What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ? [i]Posted already on the board I think...[/i]

[i]Mediovagio[/i] is a computer game that consists in a $3 \times 3$ table in which each of the nine cells has a integer number from $1$ to $n$. When one clicks a cell, the numbers in the clicked cell and in the cells that share an edge with it are increased by $1$ and the sum is evaluated${}\bmod n$. Determine the values of $n$ for which it's possible, with a finite number of clicks, obtain any combination of numbers from an given initial combination. EDIT: I corrected the statement.

[u]Round 5[/u] [b]p13.[/b] Sally is at the special glasses shop, where there are many different optical lenses that distort what she sees and cause her to see things strangely. Whenever she looks at a shape through lens $A$, she sees a shape with $2$ more sides than the original (so a square would look like a hexagon). When she looks through lens $B$, she sees the shape with $3$ fewer sides (so a hexagon would look like a triangle). How many sides are in the shape that has $200$ more diagonals when looked at from lense $A$ than from lense $B$? [b]p14.[/b] How many ways can you choose $2$ cells of a $5$ by $5$ grid such that they aren't in the same row or column? [b]p15.[/b] If $a + \frac{1}{b} = (2015)^{-1}$ and $b + \frac{1}{a} = (2016)^2$ then what are all the possible values of $b$? [u]Round 6[/u] [b]p16.[/b] In Canadian football, linebackers must wear jersey numbers from $30 -35$ while defensive linemen must wear numbers from $33 -38$ (both intervals are inclusive). If a team has $5$ linebackers and $4$ defensive linemen, how many ways can it assign jersey numbers to the $9$ players such that no two people have the same jersey number? [b]p17.[/b] What is the maximum possible area of a right triangle with hypotenuse $8$? [b]p18.[/b] $9$ people are to play touch football. One will be designated the quarterback, while the other eight will be divided into two (indistinct) teams of $4$. How many ways are there for this to be done? [u]Round 7[/u] [b]p19.[/b] Express the decimal $0.3$ in base $7$. [b]p20.[/b] $2015$ people throw their hats in a pile. One at a time, they each take one hat out of the pile so that each has a random hat. What is the expected number of people who get their own hat? [b]p21.[/b] What is the area of the largest possible trapezoid that can be inscribed in a semicircle of radius $4$? [u]Round 8[/u] [b]p22.[/b] What is the base $7$ expression of $1211_3 \cdot 1110_2 \cdot 292_{11} \cdot 20_3$ ? [b]p23.[/b] Let $f(x)$ equal the ratio of the surface area of a sphere of radius $x$ to the volume of that same sphere. Let $g(x)$ be a quadratic polynomial in the form $x^2 + bx + c$ with $g(6) = 0$ and the minimum value of $g(x)$ equal to $c$. Express $g(x)$ as a function of $f(x)$ (e.g. in terms of $f(x)$). [b]p24.[/b] In the country of Tahksess, the income tax code is very complicated. Citizens are taxed $40\%$ on their first $\$20, 000$ and $45\%$ on their next $\$40, 000$ and $50\%$ on their next $\$60, 000$ and so on, with each $5\%$ increase in tax rate aecting $\$20, 000$ more than the previous tax rate. The maximum tax rate, however, is $90\%$. What is the overall tax rate (percentage of money owed) on $1$ million dollars in income? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3157009p28696627]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3158564p28715928]here[/url]. .Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are 100 students taking an exam. The professor calls them one by one and asks each student a single person question: “How many of 100 students will have a “passed” mark by the end of this exam?” The students answer must be an integer. Upon receiving the answer, the professor immediately publicly announces the student’s mark which is either “passed” or “failed.” After all the students have got their marks, an inspector comes and checks if there is any student who gave the correct answer but got a “failed” mark. If at least one such student exists, then the professor is suspended and all the marks are replaced with “passed.” Otherwise no changes are made. Can the students come up with a strategy that guarantees a “passed” mark to each of them? [i] Denis Afrizonov [/i]

Given $n$ points $X_1,X_2,\ldots, X_n$ in the interval $0 \leq X_i \leq 1, i = 1, 2,\ldots, n$, show that there is a point $y, 0 \leq y \leq 1$, such that \[\frac{1}{n} \sum_{i=1}^{n} | y - X_i | = \frac 12.\]

4.4. Prove that there exists a set X of 1996 positive integers with the following properties: (i) the elements of X are pairwise coprime; (ii) all elements of X and all sums of two or more distinct elements of X are composite numbers

Two players play a game, starting with a pile of $N$ tokens. On each player’s turn, they must remove $2^n$ tokens from the pile for some nonnegative integer $n$. If a player cannot make a move, they lose. For how many $N$ between $ 1$ and $2019$ (inclusive) does the first player have a winning strategy?

