Define $\text{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0.$ Compute $$\sum_{n=1}^{\infty} \frac{\text{sgn}(\sin(2^n))}{2^n}.$$ (The arguments to $\sin$ are in radians.)

a) Two positive integers are chosen. The sum is revealed to logician $A$, and the sum of squares is revealed to logician $B$. Both $A$ and $B$ are given this information and the information contained in this sentence. The conversation between $A$ and $B$ goes as follows: $B$ starts B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` Now I can tell what they are.' What are the two numbers? b) When $B$ first says that he cannot tell what the two numbers are, $A$ receives a large amount of information. But when $A$ first says that he cannot tell what the two numbers are, $B$ already knows that $A$ cannot tell what the two numbers are. What good does it do $B$ to listen to $A$?

Find all pairs $(p,n)$ so that $p$ is a prime number, $n$ is a positive integer and \[p^3-2p^2+p+1=3^n \] holds.

Jacob and Laban take turns playing a game. Each of them starts with the list of square numbers $1, 4, 9, \dots, 2021^2$, and there is a whiteboard in front of them with the number $0$ on it. Jacob chooses a number $x^2$ from his list, removes it from his list, and replaces the number $W$ on the whiteboard with $W + x^2$. Laban then does the same with a number from his list, and the repeat back and forth until both of them have no more numbers in their list. Now every time that the number on the whiteboard is divisible by $4$ after a player has taken his turn, Jacob gets a sheep. Jacob wants to have as many sheep as possible. What is the greatest number $K$ such that Jacob can guarantee to get at least $K$ sheep by the end of the game, no matter how Laban plays?

Let $n$, $i$, and $j$ be integers, for which $0<i<j<n$. Is it always true that the binomial coefficients $\binom{n}{i}$ and $\binom{n}{j}$ have a common divisor greater than 1?

A sequence $(x_n)_{n\ge 0}$ is defined as follows: $x_0=a,x_1=2$ and $x_n=2x_{n-1}x_{n-2}-x_{n-1}-x_{n-2}+1$ for all $n>1$. Find all integers $a$ such that $2x_{3n}-1$ is a perfect square for all $n\ge 1$.

Find the remainder after division of $10^{10} + 10^{10^2} + 10^{10^3} + ... + 10^{10^{10}}$ by $7$.

A triangle has sides of length $48$, $55$, and $73$. Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the length of the shortest altitude of the triangle. Find the value of $a+b$.

Determine, with proof, if there exists an odd prime $p$ such that the following equation holds: $$\sum_{n = 1}^{\frac{p-1}{2}} \cot\left(\frac{\pi n^2}{p}\right) = 69\sqrt{p}$$ [i]Proposed by Chris Bao[/i]

