Olympiad Problems

Welcome to the Olympiad Problems corner of Manifolds Unfolded! This section is dedicated to math enthusiasts who love the thrill of problem-solving at the highest level. Here, I curate handpicked problems from various national and international mathematical Olympiads, accompanied by insightful hints, step-by-step solutions, and the occasional deep dive into underlying concepts. Whether you’re an aspiring competitor, a problem-solving hobbyist, or simply someone who enjoys the elegance of creative mathematics, this space will guide you through the fascinating world of Olympiad challenges—where ingenuity meets rigor.

(Harvard MIT Tournament 2018) Which one of them following is larger among the following: $(1+\frac{1}{2})(1+\frac{1}{4})(1+\frac{1}{6})\dots(1+\frac{1}{2018})\text{\space v/s\space 50}$ .
(Mathematical Olympiad of China) Given $\log_4(x+2y)+\log_4(x-2y)=1$ , find the value of $\min\{|x|-|y|\}$ .
(Harvard University Admission Interview Problem) Solve for $x$ : $2^x+x=5$ .
(Harvard MIT Tournament 2021) Compute the sum of all positive integers $n$ for which the expression $\frac{n+7}{\sqrt{n-1}}$ is an integer.
(Harvard MIT Tournament 2021) The graphs of the equations:
$y=x+8,\\173y=289x+2021$
on the Cartesian plane intersect at $(a,b)$ . Find $a+b$ .
Find all functions $f:\mathbb N\to \mathbb N$ such that $f(2)=2$ and $f(m\cdot n)=f(m)\cdot f(n)$ $\forall m,n\in \mathbb N$ and $f(m)>f(n)$ if $m>n$ .