Let $n$ be a positive integer. Initially the sequence $0,0,\cdots,0$ ($n$ times) is written on the board. In each round, Ananya choses an integer $t$ and a subset of the numbers written on the board and adds $t$ to all of them. What is the minimum number of rounds in which Ananya can make the sequence on the board strictly increasing? Proposed by Shantanu Nene

Two positive integers are called anagrams if every decimal digit occurs the same number of times in each of them (not counting the leading zeroes). Find all non-constant polynomials $P$ with non-negative integer coefficients so that whenever $a$ and $b$ are anagrams, $P(a)$ and $P(b)$ are anagrams as well. Proposed by Sutanay Bhattacharya

Let $\Delta ABC$ be an acute angled scalene triangle with circumcircle $\omega$. Let $O$ and $H$ be the circumcenter and orthocenter of $\Delta ABC,$ respectively. Let $E,F$ and $Q$ be points on segments $AB,AC$ and $\omega$, respectively, such that $$\angle BHE=\angle CHF=\angle AQH=90^\circ.$$Prove that $OQ$ and $AH$ intersect on the circumcircle of $\Delta AEF$. Proposed by Antareep Nath

For a positive integer $m$, let $f(m)$ denote the smallest power of $2024$ not less than $m$ (e.g. $f(1)=1, f(2023)=f(2024)=2024,$ and $f(2025)=2024^2$). Find all positive real numbers $c$ for which there exists a sequence $x_1,x_2,\cdots$ of real numbers in $[0,1]$ such that $$|x_m-x_n|\geq\frac{c}{f(m)}$$for all positive integers $m>n\geq1$. Proposed by Shantanu Nene

Let acute scalene $\Delta ABC$ have circumcircle $\omega$. Let $M$ be the midpoint of $BC$, define $X$ as the other intersection of $AM$ with $\omega$. Let $E,F$ be the feet of altitudes from $B,C$ to $AC, AB$ respectively. Let $Q$ be the second intersection of the circumcircle of $\Delta AEF$ and $\omega$. Let $Y\neq X$ be a point such that $MX=MY$ and $QMXY$ is cyclic. Finally, let $S$ be a point on $BC$ such that $\angle BAS=\angle MAC.$ Prove that the quadrilaterals $BFYS$ and $CEYS$ are cyclic. Proposed by Kanav Talwar and Malay Mahajan

Let $M$ be a positive integer, and let $a,b,c$ be integers in the interval $[M,M+\sqrt{\frac{M}{2}})$ such that $a^3b+b^3c+c^3a$ is divisible by $abc$. Prove that $a=b=c$. Proposed by Shantanu Nene

Rijul and Rohinee are playing a game on an $n\times n$ board alternating turns, with Rijul going first. In each turn, they fill an unfilled cell with a number from $1,2,\cdots, n^2$ such that no number is used twice. Rijul wins if there is any column such that the sum of all its elements is divisible by $n$. Rohinee wins otherwise. For what positive integers $n$ does he have a winning strategy? Proposed by Rohan Goyal

Let $ABCD$ be a trapezium with $AD||BC$; and let $X$ and $Y$ be the midpoints of $AC$ and $BD$ respectively. Prove that if $\angle DAY=\angle CAB$ then the internal bisectors of $\angle XAY$ and $\angle XBY$ meet on $XY$. Proposed by Belur Jana Venkatachala

Sunaina and Malay play a game on the coordinate plane. Sunaina has two pawns on $(0,0)$ and $(x,0)$, and Malay has a pawn on $(y,w)$, where $x,y,w$ are all positive integers. They take turns alternately, starting with Sunaina. In their turn they can move one of their pawns one step vertically up or down. Sunaina wins if at any point in time all the three pawns are colinear. Find all values of $x,y$ for which Sunaina has a winning strategy irrespective of the value of $w$. Proposed by NV Tejaswi