This is a problem from ISI BMath 2014 Subjective Solution based on Mulitple roots or Real root. Try to solve this problem. Problem: Multiple roots or real root Let $latex \mathbf { y = x^4 + ax^3 + bx^2 + cx +d , a,b,c,d,e \in \mathbb{R}}$. it is given that the functions cuts the x […]
Let PQR be a triangle. Take a point A on or inside the triangle. Let f(x, y) = ax + by + c. Show that $latex \mathbf { f(A) \le \max { f(P), f(Q) , f(R)} }$ Discussion: Basic idea is this: First we take A on a side, say PQ. We show $latex \mathbf […]
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 $n$ ) […]
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Complex Numbers and prime.
Try this beautiful problem from the Pre-RMO, 2019 based on Trigonometry Problem. You may use sequential hints to solve the problem.
Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2010 based on functional equation. You may use sequential hints.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2009 based on Triangles and sides.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2011 based on Rectangles and sides.
Try this beautiful problem from Algebra based on quadratic equation from PRMO 2016. You may use sequential hints to solve the problem.
Try this beautiful problem from algebra, based on Arithmetic Progression from AMC-10B, 2004. You may use sequential hints to solve the problem
Try this beautiful problem from PRMO, 2016 based on Triangle You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry: Area of triangle from AMC-10A, 2009, Problem-10. You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry:Area of Trapezium.AMC-10A, 2018. You may use sequential hints to solve the problem