Given is an arithmetic progression {$a_n$} of positive integers. Prove that there exist infinitely many $k$, such that $\omega (a_k)$ is even and $\omega (a_{k+1})$ is odd ($\omega (n)$ is the number of distinct prime factors of $n$). $\textit {Proposed by Viktor Simjanoski and Nikola Velov}$

[u]Round 1[/u] [b]p1.[/b] Alice and Bob compiled a list of movies that exactly one of them saw, then Cindy and Dale did the same. To their surprise, these two lists were identical. Prove that if Alice and Cindy list all movies that exactly one of them saw, this list will be identical to the one for Bob and Dale. [b]p2.[/b] Several whole rounds of cheese were stored in a pantry. One night some rats sneaked in and consumed $10$ of the rounds, each rat eating an equal portion. Some were satisfied, but $7$ greedy rats returned the next night to finish the remaining rounds. Their portions on the second night happened to be half as large as on the first night. How many rounds of cheese were initially in the pantry? [b]p3.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake, and call that number N. Pick up the stack of the top N pancakes, and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [b]p4.[/b] There are two lemonade stands along the $4$-mile-long circular road that surrounds Sour Lake. $100$ children live in houses along the road. Every day, each child buys a glass of lemonade from the stand that is closest to her house, as long as she does not have to walk more than one mile along the road to get there. A stand's [u]advantage [/u] is the difference between the number of glasses it sells and the number of glasses its competitor sells. The stands are positioned such that neither stand can increase its advantage by moving to a new location, if the other stand stays still. What is the maximum number of kids who can't buy lemonade (because both stands are too far away)? [b]p5.[/b] Merlin uses several spells to move around his $64$-room castle. When Merlin casts a spell in a room, he ends up in a different room of the castle. Where he ends up only depends on the room where he cast the spell and which spell he cast. The castle has the following magic property: if a sequence of spells brings Merlin from some room $A$ back to room $A$, then from any other room $B$ in the castle, that same sequence brings Merlin back to room $B$. Prove that there are two different rooms $X$ and $Y$ and a sequence of spells that both takes Merlin from $X$ to $Y$ and from $Y$ to $X$. [u]Round 2[/u] [b]p6.[/b] Captains Hook, Line, and Sinker are deciding where to hide their treasure. It is currently buried at the $X$ in the map below, near the lairs of the three pirates. Each pirate would prefer that the treasure be located as close to his own lair as possible. You are allowed to propose a new location for the treasure to the pirates. If at least two out of the three pirates prefer the new location (because it moves closer to their own lairs), then the treasure will be moved there. Assuming the pirates’ lairs form an acute triangle, is it always possible to propose a sequence of new locations so that the treasure eventually ends up in your backyard (wherever that is)? [img]https://cdn.artofproblemsolving.com/attachments/c/c/a9e65624d97dec612ef06f8b30be5540cfc362.png[/img] [b]p7.[/b] Homer went on a Donut Diet for the month of May ($31$ days). He ate at least one donut every day of the month. However, over any stretch of $7$ consecutive days, he did not eat more than $13$ donuts. Prove that there was some stretch of consecutive days over which Homer ate exactly $30$ donuts. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given the sequence $(t_n)$ defined as $t_0 = 0$, $t_1 = 6$, $t_{n + 2} = 14t_{n + 1} - t_n$. Prove that for every number $n \ge 1$, $t_n$ is the area of a triangle whose lengths are all numbers integers. Dang Hung Thang, University of Natural Sciences, Hanoi National University.

In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.

Let $ p,q,n$ be three positive integers with $ p \plus{} q < n$. Let $ (x_{0},x_{1},\cdots ,x_{n})$ be an $ (n \plus{} 1)$-tuple of integers satisfying the following conditions : (a) $ x_{0} \equal{} x_{n} \equal{} 0$, and (b) For each $ i$ with $ 1\leq i\leq n$, either $ x_{i} \minus{} x_{i \minus{} 1} \equal{} p$ or $ x_{i} \minus{} x_{i \minus{} 1} \equal{} \minus{} q$. Show that there exist indices $ i < j$ with $ (i,j)\neq (0,n)$, such that $ x_{i} \equal{} x_{j}$.

Let $ v$, $ w$, $ x$, $ y$, and $ z$ be the degree measures of the five angles of a pentagon. Suppose $ v < w < x < y < z$ and $ v$, $ w$, $ x$, $ y$, and $ z$ form an arithmetic sequence. Find the value of $ x$. $ \textbf{(A)}\ 72 \qquad \textbf{(B)}\ 84 \qquad \textbf{(C)}\ 90 \qquad \textbf{(D)}\ 108 \qquad \textbf{(E)}\ 120$

A sequence of integers $a_1$, $a_2$, $a_3$, $\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?