[b]p1.[/b] Ten math students take a test, and the average score on the test is $28$. If five students had an average of $15$, what was the average of the other five students' scores? [b]p2.[/b] If $a\otimes b = a^2 + b^2 + 2ab$, find $(-5\otimes 7) \otimes 4$. [b]p3.[/b] Below is a $3 \times 4$ grid. Fill each square with either $1$, $2$ or $3$. No two squares that share an edge can have the same number. After filling the grid, what is the $4$-digit number formed by the bottom row? [img]https://cdn.artofproblemsolving.com/attachments/9/6/7ef25fc1220d1342be66abc9485c4667db11c3.png[/img] [b]p4.[/b] What is the angle in degrees between the hour hand and the minute hand when the time is $6:30$? [b]p5.[/b] In a small town, there are some cars, tricycles, and spaceships. (Cars have $4$ wheels, tricycles have $3$ wheels, and spaceships have $6$ wheels.) Among the vehicles, there are $24$ total wheels. There are more cars than tricycles and more tricycles than spaceships. How many cars are there in the town? [b]p6.[/b] You toss five coins one after another. What is the probability that you never get two consecutive heads or two consecutive tails? [b]p7.[/b] In the below diagram, $\angle ABC$ and $\angle BCD$ are right angles. If $\overline{AB} = 9$, $\overline{BD} = 13$, and $\overline{CD} = 5$, calculate $\overline{AC}$. [img]https://cdn.artofproblemsolving.com/attachments/7/c/8869144e3ea528116e2d93e14a7896e5c62229.png[/img] [b]p8.[/b] Out of $100$ customers at a market, $80$ purchased oranges, $60$ purchased apples, and $70$ purchased bananas. What is the least possible number of customers who bought all three items? [b]p9.[/b] Francis, Ted and Fred planned to eat cookies after dinner. But one of them sneaked o earlier and ate the cookies all by himself. The three say the following: Francis: Fred ate the cookies. Fred: Ted did not eat the cookies. Ted: Francis is lying. If exactly one of them is telling the truth, who ate all the cookies? [b]p11.[/b] Let $ABC$ be a triangle with a right angle at $A$. Suppose $\overline{AB} = 6$ and $\overline{AC} = 8$. If $AD$ is the perpendicular from $A$ to $BC$, what is the length of $AD$? [b]p12.[/b] How many three digit even numbers are there with an even number of even digits? [b]p13.[/b] Three boys, Bob, Charles and Derek, and three girls, Alice, Elizabeth and Felicia are all standing in one line. Bob and Derek are each adjacent to precisely one girl, while Felicia is next to two boys. If Alice stands before Charles, who stands before Elizabeth, determine the number of possible ways they can stand in a line. [b]p14.[/b] A man $5$ foot, $10$ inches tall casts a $14$ foot shadow. $20$ feet behind the man, a flagpole casts ashadow that has a $9$ foot overlap with the man's shadow. How tall (in inches) is the flagpole? [b]p15.[/b] Alvin has a large bag of balls. He starts throwing away balls as follows: At each step, if he has $n$ balls and 3 divides $n$, then he throws away a third of the balls. If $3$ does not divide $n$ but $2$ divides $n$, then he throws away half of them. If neither $3$ nor $2$ divides $n$, he stops throwing away the balls. If he began with $1458$ balls, after how many steps does he stop throwing away balls? [b]p16.[/b] Oski has $50$ coins that total to a value of $82$ cents. You randomly steal one coin and find out that you took a quarter. As to not enrage Oski, you quickly put the coin back into the collection. However, you are both bored and curious and decide to randomly take another coin. What is the probability that this next coin is a penny? (Every coin is either a penny, nickel, dime or quarter). [b]p17.[/b] Let $ABC$ be a triangle. Let $M$ be the midpoint of $BC$. Suppose $\overline{MA} = \overline{MB} = \overline{MC} = 2$ and $\angle ACB = 30^o$. Find the area of the triangle. [b]p18.[/b] A spirited integer is a positive number representable in the form $20^n + 13k$ for some positive integer $n$ and any integer $k$. Determine how many spirited integers are less than $2013$. [b]p19. [/b]Circles of radii $20$ and $13$ are externally tangent at $T$. The common external tangent touches the circles at $A$, and $B$, respectively where $A \ne B$. The common internal tangent of the circles at $T$ intersects segment $AB$ at $X$. Find the length of $AX$. [b]p20.[/b] A finite set of distinct, nonnegative integers $\{a_1, ... , a_k\}$ is called admissible if the integer function $f(n) = (n + a_1) ... (n + a_k)$ has no common divisor over all terms; that is, $gcd \left(f(1), f(2),... f(n)\right) = 1$ for any integer$ n$. How many admissible sets only have members of value less than $10$? $\{4\}$ and $\{0, 2, 6\}$ are such sets, but $\{4, 9\}$ and $\{1, 3, 5\}$ are not. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Do there exist 2018 positive irreducible fractions, each with a different denominator, so that the denominator of the difference of any two (after reducing the fraction) is less than the denominator of any of the initial 2018 fractions? (6 points) Maxim Didin

