Prove that there exists a real $c<\frac{3}{4}$, such that for each sequence $x_1, x_2, \ldots$ satisfying $0 \leq x_i \leq 1$ for all $i$, there exist infinitely many $(m, n)$ with $m>n$, such that $$|x_m-x_n|\leq \frac{c} {m}.$$

Suppose that $(a_n)$ is a sequence of positive integers such that $\lim\limits_{n\to \infty} \dfrac{n}{a_n}=0$ Prove that there exists $k$ such that there are at least $1990$ perfect squares between $a_1 + a_2 + ... + a_k$ and $a_1 + a_2 + ... + a_{k+1}$.

Caitlin and Donal play a game called [i]Basketball Shoot-Out[/i]. The game consists of $10$ rounds. In each round, Caitlin and Donal both throw a ball simultaneously at each other's basket. If a player's ball falls into the basket, that player scores one point; otherwise, they score zero points. The scoreboard shows the complete sequence of points scored by each player in each of the $10$ rounds of the game. It turns out that Caitlin has scored at least as many points in total as Donal after every round of the game. Prove the number of possible scoreboards is divisible by $4$ but not by $8$.

Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its sides.

Let $0<k<\frac{1}{2}$ be a real number and let $a_0, b_0$ be arbitrary real numbers in $(0,1)$. The sequences $(a_n)_{n\ge 0}$ and $(b_n)_{n\ge 0}$ are then defined recursively by $$a_{n+1} = \dfrac{a_n+1}{2} \text{ and } b_{n+1} = b_n^k$$ for $n\ge 0$. Prove that $a_n<b_n$ for all sufficiently large $n$. [i]Proposed by Michael Ma

[u]Set 1[/u] [b]p1.[/b] Farmer John has $4000$ gallons of milk in a bucket. On the first day, he withdraws $10\%$ of the milk in the bucket for his cows. On each following day, he withdraws a percentage of the remaining milk that is $10\%$ more than the percentage he withdrew on the previous day. For example, he withdraws $20\%$ of the remaining milk on the second day. How much milk, in gallons, is left after the tenth day? [b]p2.[/b] Will multiplies the first four positive composite numbers to get an answer of $w$. Jeremy multiplies the first four positive prime numbers to get an answer of $j$. What is the positive difference between $w$ and $j$? [b]p3.[/b] In Nathan’s math class of $60$ students, $75\%$ of the students like dogs and $60\%$ of the students like cats. What is the positive difference between the maximum possible and minimum possible number of students who like both dogs and cats? [u]Set 2[/u] [b]p4.[/b] For how many integers $x$ is $x^4 - 1$ prime? [b]p5.[/b] Right triangle $\vartriangle ABC$ satisfies $\angle BAC = 90^o$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 60$ and $AB = 65$, find the area of $\vartriangle ABC$. [b]p6.[/b] Define $n! = n \times (n - 1) \times ... \times 1$. Given that $3! + 4! + 5! = a^2 + b^2 + c^2$ for distinct positive integers $a, b, c$, find $a + b + c$. [u]Set 3[/u] [b]p7.[/b] Max nails a unit square to the plane. Let M be the number of ways to place a regular hexagon (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Vincent, on the other hand, nails a regular unit hexagon to the plane. Let $V$ be the number of ways to place a square (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Find the nonnegative difference between $M$ and $V$ . [b]p8.[/b] Let a be the answer to this question, and suppose $a > 0$. Find $\sqrt{a +\sqrt{a +\sqrt{a +...}}}$ . [b]p9.[/b] How many ordered pairs of integers $(x, y)$ are there such that $x^2 - y^2 = 2019$? [u]Set 4[/u] [b]p10.[/b] Compute $\frac{p^3 + q^3 + r^3 - 3pqr}{p + q + r}$ where $p = 17$, $q = 7$, and $r = 8$. [b]p11.[/b] The unit squares of a $3 \times 3$ grid are colored black and white. Call a coloring good if in each of the four $2 \times 2$ squares in the $3 \times 3$ grid, there is either exactly one black square or exactly one white square. How many good colorings are there? Consider rotations and reflections of the same pattern distinct colorings. [b]p12.[/b] Define a $k$-[i]respecting [/i]string as a sequence of $k$ consecutive positive integers $a_1$, $a_2$, $...$ , $a_k$ such that $a_i$ is divisible by $i$ for each $1 \le i \le k$. For example, $7$, $8$, $9$ is a $3$-respecting string because $7$ is divisible by $1$, $8$ is divisible by $2$, and $9$ is divisible by $3$. Let $S_7$ be the set of the first terms of all $7$-respecting strings. Find the sum of the three smallest elements in $S_7$. [u]Set 5[/u] [b]p13.[/b] A triangle and a quadrilateral are situated in the plane such that they have a finite number of intersection points $I$. Find the sum of all possible values of $I$. [b]p14.[/b] Mr. DoBa continuously chooses a positive integer at random such that he picks the positive integer $N$ with probability $2^{-N}$ , and he wins when he picks a multiple of 10. What is the expected number of times Mr. DoBa will pick a number in this game until he wins? [b]p15.[/b] If $a, b, c, d$ are all positive integers less than $5$, not necessarily distinct, find the number of ordered quadruples $(a, b, c, d)$ such that $a^b - c^d$ is divisible by $5$. PS. You had better use hide for answers. Last 4 sets have been posted [url=https://artofproblemsolving.com/community/c4h2777362p24370554]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The sequence $(x_n)$ is defined as follows: $$x_0=2,\, x_1=1,\, x_{n+2}=x_{n+1}+x_n$$ for every non-negative integer $n$. a. For each $n\geq 1$, prove that $x_n$ is a prime number only if $n$ is a prime number or $n$ has no odd prime divisors b. Find all non-negative pairs of integers $(m,n)$ such that $x_m|x_n$.

