Set of Nilpotent Matrices – TIFR 2017

Set of Nilpotent Matrices – TIFR 2017

State True or False: The set of nilpotent matrices of \( M_3 (\mathbb{R} ) \) spans \( M_3 (\mathbb{R} ) \) considered as an \( \mathbb {R} \) – vector space ( a matrix A is said to be nilpotent if there exists \( n \in \mathbb{N} \)  such that \( A^n = 0 \) )....
Limit is Euler!

Limit is Euler!

Let \( \{a_n\}_{n\ge 1} \) be a sequence of real numbers such that $$ a_n = \frac{1 + 2 + … + (2n-1)}{n!} , n \ge 1 $$ . Then \( \sum_{n \ge 1 } a_n \) converges to ____________ Hint 1 - Sum of oddsHint 2 - Break in partialsHint 3 - Something goes to e! Notice...
Real Surds – Problem 2 Pre RMO 2017

Real Surds – Problem 2 Pre RMO 2017

ProblemHint 1Hint 2Hint 3Final Answer Suppose \(a,b\) are positive real numbers such that \(a\sqrt{a}+b\sqrt{b}=183\). \(a\sqrt{b}+b\sqrt{a}=182\). Find \(\frac{9}{5}(a+b)\). This problem will use the following elementary algebraic identity: $$ (x+y)^3 = x^3 + y^3 +...
Integers in a Triangle – AMC 10A

Integers in a Triangle – AMC 10A

There is an intuitive definition of perpendicularity. It does not involve angle. Instead, it involves the notion of distance. Consider a point P and a line L not passing through it. If you wish to walk from P to L, the path of shortest distance is the perpendicular...