We define a sequence $x_1 = \sqrt{3}, x_2 =-1, x_3 =2 - \sqrt{3},$ and for all $n \geq 4$ $$(x_n + x_{n-3})(1 - x^2_{n-1}x^2_{n-2}) = 2x_{n-1}(1 + x^2_{n-2}).$$ Suppose $m$ is the smallest positive integer for which $x_m$ is undefined. Compute $m.$

Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

Consider a sequence of real numbers defined by: $a_{n + 1} = a_n + \frac{1}{a_n}$ for $n = 0, 1, 2, ...$ Prove that, for any positive real number $a_0$, is true that $a_{1996}$ is greater than $63$.

Let $a_1 \leq a_2 \leq \cdots$ be a non-decreasing sequence of positive integers. A positive integer $n$ is called [i]good[/i] if there is an index $i$ such that $n=\dfrac{i}{a_i}$. Prove that if $2013$ is [i]good[/i], then so is $20$.

In the sequence $00$, $01$, $02$, $03$, $\cdots$, $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$, $39$, or $28$, but not by $30$ or $20$). What is the maximal number of terms that could remain on their places?

Let $a_0, a_1, a_2, \ldots$ be an arbitrary infinite sequence of positive numbers. Show that the inequality $1 + a_n > a_{n-1} \sqrt[n]{2}$ holds for infinitely many positive integers $n$.

Does there exist a piecewise linear continuous function $f:\mathbb{R}\to \mathbb{R}$ such that for any two-way infinite sequence $a_n\in[0,1]$, $n\in\mathbb{Z}$, there exists an $x\in\mathbb{R}$ with \[ \limsup_{K\to \infty} \frac{\#\{k\le K\,:\, k\in\mathbb{N},f^k(x)\in[n,n+1)\}}{K}=a_n \] for all $n\in\mathbb{Z}$, where $f^k=f\circ f\circ \dots\circ f$ stands for the $k$-fold iterate of $f$?

Show an infinite sequence $a_1, a_2, \ldots$ of integers with both of the following properties: • $a_i \neq 0$ for every positive integer $i$, that is, no term in the sequence is equal to zero; • for all positive integer $n$, $a_n + a_{2n} + \ldots + a_{2023n} = 0$.

Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$. (1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded? (2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$? Justify your answer.

Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions : i) $a_0=1$, $a_1=3$ ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$ to be true that $$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$.

Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence. Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths. [asy] size(4cm); draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin); draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin); draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin); draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin); draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin); draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin); draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin); [/asy] [i]Proposed by Zixiang Zhou, Canada[/i]

Let $a, b, c, p, q, r > 0$ such that $(a,b,c)$ is a geometric progression and $(p, q, r)$ is an arithmetic progression. If \[a^p b^q c^r = 6 \quad \text{and} \quad a^q b^r c^p = 29\] then compute $\lfloor a^r b^p c^q \rfloor$. [i]Proposed by Michael Tang[/i]

Find infinitely many triples $(a, b, c)$ of positive integers such that $a$, $b$, $c$ are in arithmetic progression and such that $ab+1$, $bc+1$, and $ca+1$ are perfect squares.

[u]Round 1[/u] [b]p1.[/b] Define the operation $\clubsuit$ so that $a \,\clubsuit \, b = a^b + b^a$. Then, if $2 \,\clubsuit \,b = 32$, what is $b$? [b]p2. [/b] A square is changed into a rectangle by increasing two of its sides by $p\%$ and decreasing the two other sides by $p\%$. The area is then reduced by $1\%$. What is the value of $p$? [b]p3.[/b] What is the sum, in degrees, of the internal angles of a heptagon? [b]p4.[/b] How many integers in between $\sqrt{47}$ and $\sqrt{8283}$ are divisible by $7$? [u]Round 2[/u] [b]p5.[/b] Some mutant green turkeys and pink elephants are grazing in a field. Mutant green turkeys have six legs and three heads. Pink elephants have $4$ legs and $1$ head. There are $100$ legs and $37$ heads in the field. How many animals are grazing? [b]p6.[/b] Let $A = (0, 0)$, $B = (6, 8)$, $C = (20, 8)$, $D = (14, 0)$, $E = (21, -10)$, and $F = (7, -10)$. Find the area of the hexagon $ABCDEF$. [b]p7.[/b] In Moscow, three men, Oleg, Igor, and Dima, are questioned on suspicion of stealing Vladimir Putin’s blankie. It is known that each man either always tells the truth or always lies. They make the following statements: (a) Oleg: I am innocent! (b) Igor: Dima stole the blankie! (c) Dima: I am innocent! (d) Igor: I am guilty! (e) Oleg: Yes, Igor is indeed guilty! If exactly one of Oleg, Igor, and Dima is guilty of the theft, who is the thief?? [b]p8.[/b] How many $11$-letter sequences of $E$’s and $M$’s have at least as many $E$’s as $M$’s? [u]Round 3[/u] [b]p9.[/b] John is entering the following summation $31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39$ in his calculator. However, he accidently leaves out a plus sign and the answer becomes $3582$. What is the number that comes before the missing plus sign? [b]p10.[/b] Two circles of radius $6$ intersect such that they share a common chord of length $6$. The total area covered may be expressed as $a\pi + \sqrt{b}$, where $a$ and $b$ are integers. What is $a + b$? [b]p11.[/b] Alice has a rectangular room with $6$ outlets lined up on one wall and $6$ lamps lined up on the opposite wall. She has $6$ distinct power cords (red, blue, green, purple, black, yellow). If the red and green power cords cannot cross, how many ways can she plug in all six lamps? [b]p12.[/b] Tracy wants to jump through a line of $12$ tiles on the floor by either jumping onto the next block, or jumping onto the block two steps ahead. An example of a path through the $12$ tiles may be: $1$ step, $2$ steps, $2$ steps, $2$ steps, $1$ step, $2$ steps, $2$ steps. In how many ways can Tracy jump through these $12$ tiles? PS. You should use hide for answers. Last rounds have been posted [url=https://artofproblemsolving.com/community/c4h2784268p24464984]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1. [b]Note: Graph-Definition[/b]. A [b]graph[/b] consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x \equal{} v_{0},v_{1},v_{2},\cdots ,v_{m} \equal{} y\,$ such that each pair $ \,v_{i},v_{i \plus{} 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.