Given a positive real $C \geq 1$ and a sequence $a_1, a_2, a_3, \cdots$ satisfying for any positive integer $n,$ $a_n \geq 0$ and for any real $x \geq 1$, $$\left|x\lg x-\sum_{k=1}^{[x]}\left[\frac{x}{k}\right]a_k \right| \leq Cx,$$ where $[x]$ is defined as the largest integer that does not exceed $x$. Prove that for any real $y \geq 1$, $$\sum_{k=1}^{[y]}a_k < 3Cy.$$

The function $f(x)=x^2+ \sin x$ and the sequence of positive numbers $\{ a_n \}$ satisfy $a_1=1$, $f(a_n)=a_{n-1}$, where $n \geq 2$. Prove that there exists a positive integer $n$ such that $a_1+a_2+ \dots + a_n > 2020$.

Find all real numbers $a > 1$ such that there exists an integer $k \ge 1$ such that the sequence $\{x_n\}_{n\ge 1}$ formed with the first $k$ digits of the number $\lfloor a^n\rfloor$ is periodical.

A [i]permutation[/i] of the set of positive integers $[n] = \{1, 2, . . . , n\}$ is a sequence $(a_1 , a_2 , \ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P (n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P (n)$ is a multiple of $2010$.

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Find $k \in \mathbb{N}$ such that [b]a.)[/b] For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots, \left( \begin{array}{c} n\\ j + k - 1\end{array} \right)$ forms an arithmetic progression. [b]b.)[/b] There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots , \left( \begin{array}{c} n\\ j + k - 2\end{array} \right)$ forms an arithmetic progression. Find all $n$ which satisfies part [b]b.)[/b]

Use the following description of a machine to solve the first 4 problems in the round. A machine displays four digits: $0000$. There are two buttons: button $A$ moves all digits one position to the left and fills the rightmost position with $0$ (for example, it changes $1234$ to $2340$), and button $B$ adds $11$ to the current number, displaying only the last four digits if the sum is greater than $9999$ (for example, it changes $1234$ to $1245$, and changes $9998$ to $0009$). We can denote a sequence of moves by writing down the buttons pushed from left to right. A sequence of moves that outputs $2100$, for example, is $BABAA$. [b]p1[/b]. Give a sequence of $17$ or less moves so that the machine displays $2020$. [b]p2.[/b] Using the same machine, how many outputs are possible if you make at most three moves? [b]p3.[/b] Button $ B$ now adds n to the four digit display, while button $ A$ remains the same. For how many positive integers $n \le 20$ (including $11$) can every possible four-digit output be reached? [b]p4.[/b] Suppose the function of button $ A$ changes to: move all digits one position to the right and fill the leftmost position with $2$. Then, what is the minimum number of moves required for the machine to display $2020$, if it initially displays $0000$? [b]p5.[/b] In the figure below, every inscribed triangle has vertices that are on the midpoints of its circumscribed triangle’s sides. If the area of the largest triangle is $64$, what is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/6/f/fe17b6a6d0037163f0980a5a5297c1493cc5bb.png[/img] [b]p6.[/b] A bee flies $10\sqrt2$ meters in the direction $45^o$ clockwise of North (that is, in the NE direction). Then, the bee turns $135^o$ clockwise, and flies $20$ forward meters. It continues by turning $60^o$ counterclockwise, and flies forward $14$ meters. Finally, the bee turns $120^o$ clockwise and flies another $14$ meters forward before finally finding a flower to pollinate. How far is the bee from its starting location in meters? [b]p7.[/b] All the digits of a $15$-digit number are either $p$ or $c$. $p$ shows up $3$ more times than $c$ does, and the average of the digits is $c - p$. What is $p + c$? [b]p8.[/b] Let $m$ be the sum of the factors of $75$ (including $1$ and $75$ itself). What is the ones digit of $m^{75}$ ? [b]p9.[/b] John flips a coin twice. For each flip, if it lands tails, he does nothing. If it lands heads, he rolls a fair $4$-sided die with sides labeled 1 through $4$. Let $a/b$ be the probability of never rolling a $3$, in simplest terms. What is $a + b$? [b]p10.[/b] Let $\vartriangle ABC$ have coordinates $(0, 0)$, $(0, 3)$,$(18, 0)$. Find the number of integer coordinates interior (excluding the vertices and edges) of the triangle. [b]p11.[/b] What is the greatest integer $k$ such that $2^k$ divides the value $20! \times 20^{20}$? [b]p12.[/b] David has $n$ pennies, where $n$ is a natural number. One apple costs $3$ pennies, one banana costs $5$ pennies, and one cranberry costs $7$ pennies. If David spends all his money on apples, he will have $2$ pennies left; if David spends all his money on bananas, he will have $4$ pennies left; is David spends all his money on cranberries, he will have $6$ pennies left. What is the second least possible amount of pennies that David can have? [b]p13.[/b] Elvin is currently at Hopperville which is $40$ miles from Waltimore and $50$ miles from Boshington DC. He takes a taxi back to Waltimore, but unfortunately the taxi gets lost. Elvin now finds himself at Kinsville, but he notices that he is still $40$ miles from Waltimore and $50$ miles from Boshington $DC$. If Waltimore and Boshington DC are $30$ miles apart, What is the maximum possible distance between Hopperville and Kinsville? [b]p14.[/b] After dinner, Rick asks his father for $1000$ scoops of ice cream as dessert. Rick’s father responds, “I will give you $2$ scoops of ice cream, plus $ 1$ additional scoop for every ordered pair $(a, b)$ of real numbers satisfying $\frac{1}{a + b}= \frac{1}{a}+ \frac{1}{b}$ you can find.” If Rick finds every solution to the equation, how many scoops of ice cream will he receive? [b]p15.[/b] Esther decides to hold a rock-paper-scissors tournament for the $56$ students at her school. As a rule, competitors must lose twice before they are eliminated. Each round, all remaining competitors are matched together in best-of-1 rock-paper-scissors duels. If there is an odd number of competitors in a round, one random competitor will not compete that round. What is the maximum number of matches needed to determine the rock-paper-scissors champion? [b]p16.[/b] $ABCD$ is a rectangle. $X$ is a point on $\overline{AD}$, $Y$ is a point on $\overline{AB}$, and $N$ is a point outside $ABCD$ such that $XYNC$ is also a rectangle and $YN$ intersects $\overline{BC}$ at its midpoint $M$. $ \angle BYM = 45^o$. If $MN = 5$, what is the sum of the areas of $ABCD$ and $XYNC$? [b]p17. [/b] Mr. Brown has $10$ identical chocolate donuts and $15$ identical glazed donuts. He knows that Amar wants $6$ donuts, Benny wants $9$ donuts, and Callie wants $9$ donuts. How many ways can he distribute out his $25$ donuts? [b]p18.[/b] When Eric gets on the bus home, he notices his $ 12$-hour watch reads $03: 30$, but it isn’t working as expected. The second hand makes a full rotation in $4$ seconds, then makes another in $8$ seconds, then another in $ 12$ seconds, and so on until it makes a full rotation in $60$ seconds. Then it repeats this process, and again makes a full rotation in $4$ second, then $8$ seconds, etc. Meanwhile, the minute hand and hour hand continue to function as if every full rotation of the second hand represents $60$ seconds. When Eric gets off the bus $75$ minutes later, his watch reads $AB: CD$. What is $A + B + C + D$? [b]p19.[/b] Alex and Betty want to meet each other at the airport. Alex will arrive at the airport between $12: 00$ and $13: 15$, and will wait for Betty for $15$ minutes before he leaves. Betty will arrive at the airport between $12: 30$ and $13: 10$, and will wait for Alex for $10$ minutes before she leaves. The chance that they arrive at any time in their respective time intervals is equally likely. The probability that they will meet at the airport can be expressed as $a/b$ where $a/b$ is a fraction written in simplest form. What is $a + b$? [b]p20.[/b] Let there be $\vartriangle ABC$ such that $A = (0, 0)$, $B = (23, 0)$, $C = (a, b)$. Furthermore, $D$, the center of the circle that circumscribes $\vartriangle ABC$, lies on $\overline{AB}$. Let $\angle CDB = 150^o$. If the area of $\vartriangle ABC$ is $m/n$ where $m, n$ are in simplest integer form, find the value of $m \,\, \mod \,\,n$ (The remainder of $m$ divided by $n$). PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $P$ is the product of $n$ quantities in Geometric Progression, $S$ their sum, and $S'$ the sum of their reciprocals, then $P$ in terms of $S$, $S'$, and $n$ is $\textbf{(A) }(SS')^{\frac{1}{2}n}\qquad\textbf{(B) }(S/S')^{\frac{1}{2}n}\qquad\textbf{(C) }(SS')^{n-2}\qquad\textbf{(D) }(S/S')^n\qquad \textbf{(E) }(S/S')^{\frac{1}{2}(n-1)}$