Let $n\ge 3$ be an integer, and suppose $x_1,x_2,\cdots ,x_n$ are positive real numbers such that $x_1+x_2+\cdots +x_n=1.$ Prove that $$x_1^{1-x_2}+x_2^{1-x_3}\cdots+x_{n-1}^{1-x_n}+x_n^{1-x_1}<2.$$ ~Sutanay Bhattacharya

Let $a,b$ be arbitrary co-prime natural numbers. Alice writes the natural number $t < b$ on a blackboard. Every second she replaces the number on the blackboard, say $x$, with the smallest natural number in $\{x \pm a, x \pm b \}$ that she has not yet ever written. She keeps doing this as long as possible. Prove that this process goes on indefinitely and that Alice will write down every natural number. ~Pranjal Srivastava and Rohan Goyal

Let $I$ be incentre of scalene $\triangle ABC$ and let $L$ be midpoint of arc $BAC$. Let $M$ be midpoint of $BC$ and let the line through $M$ parallel to $AI$ intersect $LI$ at point $P$. Let $Q$ lie on $BC$ such that $PQ\perp LI$. Let $S$ be midpoint of $AM$ and $T$ be midpoint of $LI$. Prove that $IS\perp BC$ if and only if $AQ\perp ST$. ~Mahavir Gandhi

Let $N$ be a positive integer. Suppose given any real $x\in (0,1)$ with decimal representation $0.a_1a_2a_3a_4\cdots$, one can color the digits $a_1,a_2,\cdots$ with $N$ colors so that the following hold: 1. each color is used at least once; 2. for any color, if we delete all the digits in $x$ except those of this color, the resulting decimal number is rational. Find the least possible value of $N$. ~Sutanay Bhattacharya

Let $I$ and $I_A$ denote the incentre and excentre opposite to $A$ of scalene $\triangle ABC$ respectively. Let $A'$ be the antipode of $A$ in $\odot (ABC)$ and $L$ be the midpoint of arc $(BAC)$. Let $LB$ and $LC$ intersect $AI$ at points $Y$ and $Z$ respectively. Prove that $\odot (LYZ)$ is tangent to $\odot (A'II_A)$. ~Mahavir Gandhi

Suppose $P(x)$ is a non-constant polynomial with real coefficients, and even degree. Bob writes the polynomial $P(x)$ on a board. At every step, if the polynomial on the board is $f(x)$, he can replace it with 1. $f(x)+c$ for a real number $c$, or 2. the polynomial $P(f(x))$. Can he always find a finite sequence of steps so the final polynomial on the board has exactly $2020$ real roots? What about $2021$? ~Sutanay Bhattacharya