Prove that sum of any 12 consecutive integers cannot be perfect square. Give an example where sum of 11 consecutive integers is a perfect square Discussion: Suppose a, a+1, a+2 , ... , a+ 11 are 12 consecutive integers. Sum of these 12 integers are 6(2a + 11). This is an even integer. If it […]
Here, you will find all the questions of ISI Entrance Paper 2014 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems. Problem 1: In a class there are $100$ student. We define $\mathbf { A_i} $ as the number of friends of $\mathbf { i^{th} […]
Overview of Math Olympiads in United States The American Mathematics Competitions (AMC) are the first of a series of competitions in middle school and high school mathematics that lead to the United States team for the International Mathematical Olympiad (IMO). AMC has three levels: AMC 8 - grade 8 and below AMC 10 - grades 10 and […]
This is a subjective problem number 3 from ISI BMath 2011 based on inequality of a product expression. Try out this problem. Problem: Inequality of a product expression For $latex \mathbf{n\in\mathbb{N}}$ prove that $latex \mathbf{\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\leq\frac{1}{\sqrt{2n+1}}}$ Discussion Note that $latex \mathbf{ \frac{2n}{2n+1} \ge \frac{2n-1}{2n} }$ since simple cross multiplication gives $latex \mathbf{ 4n^2 \ge 4n^2 - […]
ISI Entrance Paper BMath 2011 - from Indian Statistical Institute's Entrance Also see: ISI and CMI Entrance Course at Cheenta Given $latex \mathbf{ a,x\in\mathbb{R}}$ and $latex \mathbf{x\geq 0,a\geq 0}$ . Also $latex \mathbf{sin(\sqrt{x+a})=sin(\sqrt{x})}$ . What can you say about a? Justify your answer. Given two cubes R and S with integer sides of lengths r […]
This is a problem number 8 from ISI BMath 2007 based on the Continuity and composition of a function. Try this out. Problem: Continuity and composition of a function Let $ \mathbf{P:\mathbb{R} \to \mathbb{R}}$ be a continuous function such that $P(x)=x$ has no real solution. Prove that $P(P(x))=x$ has no real solution. Discussion: Hunch: There […]
This is a problem number 7 from ISI B.Math 2007 based on an inequality related to (sin x)/x function. Try out this problem. Problem: An inequality related to (sin x)/x function Let $ \mathbf{0\leq \theta\leq \frac{\pi}{2}}$ . Prove that $\mathbf{\sin \theta \geq \frac{2\theta}{\pi}}$. Discussion: We consider the function $ \mathbf{ f(x) = \frac{\sin x }{x} […]
Here, you will find all the questions of ISI Entrance Paper 2007 from Indian Statistical Institute's B. Math Entrance. You will also get the solutions soon of all the previous year problems. Problem 1: Let $n$ be a positive integer . If $n$ has odd number of divisors ( other than $1$ and $n$ ) […]
Here, you will find all the questions of ISI Entrance Paper 2006 from Indian Statistical Institute's B. Math Entrance. You will also get the solutions soon of all the previous year problems. Problem 1: Bishops on a chessboard move along the diagonals ( that is, on lines parallel to the two main diagonals). Prove that […]
Here, you will find all the questions of ISI Entrance Paper 2005 from the Indian Statistical Institute's B.Math Entrance. You will also get the solutions soon to all the previous year's problems. Problem 1 : For any \( k \in\mathbb{Z}^+ \) , prove that:-$$ \displaystyle{ 2(\sqrt{k+1}-\sqrt{k})<\frac{1}{\sqrt{k}}<\\2(\sqrt{k}-\sqrt{k-1})}$$Also compute integral part of \(\displaystyle{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{10000}}}\). Problem 2 : Let […]
Try this beautiful problem from the Pre-RMO, 2017 based on Integers and Inequality. You may use sequential hints to solve the problem.
Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2011 based on Permutation. You may use sequential hints to solve the problem.
Try this beautiful problem from AMC-10A, 2009 based on Diamond Pattern. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on GCD and Ordered pair.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1997 based on Trigonometry and greatest integer.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on Squares and Triangles.
Try this beautiful problem from the Pre-RMO, 2017 based on Series Problem. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Trigonometry and positive integers.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1997 based on Odd and Even integers.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1997 based on Two and Three-digit numbers.