Compute the number of $10$-bit sequences of $0$’s and $1$’s do not contain $001$ as a subsequence.

[b]p1.[/b] Find the smallest whole number $n \ge 2$ such that the product $(2^2 - 1)(3^2 - 1) ... (n^2 - 1)$ is the square of a whole number. [b]p2.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p3.[/b] Three cars are racing: a Ford $[F]$, a Toyota $[T]$, and a Honda $[H]$. They began the race with $F$ first, then $T$, and $H$ last. During the race, $F$ was passed a total of $3$ times, $T$ was passed $5$ times, and $H$ was passed $8$ times. In what order did the cars finish? [b]p4.[/b] There are $11$ big boxes. Each one is either empty or contains $8$ medium-sized boxes inside. Each medium box is either empty or contains $8$ small boxes inside. All small boxes are empty. Among all the boxes, there are a total of $102$ empty boxes. How many boxes are there altogether? [b]p5.[/b] Ann, Mary, Pete, and finally Vlad eat ice cream from a tub, in order, one after another. Each eats at a constant rate, each at his or her own rate. Each eats for exactly the period of time that it would take the three remaining people, eating together, to consume half of the tub. After Vlad eats his portion there is no more ice cream in the tube. How many times faster would it take them to consume the tub if they all ate together? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Juhan wants to order by weight five balls of pairwise different weight, using only a balance scale. First, he labels the balls with numbers 1 to 5 and creates a list of weighings, such that each element in the list is a pair of two balls. Then, for every pair in the list, he weighs the two balls against each other. Can Juhan sort the balls by weight, using a list with less than 10 pairs?

A list of natural numbers is written on a blackboard. The following operation is performed and repeated: choose any two numbers $a, b$, wipe them out and instead write gcd$(a, b)$ and lcm$(a, b)$. Show that the content of the list no longer changed after a certain point in time.

There are infinitely many boxes - initially one of them contains $n$ balls and all others are empty. On a single move we take some balls from a non-empty box and put them into an empty box and on a sheet of paper we write down the product of the resulting amount of balls in the two boxes. After some moves, the sum of all numbers on the sheet of paper became $2023$. What is the smallest possible value of $n$?

A sequence of $2n + 1$ non-negative integers $a_1, a_2, ..., a_{2n + 1}$ is given. There's also a sequence of $2n + 1$ consecutive cells enumerated from $1$ to $2n + 1$ from left to right, such that initially the number $a_i$ is written on the $i$-th cell, for $i = 1, 2, ..., 2n + 1$. Starting from this initial position, we repeat the following sequence of steps, as long as it's possible: [i]Step 1[/i]: Add up the numbers written on all the cells, denote the sum as $s$. [i]Step 2[/i]: If $s$ is equal to $0$ or if it is larger than the current number of cells, the process terminates. Otherwise, remove the $s$-th cell, and shift shift all cells that are to the right of it one position to the left. Then go to Step 1. Example: $(1, 0, 1, \underline{2}, 0) \rightarrow (1, \underline{0}, 1, 0) \rightarrow (1, \underline{1}, 0) \rightarrow (\underline{1}, 0) \rightarrow (0)$. A sequence $a_1, a_2,. . . , a_{2n+1}$ of non-negative integers is called balanced, if at the end of this process there’s exactly one cell left, and it’s the cell that was initially enumerated by $(n + 1)$, i.e. the cell that was initially in the middle. Find the total number of balanced sequences as a function of $n$. [i]Proposed by Viktor Simjanoski, North Macedonia[/i]

There are $110$ points in a unit square. Show that some four of these points reside in a circle whose radius is $1/8.$

