Let $a_1,a_2,a_3,\dots$ be an infinite sequence of non-zero reals satisfying \[a_{i} = \frac{a_{i-1}a_{i-2}-2}{a_{i-3}}\]for all $i\geq 4$. Determine all positive integers $n$ such that if $a_1,a_2,\dots,a_n$ are integers, then all elements of the sequence are integers.

For positive integer $n$, note $S_n=1^{2024}+2^{2024}+ \cdots +n^{2024}$. Prove that there exists infinitely many positive integers $n$, such that $S_n$ isn’t divisible by $1865$ but $S_{n+1}$ is divisible by $1865$

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Snowhite wrote on a piece of paper a whole number greater than $1$ and multiplied it by itself. She obtained a number, all digits of which are $1$: $n^2 = 111...111$ Does she know how to multiply? [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a bishop on an arbitrary square. Then the second player can put another bishop on a free square that is not controlled by the first bishop. Then the first player can put a new bishop on a free square that is not controlled by the bishops on the board. Then the second player can do the same, etc. A player who cannot put a new bishop on the board loses the game. Who has a winning strategy? [b]p4.[/b] Four girls Marry, Jill, Ann and Susan participated in the concert. They sang songs. Every song was performed by three girls. Mary sang $8$ songs, more then anybody. Susan sang $5$ songs less then all other girls. How many songs were performed at the concert? [b]p5.[/b] Pinocchio has a $10\times 10$ table of numbers. He took the sums of the numbers in each row and each such sum was positive. Then he took the sum of the numbers in each columns and each such sum was negative. Can you trust Pinocchio's calculations? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 9[/u] [b]p25.[/b] Let $S$ be the region bounded by the lines $y = x/2$, $y = -x/2$, and $x = 6$. Pick a random point $P = (x, y)$ in $S$ and translate it $3$ units right to $P' = (x + 3, y)$. What is the probability that $P'$ is in $S$? [b]p26.[/b] A triangle with side lengths $17$, $25$, and $28$ has a circle centered at each of its three vertices such that the three circles are mutually externally tangent to each other. What is the combined area of the circles? [b]p27.[/b] Find all ordered pairs $(x, y)$ of integers such that $x^2 - 2x + y^2 - 6y = -9$. [u]Round 10[/u] [b]p28.[/b] In how many ways can the letters in the word $SCHAFKOPF$ be arranged if the two $F$’s cannot be next to each other and the $A$ and the $O$ must be next to each other? [b]p29.[/b] Let a sequence $a_0, a_1, a_2, ...$ be defined by $a_0 = 20$, $a_1 = 11$, $a_2 = 0$, and for all integers $n \ge 3$, $$a_n + a_{n-1 }= a_{n-2} + a_{n-3}.$$ Find the sum $a_0 + a_1 + a_2 + · · · + a_{2010} + a_{2011}$. [b]p30.[/b] Find the sum of all positive integers b such that the base $b$ number $190_b$ is a perfect square. [u]Round 11[/u] [b]p31.[/b] Find all real values of x such that $\sqrt[3]{4x -1} + \sqrt[3]{4x + 1 }= \sqrt[3]{8x}$. [b]p32.[/b] Right triangle $ABC$ has a right angle at B. The angle bisector of $\angle ABC$ is drawn and extended to a point E such that $\angle ECA = \angle ACB$. Let $F$ be the foot of the perpendicular from $E$ to ray $\overrightarrow{BC}$. Given that $AB = 4$, $BC = 2$, and $EF = 8$, find the area of triangle $ACE$. [b]p33.[/b] You are the soul in the southwest corner of a four by four grid of distinct souls in the Fields of Asphodel. You move one square east and at the same time all the other souls move one square north, south, east, or west so that each square is now reoccupied and no two souls switched places directly. How many end results are possible from this move? [u]Round 12[/u] [b]p34.[/b] A [i]Pythagorean [/i] triple is an ordered triple of positive integers $(a, b, c)$ with $a < b < c $and $a^2 + b^2 = c^2$ . A [i]primitive [/i] Pythagorean triple is a Pythagorean triple where all three numbers are relatively prime to each other. Find the number of primitive Pythagorean triples in which all three members are less than $100,000$. If $P$ is the true answer and $A$ is your team’s answer to this problem, your score will be $max \left\{15 -\frac{|A -P|}{500} , 0 \right\}$ , rounded to the nearest integer. [b]p35.[/b] According to the Enable2k North American word list, how many words in the English language contain the letters $L, M, T$ in order but not necessarily together? If $A$ is your team’s answer to this problem and $W$ is the true answer, the score you will receive is $max \left\{15 -100\left| \frac{A}{W}-1\right| , 0 \right\}$, rounded to the nearest integer. [b]p36.[/b] Write down $5$ positive integers less than or equal to $42$. For each of the numbers written, if no other teams put down that number, your team gets $3$ points. Otherwise, you get $0$ points. Any number written that does not satisfy the given requirement automatically gets $0$ points. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2952214p26434209]here[/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3133709p28395558]here[/url]. Rest Rounds soon. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?

Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

To celebrate $2019$, Faraz gets four sandwiches shaped in the digits $2$, $0$, $1$, and $9$ at lunch. However, the four digits get reordered (but not ipped or rotated) on his plate and he notices that they form a $4$-digit multiple of $7$. What is the greatest possible number that could have been formed?

Find a number $n \geq 9$ such that for any $n$ numbers, not necessarily distinct, $a_1,a_2, \ldots , a_n$, there exists 9 numbers $a_{i_1}, a_{i_2}, \ldots , a_{i_9}, (1 \leq i_1 < i_2 < \ldots < i_9 \leq n)$ and $b_i \in {4,7}, i =1, 2, \ldots , 9$ such that $b_1a_{i_1} + b_2a_{i_2} + \ldots + b_9a_{i_9}$ is a multiple of 9.

Anna, Ben and Chris sit at the round table passing and eating nuts. At first only Anna has the nuts that she divides equally between Ben and Chris, eating a leftover (if there is any). Then Ben does the same with his pile. Then Chris does the same with his pile. The process repeats itself: each of the children divides his/her pile of nuts equally between his/her neighbours eating the leftovers if there are any. Initially, the number of nuts is large enough (more than 3). Prove that a) at least one nut is eaten; [i](3 points)[/i] b) all nuts cannot be eaten. [i](3 points)[/i]

Prove that there exists a positive integer $K$ that satisfies the following condition. Condition: For any prime $p > K$, the number of positive integers $a \le p$ that $p^2 \mid a^{p-1} - 1$ is less than $\frac{p}{2^{2024}}$

Suppose that positive integers $m,n,k$ satisfy the equations $$m^2+1=2n^2, 2m^2+1=11k^2.$$ Find the residue when $n$ is divided by $17$.

There are many $a\times b$ rectangular cardboard pieces ($a,b\in\mathbb{N}$ such that $a<b$). It is given that by putting such pieces together without overlapping one can make $49\times 51$ rectangle, and $99\times 101$ rectangle. Can one uniquely determine $a,b$ from this?

$n_{1},\ldots ,n_{k}, a$ are integers that satisfies the above conditions A)For every $i\neq j$, $(n_{i}, n_{j})=1$ B)For every $i, a^{n_{i}}\equiv 1 (mod n_{i})$ C)For every $i, X^{a-1}\equiv 0(mod n_{i})$. Prove that $a^{x}\equiv 1(mod x)$ congruence has at least $2^{k+1}-2$ solutions. ($x>1$)

Find all positive integers $n$ such that $6(n^4-1)$ is a square of an integer.

Let $p$ be a prime and $n$ a positive integer below $100$. What’s the probability that $p$ divides $n$?

Find all positive integers $n$ for which there exist distinct positive integers $a_1,a_2,\ldots,a_n$ such that \[ \frac 1{a_1} + \frac 2{a_2} + \cdots + \frac n { a_n} = \frac { a_1 + a_2 + \cdots + a_n } n. \]

Does there exist a quadratic trinomial $p(x)$ with integer coefficients such that, for every natural number $n$ whose decimal representation consists of digits $1$, $p(n)$ also consists only of digits $1$?

Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?

Let $p$ be a prime number, and let $0 \le a_1<a_2<...<a_m<p$ and $0 \le b_1<b_2<...<b_n<p$ be arbitrary integers. Denote by $k$ the number of different remainders of $a_i+b_j$, $1 \le i \le m$ and $1 \le j \le n$, modulo $p$. Prove that (i) if $m+n>p$, then $k=p$ (ii) if $m+n \le p$, then $k \ge m+n-1$