Let $b_0, b_1, b_2, \ldots$ be a sequence of pairwise distinct nonnegative integers such that $b_0=0$ and $b_n<2n$ for all positive integers $n$. Prove that for each nonnegative integer $m$ there exist nonnegative integers $k, \ell$ such that \begin{align*} b_k+b_{\ell}=m. \end{align*}

Consider a finite binary string $b$ with at least $2017$ ones. Show that one can insert some plus signs in between pairs of digits such that the resulting sum, when performed in base $2$, is equal to a power of two. [i]Proposed by David Stoner

We are given a chessboard 100 x 100, $k$ barriers (each with length 1), and one ball. We want to put the barriers between the cells of the board and put the ball in some cell, in such way that the ball can get to each possible cell on the board. The only way that the ball can move is by lifting the board so it can go only forward, backward, to the left or to the right. The ball passes all cells on its way until it reaches a barrier or the edge of the board where it stops. What’s the least number of barriers we need so we can achieve that?

Given an expression $$x_1 : x_2 : ... : x_n$$ ( $:$ means division). We can put the braces as we want. How many expressions can we obtain?

A storage depot is a pyramid with height $30$ and a square base with side length $40$. Determine how many cubical $3\times 3\times 3$ boxes can be stored in this depot if the boxes are always packed so that each of their edges is parallel to either an edge of the base or the altitude of the pyramid.

Endre wrote $n$ (not necessarily distinct) integers on a paper. Then for each of the $2^n$ subsets, Kelemen wrote their sum on the blackboard. a) For which values of $n$ is it possible that two different $n$-tuples give the same numbers on the blackboard? b) Prove that if Endre only wrote positive integers on the paper and Ferenc only sees the numbers on the blackboard, then he can determine which integers are on the paper.

Let $S(n)$ denote the number of sequences of length $n$ formed by the three letters $A,B,C$ with the restriction that the $C$'s (if any) all occur in a single block immediately following the first $B$ (if any). For example $ABCCAA$, $AAABAA$, and $ABCCCC$ are counted in, but $ACACCB$ and $CAAAAA$ are not. Derive a simple formula for $S(n)$ and use it to calculate $S(10)$.

$n$ points were chosen on a circle. Two players are playing the following game: On every move a point is chosen and it is connected with an edge to an adjacent point or with the center of the circle. The winner is the player, after whose move each point can be reached by any other (including the center) by moving on the constructed edges. Find who of the two players has a winning strategy.

There are $13$ stones each of which weighs an integer number of grams. It is known that any $12$ of them can be put on two pans of a balance scale, six on each pan, so that they are in equilibrium (i.e., each pan will carry an equal total weight). Prove that either all stones weigh an even number of grams or all stones weigh an odd number of grams.

