Let $a_{1}, a_{2}, a_{3}, \cdots$ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$, where $i, j, $ and $k$ are not necessarily distinct. Determine $a_{1998}$.

Let $f(x)$ be a function acting on a string of $0$s and $1$s, defined to be the number of substrings of $x$ that have at least one $1$, where a substring is a contiguous sequence of characters in $x$. Let $S$ be the set of binary strings with $24$ ones and $100$ total digits. Compute the maximum possible value of $f(s)$ over all $s\in S$. For example, $f(110) = 5$ as $\underline{1}10$, $1\underline{1}0$, $\underline{11}0$, $1\underline{10}$, and $\underline{110}$ are all substrings including a $1$. Note that $11\underline{0}$ is not such a substring.

a) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $x_n-2x_{n+1}+x_{n+2} \le 0$ for any $n$ . Moreover $x_o=1,x_{20}=9,x_{200}=6$. What is the maximal value of $x_{2009}$ can be? b) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $2x_n-3x_{n+1}+x_{n+2} \le 0$ for any $n$. Moreover $x_o=1,x_1=2,x_3=1$. Can $x_{2009}$ be greater then $0,678$ ?

Dima invented a secret code in which every letter is replaced by a word no longer than $10$ letters. A code is called “good” if every encoded word can be decoded in only one way. Serjozha (with the help of a computer) checked that for Dima’s code, every possible word of at most $10000$ letters can be decoded in only one way. Does it follow that Dima’s code is good? (Note that Dima and Serjozha are Russian, so they use the Cyrillic alphabet, which has $ 33$ letters! A word is any sequence of letters.) (D Piontkovskiy, S Shalunov)

Tireless Thomas and Jeremy construct a sequence. At the beginning there is one positive integer in the sequence. Then they successively write new numbers in the sequence in the following way: Thomas obtains the next number by adding to the previous number one of its (decimal) digits, while Jeremy obtains the next number by subtracting from the previous number one of its digits. Prove that there is a number in this sequence which will be repeated at least $100$ times. (A Shapovalov)

Let's say a language $L \subseteq \{0,1\}^*$ is in $\textbf{P}_{angel}$ if there exists a polynomial $p : \mathbb{N} \mapsto \mathbb{N}$, a sequence of strings $\{\alpha_n\}_{n \in \mathbb{N}}$ with $\alpha_n \in \{0,1\}^{p(n)}$, and a deterministic polynomial time Turing Machine $M$ such that for every $x \in \{0,1\}^n$ $$x \in L \Leftrightarrow M(x, \alpha_n) = 1$$ Let us call $\alpha_n$ to be the [i]angel string [/i]for all $x$ of the length $n$. Note that the [i]angel string[/i] is $\textbf{not}$ similar to a [i]witness[/i] or [i]certificate [/i]as used in the definition of $\textbf{NP}$ For example, all unary languages, even $UHALT$ which is undecidable, are in $\textbf{P}_{angel}$ because the \textit{angel string} can simply be a single bit that tells us if the given unary string is in $UHALT$ or not. \\\\ A set $S \subseteq \Sigma^*$ is said to be [b]sparse[/b] if there exists a polynomial $p : \mathbb{N} \mapsto \mathbb{N}$ such that for each $n \in \mathbb{N}$, the number of strings of length $n$ in $S$ is bounded by $p(n)$. In other words, $|S^{=n}| \leq p(n)$, where $S^{=n} \subseteq S$ contains all the strings in $S$ that are of length $n$. [list=1] [*] Given $k \in \mathbb{N}$ sparse sets $S_1, S_2 \ldots S_k$, show that there exists a sparse set $S$ and a deterministic polynomial time TM $M$ with oracle access to $S$ such that given an input $\langle x,i \rangle$ the TM $M$ will accept it if and only if $x \in S_i$. \\Define the set $S$ (note that it need not be computable), and give the description of $M$ with oracle $S$. \\Note that a TM $M$ with oracle access to $S$ can query whether $s \in S$ and get the correct answer in return in constant time. [/*] [*] Let us define a variant of $\textbf{P}_{angel}$ called $\textbf{P}_{bad-angel}$ with a constraint that there should exists a polynomial time algorithm that can [b]compute[/b] the angel string for any length $n \in \mathbb{N}$. In other words, there is a poly-time algorithm $A$ such that $\alpha_n = A(n)$. \\Is $\textbf{P} =\textbf{P}_{bad-angel}$? Is $\textbf{NP}=\textbf{P}_{bad-angel}$? Justify. [/*] [*] Let the language $L \in$ $\textbf{P}_{angel}$. Show that there exists a sparse set $S_L$ and a deterministic polynomial time TM $M$ with oracle access to $S_L$ that can decide the language $L$. [/*]

The pattern of figures $\triangle$ $ \bullet$ $ \square$ $\blacktriangle$ $\circ$ is repeated in the sequence $$\triangle,\bullet, \square, \blacktriangle, \circ, \triangle, \bullet, \square, \blacktriangle, \circ$$ The 214th figure in the sequence is (A) $\triangle$ (B) $\bullet$ (C) $\square$ (D) $\blacktriangle$ (E) $\circ$

Given a finite sequence of complex numbers $c_1, c_2, \ldots , c_n$, show that there exists an integer $k$ ($1 \leq k \leq n$) such that for every finite sequence $a_1, a_2, \ldots, a_n$ of real numbers with $1 \geq a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$, the following inequality holds: \[\left| \sum_{m=1}^n a_mc_m \right| \leq \left| \sum_{m=1}^k c_m \right|.\]

Consider the sequences of six positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ with the properties that $a_1=1$, and if for some $j > 1$, $a_j = m > 1$, then $m-1$ appears in the sequence $a_1,a_2,\dots,a_{j-1}$. Such sequences include $1,1,2,1,3,2$ and $1,2,3,1,4,1$ but not $1,2,2,4,3,2$. How many such sequences of six positive integers are there?

Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$ [i]Proposed by Carl Schildkraut[/i]

The sequence $\left(a_n \right)$ is defined by $a_1=1, \ a_2=2$ and $$a_{n+2} = 2a_{n+1}-pa_n, \ \forall n \ge 1,$$ for some prime $p.$ Find all $p$ for which there exists $m$ such that $a_m=-3.$

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows: [list] [*] $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and [*] $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$. [/list] Show that $a_n$ is always a strictly positive integer.

Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$

Let $ f : (0,\infty) \to \mathbb R$ be a continous function such that the sequences $ \{f(nx)\}_{n\geq 1}$ are nondecreasing for any real number $ x$. Prove that $ f$ is nondecreasing.

In a tennis tournament where $ n$ players $ A_1,A_2,\dots,A_n$ take part, any two players play at most one match, and $ k \leq \frac {n(n \minus{} 1)}{2}$ $ 2$ matches are played. The winner of a match gets $ 1$ point while the loser gets $ 0$. Prove that a sequence $ d_1,d_2,\dots,d_n$ of nonnegative integers can be the sequence of scores of the players ($ d_i$ being the score of$ A_i$) if and only if $ (i)\ \ d_1 \plus{} d_2 \plus{} \dots \plus{} d_n \equal{} k$, and $ (ii)\ \text{for any} X\subset\{A_1,\dots,A_n\}$, the number of matches between the players in $ X$ is at most $ \sum_{A_j\in X}d_j$