[u]Round 5 [/u] [b]p13.[/b] How many ways can we form a group with an odd number of members (plural) from $99$ people? Express your answer in the form $a^b + c$, where $a, b$, and $c$ are integers and $a$ is prime. [b]p14.[/b] A cube is inscibed in a right circular cone such that the ratio of the height of the cone to the radius is $2:1$. Compute the fraction of the cone’s volume that the cube occupies. [b]p15.[/b] Let $F_0 = 1$, $F_1 = 1$ and $F_k = F_{k-1} + F_{k-2}$. Let $P(x) = \sum^{99}_{k=0} x^{F_k}$ . The remainder when $P(x)$ is divided by $x^3 - 1$ can be expressed as $ax^2 + bx + c$. Find $2a + b$. [u]Round 6 [/u] [b]p16.[/b] Ankit finds a quite peculiar deck of cards in that each card has n distinct symbols on it and any two cards chosen from the deck will have exactly one symbol in common. The cards are guaranteed to not have a certain symbol which is held in common with all the cards. Ankit decides to create a function f(n) which describes the maximum possible number of cards in a set given the previous constraints. What is the value of $f(10)$? [b]p17.[/b] If $|x| <\frac14$ and $$X = \sum^{\infty}_{N=0} \sum^{N}_{n=0} {N \choose n}x^{2n}(2x)^{N-n}.$$ then write $X$ in terms of $x$ without any summation or product symbols (and without an infinite number of ‘$+$’s, etc.). [b]p18.[/b] Dietrich is playing a game where he is given three numbers $a, b, c$ which range from $[0, 3]$ in a continuous uniform distribution. Dietrich wins the game if the maximum distance between any two numbers is no more than $1$. What is the probability Dietrich wins the game? [u]Round 7 [/u] [b]p19.[/b] Consider f defined by $$f(x) = x^6 + a_1x^5 + a_2x^4 + a_3x^3 + a_4x^2 + a_5x + a_6.$$ How many tuples of positive integers $(a_1, a_2, a_3, a_4, a_5, a_6)$ exist such that $f(-1) = 12$ and $f(1) = 30$? [b]p20.[/b] Let $a_n$ be the number of permutations of the numbers $S = \{1, 2, ... , n\}$ such that for all $k$ with $1 \le k \le n$, the sum of $k$ and the number in the $k$th position of the permutation is a power of $2$. Compute $a_1 + a_2 + a_4 + a_8 + ... + a_{1048576}$. [b]p21.[/b] A $4$-dimensional hypercube of edge length $1$ is constructed in $4$-space with its edges parallel to the coordinate axes and one vertex at the origin. Its coordinates are given by all possible permutations of $(0, 0, 0, 0)$,$(1, 0, 0, 0)$,$(1, 1, 0, 0)$,$(1, 1, 1, 0)$, and $(1, 1, 1, 1)$. The $3$-dimensional hyperplane given by $x+y+z+w = 2$ intersects the hypercube at $6$ of its vertices. Compute the 3-dimensional volume of the solid formed by the intersection. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2781335p24424563]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does there exist an infinite set $ S$ such that for every $ a, b \in S$ we have $ a^2 \plus{} b^2 \minus{} ab \mid (ab)^2$.

Fred chooses a positive two-digit number with distinct nonzero digits. Laura takes Fred’s number and swaps its digits. She notices that the sum of her number and Fred’s number is a perfect square and the positive difference between them is a perfect cube. Find the greater of the two numbers.

[b]p1.[/b] An animal farm has geese and pigs with a total of $30$ heads and $84$ legs. Find the number of pigs and geese on this farm. [b]p2.[/b] What is the maximum number of $1 \times 1$ squares of a $7 \times 7$ board that can be colored black in such a way that the black squares don’t touch each other even at their corners? Show your answer on the figure below and explain why it is not possible to get more black squares satisfying the given conditions. [img]https://cdn.artofproblemsolving.com/attachments/d/5/2a0528428f4a5811565b94061486699df0577c.png[/img] [b]p3.[/b] Decide whether it is possible to divide a regular hexagon into three equal not necessarily regular hexagons? A regular hexagon is a hexagon with equal sides and equal angles. [img]https://cdn.artofproblemsolving.com/attachments/3/7/5d941b599a90e13a2e8ada635e1f1f3f234703.png[/img] [b]p4.[/b] A rectangle is subdivided into a number of smaller rectangles. One observes that perimeters of all smaller rectangles are whole numbers. Is it possible that the perimeter of the original rectangle is not a whole number? [b]p5.[/b] Place parentheses on the left hand side of the following equality to make it correct. $$ 4 \times 12 + 18 : 6 + 3 = 50$$ [b]p6.[/b] Is it possible to cut a $16\times 9$ rectangle into two equal parts which can be assembled into a square? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine for which positive integers $n$ there exists a positive integer $A$ such that • $A$ is divisible by $2022$, • the decimal expression of $A$ contains only digits $0$ and $7$, • the decimal expression of $A$ contains [i]exactly[/i] $n$ times the digit $7$.