[u]Round 5[/u] [b]p13.[/b] Determine the number of different circular bracelets can be made with $7$ beads, all either colored red or black. [b]p14.[/b] The product of $260$ and $n$ is a perfect square. The $2020$th least possible positive integer value of $n$ can be written as$ p^{e_1}_1 \cdot p^{e_2}_2\cdot p^{e_3}_3\cdot p^{e_4}_4$ . Find the sum $p_1 +p_2 +p_3 +p_4 +e_1 +e_2 +_e3 +e_4$. [b]p15.[/b] Let $B$ and $C$ be points along the circumference of circle $\omega$. Let $A$ be the intersection of the tangents at $B$ and $C$ and let $D \ne A$ be on $\overrightarrow{AC}$ such that $AC =CD = 6$. Given $\angle BAC = 60^o$, find the distance from point $D$ to the center of $\omega$. [u]Round 6[/u] [b]p16.[/b] Evaluate $\sqrt{2+\sqrt{2+\sqrt{2+...}}}$. [b]p17.[/b] Let $n(A)$ be the number of elements of set $A$ and $||A||$ be the number of subsets of set $A$. Given that $||A||+2||B|| = 2^{2020}$, find the value of $n(B)$. [b]p18.[/b] $a$ and $b$ are positive integers and $8^a9^b$ has $578$ factors. Find $ab$. [u]Round 7[/u] [b]p19.[/b] Determine the probability that a randomly chosen positive integer is relatively prime to $2019$. [b]p20.[/b] A $3$-by-$3$ grid of squares is to be numbered with the digits $1$ through $9$ such that each number is used once and no two even-numbered squares are adjacent. Determine the number of ways to number the grid. [b]p21.[/b] In $\vartriangle ABC$, point $D$ is on $AC$ so that $\frac{AD}{DC}= \frac{1}{13}$ . Let point $E$ be on $BC$, and let $F$ be the intersection of $AE$ and $BD$. If $\frac{DF}{FB}=\frac{2}{7}$ and the area of $\vartriangle DBC$ is $26$, compute the area of $\vartriangle F AB$. [u]Round 8[/u] [b]p22.[/b] A quarter circle with radius $1$ is located on a line with its horizontal base on the line and to the left of the vertical side. It is then rolled to the right until it reaches its original orientation. Determine the distance traveled by the center of the quarter circle. [b]p23.[/b] In $1734$, mathematician Leonhard Euler proved that $\frac{\pi^2}{6}=\frac11+\frac14+\frac19+\frac{1}{16}+...$. With this in mind, calculate the value of $\frac11-\frac14+\frac19-\frac{1}{16}+...$ (the series obtained by negating every other term of the original series). [b]p24.[/b] Billy the biker is competing in a bike show where he can do a variety of tricks. He knows that one trick is worth $2$ points, $1$ trick is worth $3$ points, and 1 is worth $5$ points, but he doesn’t remember which trick is worth what amount. When it’s Billy’s turn to perform, he does $6$ tricks, randomly choosing which trick to do. Compute the sum of all the possible values of points that Billy could receive in total. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166016p28809598]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166115p28810631]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Alice and Bob take turns removing balls from a bag containing $10$ black balls and $10$ white balls, with Alice going first. Alice always removes a black ball if there is one, while Bob removes one of the remaining balls uniformly at random. Once all balls have been removed, the expected number of black balls which Bob has can be expressed as $a/b$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

$10$ people are sitting at a round table. There are some nuts in front of each of them, $100$ nuts altogether. After a certain signal each person passes some of his nuts to the person sitting to his right . If he has an even number of nuts, he passes half of them; otherwise he passes one nut plus half of the remaining nuts. This procedure is repeated over and over again. Prove that eventually everyone will have exactly $10$ nuts. (A Shapovalov)

Prove that from every set of $200$ integers you can choose a subset of $100$ with the total sum divisible by $100$.

A large pond contains infinitely many lily pads labelled $1$, $2$, $3$,$ ... $, placed in a line, where for each $k$, lily pad $k + 1$ is one unit to the right of lily pad $k$. A frog starts at lily pad $100$. Each minute, if the frog is at lily pad $n$, it hops to lily pad $n + 1$ with probability $\frac{n-1}{n}$ , and hops all the way back to lily pad $1$ with probability $\frac{1}{n}$. Let $N$ be the position of the frog after $1000$ minutes. What is the expected value of $N$?

The game of $pebbles$ is played on an infinite board of lattice points $(i,j)$. Initially there is a $pebble$ at $(0,0)$. A move consists of removing a $pebble$ from point $(i,j)$and placing a $pebble$ at each of the points $(i+1,j)$ and $(i,j+1)$ provided both are vacant. Show taht at any stage of the game there is a $pebble$ at some lattice point $(a,b)$ with $0 \leq a+b \leq 3$

For every set $S$ with $n(\ge3)$ distinct integers, show that there exists a function $f:\{1,2,\dots,n\}\rightarrow S$ satisfying the following two conditions. (i) $\{ f(1),f(2),\dots,f(n)\} = S$ (ii) $2f(j)\neq f(i)+f(k)$ for all $1\le i<j<k\le n$.

Rațiu and Horațiu are playing a game on a $100\times 100$ grid. They make moves alternatively, starting with Rațiu. At a move, a player places a token on an empty cell of the grid. If a player places a token on a cell which is adjacent to another cell with a token, he loses. Determine who has a winning strategy.

Gabriel plays to draw triangles using the vertices of a regular polygon with $2019$ sides, following these rules: (i) The vertices used by each triangle must not have been previously used. (ii) The sides of the triangle to be drawn must not intersect with the sides of the triangles previously drawn. If Gabriel continues to draw triangles until it is no longer possible, determine the minimum number of triangles that he drew.

[u]Set 6[/u] [b]p16.[/b] Let $x$ and $y$ be non-negative integers. We say point $(x, y)$ is square if $x^2 + y$ is a perfect square. Find the sum of the coordinates of all distinct square points which also satisfy $x^2 + y \le 64$. [b]p17.[/b] Two integers $a$ and $b$ are randomly chosen from the set $\{1, 2, 13, 17, 19, 87, 115, 121\}$, with $a > b$. What is the expected value of the number of factors of $ab$? [b]p18.[/b] Marnie the Magical Cello is jumping on nonnegative integers on number line. She starts at $0$ and jumps following two specific rules. For each jump she can either jump forward by $1$ or jump to the next multiple of $4$ (the next multiple must be strictly greater than the number she is currently on). How many ways are there for her to jump to $2022$? (Two ways are considered distinct only if the sequence of numbers she lands on is different.) PS. You should use hide for answers.

