Show that there exists a sequence of positive integers $x_1, x_2,…x_n,…$ that satisfies the following two conditions: (i) Every positive integer appears exactly once, (ii) For every $n=1,2,…$ the partial sum $x_1+x_2+…+x_n$ is divisible by $n^n$.

Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let \[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \] Prove that \[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \] where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?

Determine the maximum value of the sum \[ S=\sum_{n=1}^{\infty}\frac{n}{2^n}(a_1 a_2 \dots a_n)^{\frac{1}{n}} \] over all sequences $a_1,a_2,a_3,\dots$ of nonnegative real numbers satisfying \[ \sum_{k=1}^{\infty}a_k=1. \]

Let $p$ be a prime number. Determine all positive integers $a$ such that the sequence $(a_n)_{n\geq 0}$ defined by $a_0=a$ and $a_{n+1}=pa_n-(p-1)\lfloor \sqrt[p]{a_ n} \rfloor^p$, for every $n\geq0$, is eventually constant.

Consider the iterated sequence (1) $ x_0,x_1 \equal{} f(x_0),\dots,x_{n \plus{} 1} \equal{} f(x_n),\dots$, where $ f(x) \equal{} 4x \minus{} x^2$. Determine the points $ x_0$ of $ [0,1]$ for which (1) converges and find the limit of (1).

In a country, some pairs of cities are connected by one-way roads. It turns out that every city has at least two out-going and two in-coming roads assigned to it, and from every city one can travel to any other city by a sequence of roads. Prove that it is possible to delete a cyclic route so that it is still possible to travel from any city to any other city.

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that \[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\] Prove there exist positive integers $\ell \leq s$ and $N$, such that \[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\] [i]Proposed by Morteza Saghafiyan, Iran[/i]

For an integer $a$ and positive integers $n$ and $k$, let $f_k(a, n)$ be the remainder when $a^k$ is divided by $n$. Find the largest composite integer $n\leq 100$ that guarantees the infinite sequence \[ f_1(a,n),f_2(a,n),f_3(a,n),\ldots,f_i(a,n),\ldots \] to be periodic for all integers $a$ (i.e., for each choice of $a$, there is some positive integer $T$ such that $f_k(a,n) = f_{k+T}(a,n)$ for all $k$).

Every time Josh and Ron tap their screens, one of three emojis appears, each with equal probability: barbecue, bacon, or burger. Josh taps his screen until he gets a sequence of barbecue, bacon, and burger consecutively (in that specific order.) Ron taps his screen until he gets a sequence of three bacons in a row. Let $M$ and $N$ be the expected number of times Josh and Ron tap their screens, respectively. What is $|M-N|$?

A sequence $(a_n)_{n \in N}$ of squares of nonzero integers is such that for each $n$ the difference $a_{n+1} - a_n$ is a prime or the square of a prime. Show that all such sequences are finite and determine the longest sequence.

Call a $ 3$-digit number [i]geometric[/i] if it has $ 3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.

Suppose $f$ is a nonconstant polynomial with integer coefficients with the following property: [list] [*]$f(0)$ and $f(1)$ are both odd. [*]Define a sequence of integers with $a_k = f(1)f(2) \cdots f(k)+1$ [/list] Prove that there are infinitely many prime numbers dividing at least one element of the sequence. [i]Proposed by Sayandeep Shee[/i]

Let $X$ be a finite set of real numbers. For any $x,x' \in X$ with $x<x'$, define a function $f(x,x')$, then $f$ is called an ordered pair function on $X$. For any given ordered pair function $f$ on $X$, if there exist elements $x_1 <x_2 <\cdots<x_k$ in $X$ such that $f(x_1 ,x_2 ) \le f(x_2 ,x_3 ) \le \cdots \le f(x_{k-1} ,x_k )$, then $x_1 ,x_2 ,\cdots,x_k$ is called an $f$-ascending sequence of length $k$ in $X$. Similarly, define an $f$-descending sequence of length $l$ in $X$. For integers $k,l \ge 3$, let $h(k,l)$ denote the smallest positive integer such that for any set $X$ of $s$ real numbers and any ordered pair function $f$ on $X$, there either exists an $f$-ascending sequence of length $k$ in $X$ or an $f$-descending sequence of length $l$ in $X$ if $s \ge h(k,l)$. Prove: 1.For $k,l>3,h(k,l) \le h(k-1,l)+h(k,l-1)-1$; 2.$h(k,l) \le \binom{l-2}{k+l-4} +1$.

The positive integers are partitioned into 2 sequences $a_1<a_2<\dots$ and $b_1<b_2<\dots$ such that $b_n=a_n+n$ for every positive integer $n$. Show that $a_n+b_n=a_{b_n}$.

Source: 1976 Euclid Part A Problem 2 ----- The sum of the series $2+5+8+11+14+...+50$ equals $\textbf{(A) } 90 \qquad \textbf{(B) } 425 \qquad \textbf{(C) } 416 \qquad \textbf{(D) } 442 \qquad \textbf{(E) } 495$