Prove that there exists a sequence of positive integers $a_1,a_2,a_3, ...$ such that $a_1^2+a_2^2+...+a_n^2$ is a perfect square for all positive integers $n$.

Each term of an infinite sequence $a_1 ,a_2 ,a_3 , \dots$ is equal to 0 or 1. For each positive integer $n$, $$a_n + a_{n+1} \neq a_{n+2} + a_{n+3},\, \text{and}$$ $$a_n + a_{n+1} + a_{n+2} \neq a_{n+3} + a_{n+4} + a_{n+5}.$$ Prove that if $a_1 = 0$, then $a_{2020} = 1$.

Let $\{a_k\}_{k\geq 1}$ be a sequence of non-negative integers, such that $a_k \geq a_{2k} + a_{2k+1}$, for all $k\geq 1$. a) Prove that for all positive integers $n\geq 1$ there exist $n$ consecutive terms equal with 0 in the sequence $\{a_k\}_k$; b) State an example of sequence with the property in the hypothesis which contains an infinite number of non-zero terms.

Let $F_1, F_2, F_3 \cdots$ be the Fibonacci sequence, the sequence of positive integers satisfying $$F_1 =F_2=1$$ and $$F_{n+2} = F_{n+1} + F_n$$ for all $n \ge 1$. Does there exist an $n \ge 1$ such that $F_n$ is divisible by $2014$? Prove your answer.

The sequence of positive integers $a_1,a_2,a_3,\dots$ is [i]brazilian[/i] if $a_1=1$ and $a_n$ is the least integer greater than $a_{n-1}$ and $a_n$ is [b]coprime[/b] with at least half elements of the set $\{a_1,a_2,\dots, a_{n-1}\}$. Is there any odd number which does [b]not[/b] belong to the brazilian sequence?

