{n-1}+F_{n-2}$; we define the [i]Stirling number of the second kind[/i] $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets. For every positive integer $n$, let $t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$. [i]Proposed by Victor Wang[/i]

For real numbers, $a,b,c$ with $bc \ne 0$ we have to $\frac{1-c^2}{bc} \ge 0$. Prove that $$5( a^2+b^2+c^2 -bc^3) \ge ab.$$

Within an arithmetic progression of length $ 2005, $ find the number of arithmetic subprogressions of length $ 501 $ that don't contain the $ \text{1000-th} $ term of the progression.

A store had $376$ chocolate bars. Min bought some of the bars, and Max bought $41$ more of the bars than Min bought. After that, the store still had three times as many chocolate bars as Min bought. Find the number of chocolate bars that Min bought.

2) all elements in {0,1,2}; B[n] = number of rows with no 2 sequent 0's; A[n] with no 3 sequent elements the same; prove |A[n+1]|=3.|B[n]|