Here is the post for the Regional Mathematics Olympiad (India) RMO Number Theory Problems. These are problems from previous year papers. (This is a work in progress. More problems will be added soon). RMO Number Theory Problems: Find all triples (p, q, r) of primes such that pq = r + 1 and 2(p 2 […]
Problems Problem 1 Let \( a, b, c \) be positive real numbers such that $$ \frac{a}{1+a} + \frac{b}{1+b} + \frac{c}{1+c} = 1 $$ Prove that \( abc \leq \frac{1}{8} \)
prmo2016 CLICK ON THE ABOVE LINK to get the WB PRE-RMO 2016 PAPER AND ANSWERS.
Cauchy Schwarz Problem: Let be a polynomial with non-negative coefficients.Prove that if for ,then the same inequality holds for each . Discussion: Cauchy Schwarz's Inequality: Suppose for real numbers (\ a_{i},b_{i}), where (\ i\in{1,2,\dots,n}) we can say that $${\sum_{i=1}^{n}a_{i}^2}{\sum_{i=1}^{n}b_{i}^2}=\sum_{i=1}^{n}{a_{i}b_{i}}^2$$. Titu's Lemma: Let (\ a_{i},b_{i}\in{\mathbb{R}}) and let (\ a_{i},b_{i}>0) for (\ i\in{1,2,\dots,n}) $$\sum_{i=1}^{n}\frac{a_{i}^2}{b_{i}}\ge\frac{{\sum_{i=1}^{n}a_{i}}^2}{\sum_{i=1}^{n}b_{i}}$$ Proof of Cauchy Schwarz's Inequality: We […]
Problem 1 Find the sum 𝑆=Σ2015𝑘=1(−1)𝑘(𝑘+1)2⋅𝑘 Problem 2 Suppose in $\triangle A B C$, $A B=\sqrt{3}$, $B C=1$, $C A=2$. Suppose there exists a point $P_{0}$ in the plane of $\triangle A B C$ such that $A P_{0}$+$B P_{0}$+$C P_{0} \leq A P+B P+C P$ for all points $P$ in the plane of $\triangle A […]
This is a beautiful problem ISI MStat 2016 (sample) PSB based on order statistics . We provide detailed solution with the prerequisites mentioned explicitly.
Try this beautiful problem from algebra, based on Sum of the digits from AMC-10A, 2007. You may use sequential hints to solve the problem
Try this beautiful problem based on Integers and Divisors from TOMATO useful for ISI B.Stat Entrance. You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry:Radius of a circle.AMC-10A, 2003. You may use sequential hints to solve the problem
Try this beautiful problem from Geometry based on medians of triangle from PRMO 2018. You may use sequential hints to solve the problem.
Try this beautiful sum of Co-ordinates based on co-ordinate Geometry from AMC-10A, 2014. You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry based on Hexagon from AMC-10A, 2014. You may use sequential hints to solve the problem
This is a beautiful problem from ISI MStat 2016 (sample ) PSB Problem 1. This is based on finding the minimum value of a function subjected to the restriction .
Try this beautiful problem from the Pre-RMO, 2019 based on the Diameter of a circle. You may use sequential hints to solve the problem.
Try this beautiful problem from PRMO, 2019, problem-17, based on Largest Possible Value Problem. You may use sequential hints to solve the problem.