There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that \[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\] What is $k?$ $\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306$

Find all the sequences of natural $k_n$ with two properties: a) $k_n \le n \sqrt {n}$ for all $n$ b) $(k_n - k_m)$ is divisible by $(m-n)$ for all $m>n$

For a positive integer $n$, a [i]sum-friendly odd partition[/i] of $n$ is a sequence $(a_1, a_2, \ldots, a_k)$ of odd positive integers with $a_1 \le a_2 \le \cdots \le a_k$ and $a_1 + a_2 + \cdots + a_k = n$ such that for all positive integers $m \le n$, $m$ can be [b]uniquely[/b] written as a subsum $m = a_{i_1} + a_{i_2} + \cdots + a_{i_r}$. (Two subsums $a_{i_1} + a_{i_2} + \cdots + a_{i_r}$ and $a_{j_1} + a_{j_2} + \cdots + a_{j_s}$ with $i_1 < i_2 < \cdots < i_r$ and $j_1 < j_2 < \cdots < j_s$ are considered the same if $r = s$ and $a_{i_l} = a_{j_l}$ for $1 \le l \le r$.) For example, $(1, 1, 3, 3)$ is a sum-friendly odd partition of $8$. Find the number of sum-friendly odd partitions of $9999$.

Let $p$ be a fixed prime number. Jomland has $p$ cities labelled $0,1,\dots,p-1$. Navi is a traveller and JomAirlines only has flights between two cities with labels $a$ and $b$ (flights are available in both directions) iff there exist positive integers $x$ and $y$ such that \[ \begin{cases} a \equiv x^2 + 2025xy + y^2\pmod{p}\\ b \equiv 20x^2 + xy + 25y^2\pmod{p} \end{cases} \] Prove that: i) There exist infinitely many primes $p$ such that there exist $2$ cities where Navi cannot start from one city and get to the other through a sequence of flights; ii) There exist infinitely many primes $p$ such that for any $2$ cities, Navi can start from one city and get to the other through a sequence of flights. [i](Proposed by Ivan Chan Guan Yu)[/i]

The sequence $(x_n)$ is given by $x_1=\frac{1}{2},$ $x_n=\frac{2n-3}{2n} \cdot x_{n-1}$ for $n=2,3,... .$ Prove that for all natural numbers $n \geq 1$ the following inequality holds $x_1+x_2+...+x_n < 1$.

For the sequence \[S_n=\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}},\]find the limit \[\lim_{n\to\infty}n\left(n\cdot\left(\log(1+\sqrt{2})-S_n\right)-\frac{1}{2\sqrt{2}(1+\sqrt{2})}\right).\]

There is a sequence with $a(2)=0$, $a(3)=1$ and $a(n)=a\left(\left\lfloor\dfrac n2\right\rfloor\right)+a\left(\left\lceil\dfrac n2\right\rceil\right)$ for $n\geq 4$. Find $a(2014)$. [Note that $\left\lfloor\dfrac n2\right\rfloor$ and $\left\lceil\dfrac n2\right\rceil$ denote the floor function (largest integer $\leq\tfrac n2$) and the ceiling function (smallest integer $\geq\tfrac n2$), respectively.]

Let $ n$ be a natural number equal or greater than 3 . A sequence of non-negative numbers $ (c_0,c_1,\ldots,c_n)$ satisfies the condition: $ c_{p}c_{s}\plus{}c_{r}c_{t}\equal{} c_{p\plus{}r}c_{r\plus{}s}$ for all non-negative $ p,q,r,s$ such that $ p\plus{}q\plus{}r\plus{}s\equal{}n$. Determine all possible values of $ c_2$ when $ c_1\equal{}1$.

The sequence of real numbers $a_1,a_2,\dots$ is defined as follows: $a_1=56$ and $a_{n+1}=a_n-\frac{1}{a_n}$ for $n\ge 1$. Show that there is an integer $1\leq{k}\leq2002$ such that $a_k<0$.

A list of numbers $a_1,a_2,\ldots,a_m$ contains an arithmetic trio $a_i, a_j, a_k$ if $i < j < k$ and $2a_j = a_i + a_k$. Let $n$ be a positive integer. Show that the numbers $1, 2, 3, \ldots, n$ can be reordered in a list that does not contain arithmetic trios.

Let $R_1 =1$ and $R_{n+1}= 1+ n\slash R_n$ for $n\geq 1.$ Show that for $n\geq 1,$ $$ \sqrt{n} \leq R_n \leq \sqrt{n} +1.$$

Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]

For any sequence ($a_1,a_2,...,a_{2013}$) of integers, we call a triple ($i,j, k$) satisfying $1 \le i < j < k \le 2013$ to be [i]progressive [/i] if $a_k-a_j = a_j -a_i = 1$. Determine the maximum number of progressive triples that a sequence of $2013$ integers could have.

Let be given the sequence $(x_n)$ defined by $x_1 = 1$ and $x_{n+1} = 3x_n + \lfloor x_n \sqrt5 \rfloor$ for all $n = 1,2,3,...,$ where $\lfloor x \rfloor$ denotes the greatest integer that does not exceed $x$. Prove that for any positive integer $n$ we have $$x_nx_{n+2} - x^2_{n+1} = 4^{n-1}$$ Trần Nam Dũng

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$. Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.