[b]p1.[/b] The clock is $2$ hours $20$ minutes ahead of the correct time each week. The clock is set to the correct time at midnight Sunday to Monday. What time does this clock show at 6pm correct time on Thursday? [b]p2.[/b] Five cities $A,B,C,D$, and $E$ are located along the straight road in the alphabetical order. The sum of distances from $B$ to $A,C,D$ and $E$ is $20$ miles. The sum of distances from $C$ to the other four cities is $18$ miles. Find the distance between $B$ and $C$. [b]p3.[/b] Does there exist distinct digits $a, b, c$, and $d$ such that $\overline{abc}+\overline{c} = \overline{bda}$? Here $\overline{abc}$ means the three digit number with digits $a, b$, and $c$. [b]p4.[/b] Kuzya, Fyokla, Dunya, and Senya participated in a mathematical competition. Kuzya solved $8$ problems, more than anybody else. Senya solved $5$ problem, less than anybody else. Each problem was solved by exactly $3$ participants. How many problems were there? [b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] What is $2 \cdot 24 + 20 \cdot 24 + 202 \cdot 4 + 2024$? [b]p2.[/b] Jerry has $300$ legos. Tie can either make cars, which require $17$ legos, or bikes, which require $13$ legos. Assuming he uses all of his legos, how many ordered pairs $(a, b)$ are there such that he makes $a$ cars and $b$ bikes? [b]p3.[/b] Patrick has $7$ unique textbooks: $2$ Geometry books, $3$ Precalculus books and $2$ Algebra II books. How many ways can he arrange his books on a bookshelf such that all the books of the same subjects are adjacent to each other? [b]p4.[/b] After a hurricane, a $32$ meter tall flagpole at the Act on-Boxborough Regional High School snapped and fell over. Given that the snapped part remains in contact with the original pole, and the top of the polo falls $24$ meters away from the bottom of the pole, at which height did the polo snap? (Assume the flagpole is perpendicular to the ground.) [b]p5.[/b] Jimmy is selling lemonade. Iio has $200$ cups of lemonade, and he will sell them all by the end of the day. Being the ethically dubious individual he is, Jimmy intends to dilute a few of the cups of lemonade with water to conserve resources. Jimmy sells each cup for $\$4$. It costs him $\$ 1$ to make a diluted cup of lemonade, and it costs him $\$2.75$ to make a cup of normal lemonade. What is the minimum number of diluted cups Jimmy must sell to make a profit of over $\$400$? [b]p6.[/b] Jeffrey has a bag filled with five fair dice: one with $4$ sides, one with $6$ sides, one with $8$ sides, one with $12$ sides, and one with $20$ sides. The dice are numbered from $1$ to the number of sides on the die. Now, Marco will randomly pick a die from .Jeffrey's bag and roll it. The probability that Marco rolls a $7$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$. [b]p7.[/b] What is the remainder when the sum of the first $2024$ odd numbers is divided by $6072$? [b]p8.[/b] A rhombus $ABCD$ with $\angle A = 60^o$ and $AB = 600$ cm is drawn on a piece of paper. Three ants start moving from point $A$ to the three other points on the rhombus. One ant walks from $A$ to $B$ at a leisurely speed of $10$ cm/s. The second ant runs from $A$ to $C$ at a slightly quicker pace of $6\sqrt3$ cm/s, arriving to $C$ $x$ seconds after the first ant. The third ant travels from $A$ to $B$ to $D$ at a constant speed, arriving at $D$ $x$ seconds after the second ant. The speed of the last ant can be written as $\frac{m}{n}$ cm/s, where $m$ and $n$ are relatively prime positive integers. Find $mn$. [b]p9.