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 […]
Here is a variant of clueless Sudoku that I was trying for fun. In clueless Sudoku A n*n board is given No numbers are written on the board The board is divided in some blocks (typically in some pattern). Sum of numbers in each block must be constant. Numbers 1 to n must appear in […]
Try this beautiful problem from Singapore Mathematical Olympiad, SMO, 2010 - Problem 7 based on the combination of equations.
Try this beautiful problem from PRMO, 2019, problem-17, based on Largest Possible Value Problem. You may use sequential hints to solve the problem.
Try this beautiful problem from TOMATO useful for ISI B.Stat Entrance based on Sign change. You may use sequential hints to solve the problem.
Try this beautiful problem from TOMATO useful for ISI B.Stat Entrance based on Calculus. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on combinatorics in Tournament.
Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Maximum and Minimum Element. You may use sequential hints.
Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on function. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Interior Angle.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Smallest positive Integer.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Proper divisors.