[b]p1.[/b] A teenager coining home after midnight heard the hall clock striking the hour. At some moment between $15$ and $20$ minutes later, the minute hand hid the hour hand. To the nearest second, what time was it then? [b]p2.[/b] The ratio of two positive integers $a$ and $b$ is $2/7$, and their sum is a four digit number which is a perfect cube. Find all such integer pairs. [b]p3.[/b] Given the integers $1, 2 , ..., n$ , how many distinct numbers are of the form $\sum_{k=1}^n( \pm k) $ , where the sign ($\pm$) may be chosen as desired? Express answer as a function of $n$. For example, if $n = 5$ , then we may form numbers: $ 1 + 2 + 3- 4 + 5 = 7$ $-1 + 2 - 3- 4 + 5 = -1$ $1 + 2 + 3 + 4 + 5 = 15$ , etc. [b]p4.[/b] $\overline{DE}$ is a common external tangent to two intersecting circles with centers at $O$ and $O'$. Prove that the lines $AD$ and $BE$ are perpendicular. [img]https://cdn.artofproblemsolving.com/attachments/1/f/40ffc1bdf63638cd9947319734b9600ebad961.png[/img] [b]p5.[/b] Find all polynomials $f(x)$ such that $(x-2) f(x+1) - (x+1) f(x) = 0$ for all $x$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $k$ heaps on the table, each containing a different positive number of stones. Juri and Mari make moves alternatingly, Juri starts. On each move, the player making the move has to pick a heap and remove one or more stones in it from the table; in addition, the player is allowed to distribute any number of remaining stones from that heap in any way between other non-empty heaps. The player to remove the last stone from the table wins. For which positive integers $k$ does Juri have a winning strategy for any initial state that satisfies the conditions?

We have \( n \) chips that are initially placed on the number line at position 0. On each move, we select a position \( x \in \mathbb{Z} \) where there are at least two chips; we take two of these chips, then place one at \( x-1 \) and the other at \( x+1 \). a) Prove that after a finite number of moves, regardless of how the moves are chosen, we will reach a final position where no two chips occupy the same number on the number line. b) For every possible final position, let \( \Delta \) represent the difference between the numbers where the rightmost and the leftmost chips are located. Find all possible values of \( \Delta \) in terms of \( n \).

In space are given $n\ge 2$ points, no four of which are coplanar. Some of these points are connected by segments. Let $K$ be the number of segments $(K>1)$ and $T$ be the number of formed triangles. Prove that $9T^2<2K^3$.

Let $n\geq 3$ be a positive integer and a circle $\omega$ is given. A regular polygon(with $n$ sides) $P$ is drawn and your vertices are in the circle $\omega$ and these vertices are red. One operation is choose three red points $A,B,C$, such that $AB=BC$ and delete the point $B$. Prove that one can do some operations, such that only two red points remain in the circle.

Ten square cardboards of $3$ centimeters on a side are cut by a line, as indicated in the figure. After the cuts, there are $20$ pieces: $10$ triangles and $10$ trapezoids. Assemble a square that uses all $20$ pieces without overlaps or gaps. [img]https://cdn.artofproblemsolving.com/attachments/7/9/ec2242cca617305b02eef7a5409e6a6b482d66.gif[/img]

Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied: a) For any $x \in M, n \in \mathbb{Z}$, one has that $x+n \in \mathbb{M}$. b) For any $x \in M$, one has that $-x \in M$. c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$. For any real $x$, let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$. Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$, then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.

[b]p1.[/b] If Suzie has $6$ coins worth $23$ cents, how many nickels does she have? [b]p2.[/b] Let $a * b = (a - b)/(a + b)$. If $8 * (2 * x) = 4/3$, what is $x$? [b]p3.[/b] How many digits does $x = 100000025^2 - 99999975^2$ have when written in decimal form? [b]p4.[/b] A paperboy’s route covers $8$ consecutive houses along a road. He does not necessarily deliver to all the houses every day, but he always traverses the road in the same direction, and he takes care never to skip over $2$ consecutive houses. How many possible routes can he take? [b]p5.[/b] A regular $12$-gon is inscribed in a circle of radius $5$. What is the sum of the squares of the distances from any one fixed vertex to all the others? [b]p6.[/b] In triangle $ABC$, let $D, E$ be points on $AB$, $AC$, respectively, and let $BE$ and $CD$ meet at point $P$. If the areas of triangles $ADE$, $BPD$, and $CEP$ are $5$, $8$, and $3$, respectively, find the area of triangle ABC. [b]p7.[/b] Bob has $11$ socks in his drawer: $3$ different matched pairs, and $5$ socks that don’t match with any others. Suppose he pulls socks from the drawer one at a time until he manages to get a matched pair. What is the probability he will need to draw exactly $9$ socks? [b]p8.[/b] Consider the unit cube $ABCDEFGH$. The triangle bound to $A$ is the triangle formed by the $3$ vertices of the cube adjacent to $A$ (and similarly for the other vertices of the cube). Suppose we slice a knife through each of the $8$ triangles bound to vertices of the cube. What is the volume of the remaining solid that contains the former center of the cube? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].