[/b] This year, the Apple family has harvested so many apples that they cannot sell them all! Applejack decides to make $40$ glasses of apple cider to give to her friends. If Twilight and Fluttershy each want $1$ or $2$ glasses; Pinkie Pic wants cither $2$, $14$, or $15$ glasses; Rarity wants an amount of glasses that is a power of three; and Rainbow Dash wants any odd number of glasses, then how many ways can Applejack give her apple cider to her friends? Note: $1$ is considered to be a power of $3$. [b]p10.[/b] Let $g_x$ be a geometric sequence with first term $27$ and successive ratio $2n$ (so $g_{x+1}/g_x = 2n$). Then, define a function $f$ as $f(x) = \log_n(g_x)$, where $n$ is the base of the logarithm. It is known that the sum of the first seven terms of $f(x)$ is $42$. Find $g_2$, the second term of the geometric sequence. Note: The logarithm base $b$ of $x$, denoted $\log_b(x)$ is equal to the value $y$ such that $b^y = x$. In other words, if $\log_b(x) = y$, then $b^y = x$. [b]p11.[/b] Let $\varepsilon$ be an ellipse centered around the origin, such that its minor axis is perpendicular to the $x$-axis. The length of the ellipse's major and minor axes is $8$ and $6$, respectively. Then, let $ABCD$ be a rectangle centered around the origin, such that $AB$ is parallel to the $x$-axis. The lengths of $AB$ and $BC$ are $8$ and $3\sqrt2$, respectively. The area outside the ellipse but inside the rectangle can be expressed as $a\sqrt{b}-c-d\pi$, for positive integers $a$, $b$, $c$, $d$ where $b$ is not divisible by a perfect square of any prime. Find $a + b + c + d$. [img]https://cdn.artofproblemsolving.com/attachments/e/c/9d943966763ee7830d037ef98c21139cf6f529.png[/img] [b]p12.[/b] Let $N = 2^7 \cdot 3^7 \cdot 5^5$. Find the number of ways to express $N$ as the product of squares and cubes, all of which are integers greater than $1$. [b]p13.[/b] Jerry and Eric are playing a $10$-card game where Jerry is deemed the ’’landlord" and Eric is deemed the ' peasant'’. To deal the cards, the landlord keeps one card to himself. Then, the rest of the $9$ cards are dealt out, such that each card has a $1/2$ chance to go to each player. Once all $10$ cards are dealt out, the landlord compares the number of cards he owns with his peasant. The probability that the landlord wins is the fraction of cards he has. (For example, if Jerry has $5$ cards and Eric has $2$ cards, Jerry has a$ 5/7$ ths chance of winning.) The probability that Jerry wins the game can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime. Find $p + q$. [b]p14.[/b] Define $P(x) = 20x^4 + 24x^3 + 10x^2 + 21x+ 7$ to have roots $a$, $b$, $c$, and $d$. If $Q(x)$ has roots $\frac{1}{a-2}$,$\frac{1}{b-2}$,$ \frac{1}{c-2}$, $\frac{1}{d-2}$ and integer coefficients with a greatest common divisor of $1$, then find $Q(2)$. [b]p15.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 14$, $BC = 13$, and $AC = 15$. The incircle of $\vartriangle ABC$ is drawn with center $I$, tangent to $\overline{AB}$ at $X$. The line $\overleftrightarrow{IX}$ intersects the incircle again at $Y$ and intersects $\overline{AC}$ at $Z$. The area of $\vartriangle AYZ$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are two boxes containing balls. One of them contains $m$ balls, and the other contains $n$ balls, where $m, n > 0$. Two actions are permitted: (i) Remove an equal number of balls from both boxes. (ii) Increase the number of balls in one of the boxes by a factor $k$. Is it possible to remove all of the balls from both boxes with just these two actions, 1. if $k = 2$? 2. if $k = 3$?