[u]Round 1[/u] [b]p1.[/b] Alex is writing a sequence of $A$’s and $B$’s on a chalkboard. Any $20$ consecutive letters must have an equal number of $A$’s and $B$’s, but any 22 consecutive letters must have a different number of $A$’s and $B$’s. What is the length of the longest sequence Alex can write?. [b]p2.[/b] A positive number is placed on each of the $10$ circles in this picture. It turns out that for each of the nine little equilateral triangles, the number on one of its corners is the sum of the numbers on the other two corners. Is it possible that all $10$ numbers are different? [img]https://cdn.artofproblemsolving.com/attachments/b/f/c501362211d1c2a577e718d2b1ed1f1eb77af1.png[/img] [b]p3.[/b] Pablo and Nina take turns entering integers into the cells of a $3 \times 3$ table. Pablo goes first. The person who fills the last empty cell in a row must make the numbers in that row add to $0$. Can Nina ensure at least two of the columns have a negative sum, no matter what Pablo does? [b]p4. [/b]All possible simplified fractions greater than $0$ and less than $1$ with denominators less than or equal to $100$ are written in a row with a space before each number (including the first). Zeke and Qing play a game, taking turns choosing a blank space and writing a “$+$” or “$-$” sign in it. Zeke goes first. After all the spaces have been filled, Zeke wins if the value of the resulting expression is an integer. Can Zeke win no matter what Qing does? [img]https://cdn.artofproblemsolving.com/attachments/3/6/15484835686fbc2aa092e8afc6f11cd1d1fb88.png[/img] [b]p5.[/b] A police officer patrols a town whose map is shown. The officer must walk down every street segment at least once and return to the starting point, only changing direction at intersections and corners. It takes the officer one minute to walk each segment. What is the fastest the officer can complete a patrol? [img]https://cdn.artofproblemsolving.com/attachments/0/c/d827cf26c8eaabfd5b0deb92612a6e6ebffb47.png[/img] [u]Round 2[/u] [b]p6.[/b] Prove that among any $3^{2022}$ integers, it is possible to find exactly $3^{2021}$ of them whose sum is divisible by $3^{2021}$. [b]p7.[/b] Given a list of three numbers, a zap consists of picking two of the numbers and decreasing each of them by their average. For example, if the list is $(5, 7, 10)$ and you zap $5$ and $10$, whose average is $7.5$, the new list is $(-2.5, 7, 2.5)$. Is it possible to start with the list $(3, 1, 4)$ and, through some sequence of zaps, end with a list in which the sum of the three numbers is $0$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Find $20 \cdot 18 + 20 + 18 + 1$. [b]p2.[/b] Suzie’s Ice Cream has $10$ flavors of ice cream, $5$ types of cones, and $5$ toppings to choose from. An ice cream cone consists of one flavor, one cone, and one topping. How many ways are there for Sebastian to order an ice cream cone from Suzie’s? [b]p3.[/b] Let $a = 7$ and $b = 77$. Find $\frac{(2ab)^2}{(a+b)^2-(a-b)^2}$ . [b]p4.[/b] Sebastian invests $100,000$ dollars. On the first day, the value of his investment falls by $20$ percent. On the second day, it increases by $25$ percent. On the third day, it falls by $25$ percent. On the fourth day, it increases by $60$ percent. How many dollars is his investment worth by the end of the fourth day? [b]p5.[/b] Square $ABCD$ has side length $5$. Points $K,L,M,N$ are on segments $AB$,$BC$,$CD$,$DA$ respectively,such that $MC = CL = 2$ and $NA = AK = 1$. The area of trapezoid $KLMN$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p6.[/b] Suppose that $p$ and $q$ are prime numbers. If $p + q = 30$, find the sum of all possible values of $pq$. [b]p7.[/b] Tori receives a $15 - 20 - 25$ right triangle. She cuts the triangle into two pieces along the altitude to the side of length $25$. What is the difference between the areas of the two pieces? [b]p8.[/b] The factorial of a positive integer $n$, denoted $n!$, is the product of all the positive integers less than or equal to $n$. For example, $1! = 1$ and $5! = 120$. Let $m!$ and $n!$ be the smallest and largest factorial ending in exactly $3$ zeroes, respectively. Find $m + n$. [b]p9.[/b] Sam is late to class, which is located at point $B$. He begins his walk at point $A$ and is only allowed to walk on the grid lines. He wants to get to his destination quickly; how many paths are there that minimize his walking distance? [img]https://cdn.artofproblemsolving.com/attachments/a/5/764e64ac315c950367357a1a8658b08abd635b.png[/img] [b]p10.[/b] Mr. Iyer owns a set of $6$ antique marbles, where $1$ is red, $2$ are yellow, and $3$ are blue. Unfortunately, he has randomly lost two of the marbles. His granddaughter starts drawing the remaining $4$ out of a bag without replacement. She draws a yellow marble, then the red marble. Suppose that the probability that the next marble she draws is blue is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positiveintegers. What is $m + n$? [b]p11.[/b] If $a$ is a positive integer, what is the largest integer that will always be a factor of $(a^3+1)(a^3+2)(a^3+3)$? [b]p12.[/b] What is the largest prime number that is a factor of $160,401$? [b]p13.[/b] For how many integers $m$ does the equation $x^2 + mx + 2018 = 0$ have no real solutions in $x$? [b]p14.[/b] What is the largest palindrome that can be expressed as the product of two two-digit numbers? A palindrome is a positive integer that has the same value when its digits are reversed. An example of a palindrome is $7887887$. [b]p15.[/b] In circle $\omega$ inscribe quadrilateral $ADBC$ such that $AB \perp CD$. Let $E$ be the intersection of diagonals $AB$ and $CD$, and suppose that $EC = 3$, $ED = 4$, and $EB = 2$. If the radius of $\omega$ is $r$, then $r^2 =\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $m + n$. [b]p16.[/b] Suppose that $a, b, c$ are nonzero real numbers such that $2a^2 + 5b^2 + 45c^2 = 4ab + 6bc + 12ca$. Find the value of $\frac{9(a + b + c)^3}{5abc}$ . [b]p17.[/b] Call a positive integer n spicy if there exist n distinct integers $k_1, k_2, ... , k_n$ such that the following two conditions hold: $\bullet$ $|k_1| + |k_2| +... + |k_n| = n2$, $\bullet$ $k_1 + k_2 + ...+ k_n = 0$. Determine the number of spicy integers less than $10^6$. [b]p18.[/b] Consider the system of equations $$|x^2 - y^2 - 4x + 4y| = 4$$ $$|x^2 + y^2 - 4x - 4y| = 4.$$ Find the sum of all $x$ and $y$ that satisfy the system. [b]p19.[/b] Determine the number of $8$ letter sequences, consisting only of the letters $W,Q,N$, in which none of the sequences $WW$, $QQQ$, or $NNNN$ appear. For example, $WQQNNNQQ$ is a valid sequence, while $WWWQNQNQ$ is not. [b]p20.[/b] Triangle $\vartriangle ABC$ has $AB = 7$, $CA = 8$, and $BC = 9$. Let the reflections of $A,B,C$ over the orthocenter H be $A'$,$B'$,$C'$. The area of the intersection of triangles $ABC$ and $A'B'C'$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ , where $b$ is squarefree and $a$ and $c$ are relatively prime. determine $a+b+c$. (The orthocenter of a triangle is the intersection of its three altitudes.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a one-way street, an unending sequence of cars of width $a$, length $b$ passes with velocity $v$. The cars are separated by the distance $c$. A pedestrian crosses the street perpendicularly with velocity $w$, without paying attention to the cars. [b](a)[/b] What is the probability that the pedestrian crosses the street uninjured? [b](b)[/b] Can he improve this probability by crossing the road in a direction other than perpendicular?

Peter picked a positive integer, multiplied it by 5, multiplied the result by 5,then multiplied the result by 5 again and so on. Altogether $k$ multiplications were made. It so happened that the decimal representations of the original number and of all $k$ resulting numbers in this sequence do not contain digit $7$. Prove that there exists a positive integer such that it can be multiplied $k$ times by $2$ so that no number in this sequence contains digit $7$.

The numerical sequence $x_1 , x_2 ,.. $ satisfies $x_1 = \frac12$ and $x_{k+1} =x^2_k+x_k$ for all natural integers $k$ . Find the integer part of the sum $\frac{1}{x_1+1}+\frac{1}{x_2+1}+...+\frac{1}{x_{100}+1}$ {A. Andjans, Riga)

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Let $n$ be a positive integer. A sequence of $n$ positive integers (not necessarily distinct) is called [b]full[/b] if it satisfies the following condition: for each positive integer $k\geq2$, if the number $k$ appears in the sequence then so does the number $k-1$, and moreover the first occurrence of $k-1$ comes before the last occurrence of $k$. For each $n$, how many full sequences are there ?

Let $\{u_n\}_ {n\ge 1}$ be given sequence satisfying the conditions: $u_1 = 0$, $u_2 = 1$, $u_{n+1} = u_{n-1} + 2n - 1$ for $n \ge 2$. 1) Calculate $u_5$. 2) Calculate $u_{100} + u_{101}$.

Suppose that $x_1 < x_2 < \dots < x_n$ is a sequence of positive integers such that $x_k$ divides $x_{k+2}$ for each $k = 1, 2, \dots, n-2$. Given that $x_n = 1000$, what is the largest possible value of $n$? [i]Proposed by Evan Chen[/i]

[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].