Let \( n \) be a positive integer. A function \( f : \{0, 1, \dots, n\} \to \{0, 1, \dots, n\} \) is called \( n \)-Bolivian if it satisfies the following conditions: • \( f(0) = 0 \); • \( f(t) \in \{ t-1, f(t-1), f(f(t-1)), \dots \} \) for all \( t = 1, 2, \dots, n \). For example, if \( n = 3 \), then the function defined by \( f(0) = f(1) = 0 \), \( f(2) = f(3) = 1 \) is 3-Bolivian, but the function defined by \( f(0) = f(1) = f(2) = 0 \), \( f(3) = 1 \) is not 3-Bolivian. For a fixed positive integer \( n \), Gollum selects an \( n \)-Bolivian function. Smeagol, knowing that \( f \) is \( n \)-Bolivian, tries to figure out which function was chosen by asking questions of the type: \[ \text{How many integers } a \text{ are there such that } f(a) = b? \] given a \( b \) of his choice. Show that if Gollum always answers correctly, Smeagol can determine \( f \) and find the minimum number of questions he needs to ask, considering all possible choices of \( f \).

Prove that the set $S=\{\lfloor n\pi\rfloor \mid n=0,1,2,3,\ldots\}$ contains arithmetic progressions of any finite length, but no infinite arithmetic progressions. [i]Vasile Pop[/i]

[u]Set 1[/u] [b]B1.[/b] Evaluate $2 + 0 - 2 \times 0$. [b]B2.[/b] It takes four painters four hours to paint four houses. How many hours does it take forty painters to paint forty houses? [b]B3.[/b] Let $a$ be the answer to this question. What is $\frac{1}{2-a}$? [u]Set 2[/u] [b]B4.[/b] Every day at Andover is either sunny or rainy. If today is sunny, there is a $60\%$ chance that tomorrow is sunny and a $40\%$ chance that tomorrow is rainy. On the other hand, if today is rainy, there is a $60\%$ chance that tomorrow is rainy and a $40\%$ chance that tomorrow is sunny. Given that today is sunny, the probability that the day after tomorrow is sunny can be expressed as n%, where n is a positive integer. What is $n$? [b]B5.[/b] In the diagram below, what is the value of $\angle DD'Y$ in degrees? [img]https://cdn.artofproblemsolving.com/attachments/0/8/6c966b13c840fa1885948d0e4ad598f36bee9d.png[/img] [b]B6.[/b] Christina, Jeremy, Will, and Nathan are standing in a line. In how many ways can they be arranged such that Christina is to the left of Will and Jeremy is to the left of Nathan? Note: Christina does not have to be next to Will and Jeremy does not have to be next to Nathan. For example, arranging them as Christina, Jeremy, Will, Nathan would be valid. [u]Set 3[/u] [b]B7.[/b] Let $P$ be a point on side $AB$ of square $ABCD$ with side length $8$ such that $PA = 3$. Let $Q$ be a point on side $AD$ such that $P Q \perp P C$. The area of quadrilateral $PQDB$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]B8.[/b] Jessica and Jeffrey each pick a number uniformly at random from the set $\{1, 2, 3, 4, 5\}$ (they could pick the same number). If Jessica’s number is $x$ and Jeffrey’s number is $y$, the probability that $x^y$ has a units digit of $1$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]B9.[/b] For two points $(x_1, y_1)$ and $(x_2, y_2)$ in the plane, we define the taxicab distance between them as $|x_1 - x_2| + |y_1 - y_2|$. For example, the taxicab distance between $(-1, 2)$ and $(3,\sqrt2)$ is $6-\sqrt2$. What is the largest number of points Nathan can find in the plane such that the taxicab distance between any two of the points is the same? [u]Set 4[/u] [b]B10.[/b] Will wants to insert some × symbols between the following numbers: $$1\,\,\,2\,\,\,3\,\,\,4\,\,\,6$$ to see what kinds of answers he can get. For example, here is one way he can insert $\times$ symbols: $$1 \times 23 \times 4 \times 6 = 552.$$ Will discovers that he can obtain the number $276$. What is the sum of the numbers that he multiplied together to get $276$? [b]B11.[/b] Let $ABCD$ be a parallelogram with $AB = 5$, $BC = 3$, and $\angle BAD = 60^o$ . Let the angle bisector of $\angle ADC$ meet $AC$ at $E$ and $AB$ at $F$. The length $EF$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? [b]B12.[/b] Find the sum of all positive integers $n$ such that $\lfloor \sqrt{n^2 - 2n + 19} \rfloor = n$. Note: $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. [u]Set 5[/u] [b]B13.[/b] This year, February $29$ fell on a Saturday. What is the next year in which February $29$ will be a Saturday? [b]B14.[/b] Let $f(x) = \frac{1}{x} - 1$. Evaluate $$f\left( \frac{1}{2020}\right) \times f\left( \frac{2}{2020}\right) \times f\left( \frac{3}{2020}\right) \times \times ... \times f\left( \frac{2019}{2020}\right) .$$ [b]B15.[/b] Square $WXYZ$ is inscribed in square $ABCD$ with side length $1$ such that $W$ is on $AB$, $X$ is on $BC$, $Y$ is on $CD$, and $Z$ is on $DA$. Line $W Y$ hits $AD$ and $BC$ at points $P$ and $R$ respectively, and line $XZ$ hits $AB$ and $CD$ at points $Q$ and $S$ respectively. If the area of $WXYZ$ is $\frac{13}{18}$ , then the area of $PQRS$ can be expressed as $m/n$ for relatively prime positive integers $m$ and $n$. What is $m + n$? PS. You had better use hide for answers. Last sets have been posted [url=https://artofproblemsolving.com/community/c4h2777424p24371574]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