Let $p$ be a given prime number. For positive integers $n,k\geq2$ let $S_1, S_2,\dots, S_n$ be unit square sets constructed by choosing exactly one unit square from each of the columns from $p\times k$ chess board. If $|S_i \cap S_j|=1$ for all $1\leq i < j \leq n$ and for any duo of unit squares which are located at different columns there exists $S_i$ such that both of these unit squares are in $S_i$ find all duos of $(n,k)$ in terms of $p$. Note: Here we denote the number of rows by $p$ and the number of columns by $k$.

An $ (n, k) \minus{}$ tournament is a contest with $ n$ players held in $ k$ rounds such that: $ (i)$ Each player plays in each round, and every two players meet at most once. $ (ii)$ If player $ A$ meets player $ B$ in round $ i$, player $ C$ meets player $ D$ in round $ i$, and player $ A$ meets player $ C$ in round $ j$, then player $ B$ meets player $ D$ in round $ j$. Determine all pairs $ (n, k)$ for which there exists an $ (n, k) \minus{}$ tournament. [i]Proposed by Carlos di Fiore, Argentina[/i]

[i]Round 5[/i] [b]p13.[/b] Kelvin Amphibian, a not-frog who lives on the coordinate plane, likes jumping around. Each step, he jumps either to the spot that is $1$ unit to the right and 2 units up, or the spot that is $2$ units to the right and $1$ unit up, from his current location. He chooses randomly among these two choices with equal probability. He starts at the origin and jumps for a long time. What is the probability that he lands on $(10, 8)$ at some time in his journey? [b]p14.[/b] Points $A, B, C$, and $D$ are randomly chosen on the circumference of a unit circle. What is the probability that line segments $AB$ and $CD$ intersect inside the circle? [b]p15.[/b] Let $P(x)$ be a quadratic polynomial with two consecutive integer roots. If it is also known that $\frac{P(2017)} {P(2016)} = \frac{2016}{2017}$ , find the larger root of $P(x)$. [u]Round 6[/u] [b]p16.[/b] Let $S_n$ be the sum of reciprocals of the integers between $1$ and $n$ inclusive. Find a triple $(a, b, c)$ of positive integers such that $S_{2017} \cdot S_{2017} - S_{2016} \cdot S_{2018} = \frac{S_a+S_b}{c}$ . [b]p17.[/b] Suppose that $m$ and $n$ are both positive integers. Alec has $m$ standard $6$-sided dice, each labelled $1$ to $6$ inclusive on the sides, while James has $n$ standard $12$-sided dice, each labelled $1$ to $12$ inclusive on the sides. They decide to play a game with their dice. They each toss all their dice simultaneously and then compute the sum of the numbers that come up on their dice. Whoever has a higher sum wins (if the sums are equal, they tie). Given that both players have an equal chance of winning, determine the minimum possible value of mn. [b]p18.[/b] Overlapping rectangles $ABCD$ and $BEDF$ are congruent to each other and both have area $1$. Given that $A,C,E, F$ are the vertices of a square, find the area of the square. [u]Round 7[/u] [b]p19.[/b] Find the number of solutions to the equation $$||| ... |||||x| + 1| - 2| + 3| - 4| +... - 98| + 99| - 100| = 0$$ [b]p20.[/b] A split of a positive integer in base $10$ is the separation of the integer into two nonnegative integers, allowing leading zeroes. For example, $2017$ can be split into $2$ and $017$ (or $17$), $20$ and $17$, or $201$ and $7$. A split is called squarish if both integers are nonzero perfect squares. $49$ and $169$ are the two smallest perfect squares that have a squarish split ($4$ and $9$, $16$ and $9$ respectively). Determine all other perfect squares less than $2017$ with at least one squarish split. [b]p21.[/b] Polynomial $f(x) = 2x^3 + 7x^2 - 3x + 5$ has zeroes $a, b$ and $c$. Cubic polynomial $g(x)$ with $x^3$-coefficient $1$ has zeroes $a^2$, $b^2$ and $c2$. Find the sum of coefficients of $g(x)$. [u]Round 8[/u] [b]p22.[/b] Two congruent circles, $\omega_1$ and $\omega_2$, intersect at points $A$ and $B$. The centers of $\omega_1$ and $\omega_2$ are $O_1$ and $O_2$ respectively. The arc $AB$ of $\omega_1$ that lies inside $\omega_2$ is trisected by points $P$ and $Q$, with the points lying in the order $A, P, Q,B$. Similarly, the arc $AB$ of $\omega_2$ that lies inside $\omega_1$ is trisected by points $R$ and $S$, with the points lying in the order $A,R, S,B$. Given that $PQ = 1$ and $PR =\sqrt2$, find the measure of $\angle AO_1B$ in degrees. [b]p23.[/b] How many ordered triples of $(a, b, c)$ of integers between $-10$ and $10$ inclusive satisfy the equation $-abc = (a + b)(b + c)(c + a)$? [b]p24.[/b] For positive integers $n$ and $b$ where $b > 1$, define $s_b(n)$ as the sum of digits in the base-$b$ representation of $n$. A positive integer $p$ is said to dominate another positive integer $q$ if for all positive integers $n$, $s_p(n)$ is greater than or equal to $s_q(n)$. Find the number of ordered pairs $(p, q)$ of distinct positive integers between $2$ and $100$ inclusive such that $p$ dominates $q$. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2936487p26278546]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In each square of a chessboard with $a$ rows and $b$ columns, a $0$ or $1$ is written satisfying the following conditions. [list][*]If a row and a column intersect in a square with a $0$, then that row and column have the same number of $0$s. [*]If a row and a column intersect in a square with a $1$, then that row and column have the same number of $1$s.[/list] Find all pairs $(a,b)$ for which this is possible.

Let \(n\) be a positive integer. The kingdom of Zoomtopia is a convex polygon with integer sides, perimeter \(6n\), and \(60^\circ\) rotational symmetry (that is, there is a point \(O\) such that a \(60^\circ\) rotation about \(O\) maps the polygon to itself). In light of the pandemic, the government of Zoomtopia would like to relocate its \(3n^2+3n+1\) citizens at \(3n^2+3n+1\) points in the kingdom so that every two citizens have a distance of at least \(1\) for proper social distancing. Prove that this is possible. (The kingdom is assumed to contain its boundary.) [i]Proposed by Ankan Bhattacharya, USA[/i]

Dilhan has objects of $3$ types, $A$, $B$, and $C$, and $6$ functions $$f_{A,B},f_{A,C},f_{B,A},f_{B,C},f_{C,A},f_{C,B}$$where $f_{X,Y}$ takes in an object of type $X$ and outputs an object of type $Y$. Dilhan wants to compose his $6$ functions, without repeats, such that the resulting expression is well-typed, meaning an object can be taken in by the first function, and the resulting output can then be taken in by the second function, and so on. In how many orders can he compose his $6$ functions, satisfying this constraint? [i]Proposed by Adam Bertelli[/i]