Signals used for communication are binary sequences of length $10$. Unfortunately, the receiving device got broken so that it cannot distinguish between two signals unless those differ in more than five places. What is the largest possible number of signals that can still be used to prevent ambiguities?

Find all $C\in \mathbb{R}$ such that every sequence of integers $\{a_n\}_{n=1}^{\infty}$ which is bounded from below and for all $n\geq 2$ satisfy $$0\leq a_{n-1}+Ca_n+a_{n+1}<1$$ is periodic. Proposed by Navid Safaei

$a_0, a_1, \ldots, a_{100}$ and $b_1, b_2,\ldots, b_{100}$ are sequences of real numbers, for which the property holds: for all $n=0, 1, \ldots, 99$, either $$a_{n+1}=\frac{a_n}{2} \quad \text{and} \quad b_{n+1}=\frac{1}{2}-a_n,$$ or $$a_{n+1}=2a_n^2 \quad \text{and} \quad b_{n+1}=a_n.$$ Given $a_{100}\leq a_0$, what is the maximal value of $b_1+b_2+\cdots+b_{100}$?

Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow? [i]Proposed by Gerhard Woeginger, Netherlands[/i]

The numbers $2005! + 2, 2005! + 3, ... , 2005! + 2005$ form a sequence of $2004$ consequtive integers, none of which is a prime number. Does there exist a sequence of $2004$ consequtive integers containing exactly $12$ prime numbers?

A circle with area $A_1$ is contained in the interior of a larger circle with area $A_1+A_2$. If the radius of the larger circle is $3$, and if $A_1 , A_2, A_1 + A_2$ is an arithmetic progression, then the radius of the smaller circle is $\textbf{(A) }\frac{\sqrt{3}}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }\frac{2}{\sqrt{3}}\qquad\textbf{(D) }\frac{3}{2}\qquad\textbf{(E) }\sqrt{3}$

Let $S = \{a_1, \ldots, a_n \}$ be a finite set of positive integers of size $n \ge 1$, and let $T$ be the set of all positive integers that can be expressed as sums of perfect powers (including $1$) of distinct numbers in $S$, meaning \[ T = \left\{ \sum_{i=1}^n a_i^{e_i} \mid e_1, e_2, \dots, e_n \ge 0 \right\}. \] Show that there is a positive integer $N$ (only depending on $n$) such that $T$ contains no arithmetic progression of length $N$. [i]Yang Liu[/i]

Let $u,v$ be positive integers. Define sequence $\{a_n\}$ as follows: $a_1=u+v$, and for integers $m\ge 1$, \[\begin{array}{lll} \begin{cases} a_{2m}=a_m+u, \\ a_{2m+1}=a_m+v, \end{cases} \end{array}\] Let $S_m=a_1+a_2+\ldots +a_m(m=1,2,\ldots )$. Prove that there are infinitely many perfect squares in the sequence $\{S_n\}$.

Let $ r$ be a real number such that the sequence $ (a_{n})_{n\geq 1}$ of positive real numbers satisfies the equation $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{m \plus{} 1} \leq r a_{m}$ for each positive integer $ m$. Prove that $ r \geq 4$.

The sequence $(s_n)$, where $s_n= \sum_{k=1}^n \sin k$ for $n = 1, 2,\dots$ is bounded. Find an upper and lower bound.

What conditions must the real numbers $ a $, $ b $, and $ c $ satisfy so that the equation $$ x^3 + ax^2 + bx + c = 0$$ has three distinct real roots forming a geometric progression?

A sequence $x_0=A$ and $x_1=B$ and $x_{n+2}=x_{n+1} +x_n$ is called a Fibonacci type sequence. Call a number $C$ a repeated value if $x_t=x_s=c$ for $t$ different from $s$. Prove one can choose $A$ and $B$ to have as many repeated value as one likes but never infinite.

Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$, define $b_m=\min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$. If $a_{19}=85$, determine the maximum value of \[a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}.\]

Let $D$ be an infinite (in one direction) sequence of zeros and ones. For each $n\in\mathbb{N}$, let $a_n$ denote the number of distinct subsequences of consecutive symbols in $D$ of length $n$. Does there exist a sequence $D$ for which the inequality \[ \left|\frac{a_n}{n\log_2 n} - 1\right| < \frac{1}{100} \] is satisfied for every natural number $n > 10^{10000}$?

Let $\{\varepsilon_i\}_{i\ge 1}, \{\theta_i\}_{i\ge 0}$ be two infinite sequences of real numbers, such that $\varepsilon_i \in \{-1,1\}$ for all $i$, and the numbers $\theta_i$ obey$$\tan \theta_{n+1} = \tan \theta_{n}+\varepsilon_n \sec(\theta_{n})-\tan \theta_{n-1} , \qquad n \ge 1$$and $\theta_0 = \frac{\pi}{4}, \theta_1 = \frac{2\pi}{3}$. Compute the sum of all possible values of $$\lim_{m \to \infty} \left(\sum_{n=1}^m \frac{1}{\tan \theta_{n+1} + \tan \theta_{n-1}} + \tan \theta_m - \tan \theta_{m+1}\right)$$ [i]Proposed by Grant Yu[/i]

In the tribe of Zimmer, being able to hike long distances and knowing the roads through the forest are both extremely important, so a boy who reaches the age of manhood is not designated as a man by the tribe until he completes an interesting rite of passage. The man must go on a sequence of hikes. The first hike is a $5$ kilometer hike down the main road. The second hike is a $5\frac{1}{4}$ kilometer hike down a secondary road. Each hike goes down a different road and is a quarter kilometer longer than the previous hike. The rite of passage is completed at the end of the hike where the cumulative distance walked by the man on all his hikes exceeds $1000$ kilometers. So in the tribe of Zimmer, how many roads must a man walk down, before you call him a man?