A [i]binary string[/i] is a sequence, each of whose terms is $0$ or $1$. A set $\mathcal{B}$ of binary strings is defined inductively according to the following rules. [list] [*]The binary string $1$ is in $\mathcal{B}$.[/*] [*]If $s_1,s_2,\dotsc ,s_n$ is in $\mathcal{B}$ with $n$ odd, then both $s_1,s_2,\dotsc ,s_n,0$ and $0,s_1,s_2,\dotsc ,s_n$ are in $\mathcal{B}$.[/*] [*]If $s_1,s_2,\dotsc ,s_n$ is in $\mathcal{B}$ with $n$ even, then both $s_1,s_2,\dotsc ,s_n,1$ and $1,s_1,s_2,\dotsc ,s_n$ are in $\mathcal{B}$.[/*] [*]No other binary strings are in $\mathcal{B}$.[/*] [/list] For each positive integer $n$, let $b_n$ be the number of binary strings in $\mathcal{B}$ of length $n$. [list=a] [*]Prove that there exist constants $c_1,c_2>0$ and $1.6<\lambda_1,\lambda_2<1.9$ such that $c_1\lambda_1^n<b_n<c_2\lambda_2^n$ for all positive integer $n$.[/*] [*]Determine $\liminf_{n\to \infty} {\sqrt[n]{b_n}}$ and $\limsup_{n\to \infty} {\sqrt[n]{b_n}}$[/*] [/list] [i]Note: The problem is open in the sense that no solution is currently known to part (b).[/i]

Let $(a_n)_{n\ge 1}$ be an increasing sequence of positive integers. Assume that there is a constant $M>0$ satisfying$$0<a_{n+1}-a_n<M.a_n^{5/8},\forall n\ge 1.$$ Prove that: there exists a real number $A$ such that for each $k\in \mathbb{Z}^+,[A^{3^k}]$ is an element of $(a_n)_{n\ge 1}.$

In the sequence $\{a_n\}_{n=0}^{\infty}$ we have $a_0=1$, $a_1=2$ and \[a_{n+1}=a_n+\dfrac{a_{n-1}}{1+a_{n-1}^2} \qquad \forall n \geq 1\] Prove that \[52 < a_{1371} < 65\]

Let be an increasing, infinite sequence of natural numbers $ \left( a_n \right)_{n\ge 1} . $ [b]a)[/b] Prove that if $ a_n=n, $ for any natural numbers $ n, $ then $$ -2+2\sqrt{1+n} <\frac{1}{\sqrt{a_1}} +\frac{1}{\sqrt{a_2}} +\cdots +\frac{1}{\sqrt{a_n}} <2\sqrt n , $$ for any natural numbers $ n. $ [b]b)[/b] Disprove the converse of [b]a).[/b] [i]Vasile Radu[/i]

$\{a_{n}\}$ is a sequence of natural numbers satisfying the following inequality for all natural number $n$: $$(a_{1}+\cdots+a_{n})\left(\frac{1}{a_{1}}+\cdots+\frac{1}{a_{n}}\right)\le{n^{2}}+2019$$ Prove that $\{a_{n}\}$ is constant.

Estimate the number of primes among the first thousand primes divide some term of the sequence \[2^0+1,2^1+1,2^2+1,2^3+1,\ldots.\] An estimate of $E$ earns $2^{1-0.02|A-E|}$ points, where $A$ is the actual answer. [i]2021 CCA Math Bonanza Lightning Round #5.4[/i]

Prove that among the elements of the sequence $\left( \left\lfloor n \sqrt 2 \right\rfloor + \left\lfloor n \sqrt 3 \right\rfloor \right)_{n \geq 0}$ are an infinity of even numbers and an infinity of odd numbers.

Let $ A_n $ be the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that every two neighbouring terms of each subsequence have different parity,and $ B_n $ the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that all the terms of each subsequence have the same parity ( for example,the partition $ {(1,4,5,8),(2,3),(6,9),(7)} $ is an element of $ A_9 $,and the partition $ {(1,3,5),(2,4),(6)} $ is an element of $ B_6 $ ). Prove that for every positive integer $ n $ the sets $ A_n $ and $ B_{n+1} $ contain the same number of elements.