Find all positive intengers $x,y$, satisfying: $$3^x+x^4=y!+2019.$$

Let $S$ be a set of integers (not necessarily positive) such that (a) there exist $a,b \in S$ with $\gcd(a,b)=\gcd(a-2,b-2)=1$; (b) if $x$ and $y$ are elements of $S$ (possibly equal), then $x^2-y$ also belongs to $S$. Prove that $S$ is the set of all integers.

Given $n \ge 2$ distinct positive integers $a_1, a_2, ..., a_n$ none of which is a perfect cube. Find the maximal possible number of perfect cubes among their pairwise products.

Let $n$ be a positive integer not divisible by $2$ or $3$. Prove that for all integers $k$, the number $(k+1)^n-k^n-1$ is divisible by $k^2+k+1$.

Solve in positive integers $x^yy^x=(x+y)^z$

Determine all positive integers $n$ for which $n(n + 9)$ is a perfect square.

A circle is divided in $100$ equal parts and the points of this division are colored green or yellow, such that when between two points of division $A,B$ there are exactly $4$ division points and the point $A$ is green, then the point $B$ shall be yellow. Which points are more, the green or the yellow ones?

For all positive integers $n$, let \[f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.\] Compute $f(2019) - f(2018)$. Here $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$.

Let $p$ be a prime number and $a_1, a_2, \ldots, a_{(p+1)/2}$ different natural numbers less than or equal to $p.$ Prove that for each natural number $r$ less than or equal to $p$, there exist two numbers (perhaps equal) $a_i$ and $a_j$ such that \[p \equiv a_i a_j \pmod r.\]

Let $n \geq 2$ be natural number. Prove, that $(1^{n-1}+2^{n-1}+....+(n-1)^{n-1})+1$ divided by $n$ iff for any prime divisor $p$ of $n$ $p| \frac{n}{p}-1 $ and $(p-1)| \frac{n}{p}-1$.