The sequence $a_1, a_2, \ldots, a_n$ of positive real numbers satisfies the following conditions: \begin{align*} \sum_{i=1}^n \frac{1}{a_i} \le 1 \ \ \ \ \hbox{and} \ \ \ \ a_i \le a_{i-1}+1 \end{align*} for all $i\in \lbrace 1, 2, \ldots, n \rbrace$, where $a_0$ is an integer. Prove that \begin{align*} n \le 4a_0 \cdot \sum_{i=1}^n \frac{1}{a_i} \end{align*}

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

Let $x_1, x_2 \dots, x_{2024}$ be non-negative real numbers such that $x_1 \le x_2\cdots \le x_{2024}$, and $x_1^3 + x_2^3 + \dots + x_{2024}^3 = 2024$. Prove that \[\sum_{1 \le i < j \le 2024} (-1)^{i+j} x_i^2 x_j \ge -1012.\] [i]Proposed by Shantanu Nene[/i]

Determine the number of permutations $a_1, a_2, \dots, a_n$ of $1, 2, \dots, n$ such that for every positive integer $k$ with $1 \le k \le n$, there exists an integer $r$ with $0 \le r \le n - k$ which satisfies \[ 1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}. \]

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

Let $a$ and $b$ be positive integers with $a>1$ and $b>2$. Prove that $a^b+1\ge b(a+1)$ and determine when there is inequality.

Let \(\mathbb R_{>0}\) denote the set of positive real numbers. Find all functions \(f:\mathbb R_{>0}\to\mathbb R_{>0}\) such that for all positive real numbers \(x\) and \(y\), \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\] [i]Proposed by Luke Robitaille[/i]

Prove that any triangle can be cut into $2019$ quadrilaterals such that each quadrilateral is both inscribed and circumscribed. (Nairi Sedrakyan)

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.

A positive integer is written on a blackboard. Players $A$ and $B$ play the following game: in each move one has to choose a proper divisor $m$ of the number $n$ written on the blackboard ($1<m<n$) and replaces $n$ with $n-m$. Player $A$ makes the first move, then players move alternately. The player who can't make a move loses the game. For which starting numbers is there a winning strategy for player $B$?

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

Sequence $a_1, a_2, a_3, \cdots$ satisfies the following condition. [b](Condition)[/b] For all positive integer $n$, $\sum_{k=1}^{n}\frac{1}{2}\left(1 - (-1)^{\left[\frac{n}{k}\right]}\right)a_k=1$ holds. For a positive integer $m = 1001 \cdot 2^{2025}$, compute $a_m$.

Let $\mathbb{Z}_{\ge 0}$ be the set of non-negative integers and $\mathbb{R}^+$ be the set of positive real numbers. Let $f: \mathbb{Z}_{\ge 0}^2 \rightarrow \mathbb{R}^+$ be a function such that $f(0, k) = 2^k$ and $f(k, 0) = 1$ for all integers $k \ge 0$, and $$f(m, n) = \frac{2f(m-1, n) \cdot f(m, n-1)}{f(m-1, n)+f(m, n-1)}$$ for all integers $m, n \ge 1$. Prove that $f(99, 99)<1.99$. [i]Proposed by Navilarekallu Tejaswi[/i]

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom