by Ashani Dasgupta

The golden ratio is arguably the third most interesting number in mathematics. The first two slots are of course reserved for \( \pi \) and \( e \). Among other things, golden ratio has the uncanny habit of appearing unexpectedly in nature and geometry. What is golden... by Ashani Dasgupta

This is an I.S.I. Entrance Solution Problem: P is a variable point on a circle C and Q is a fixed point on the outside of C. R is a point in PQ dividing it in the ratio p:q, where p> 0 and q > 0 are fixed. Then the locus of R is (A) a circle; (B) an ellipse; ... by Ashani Dasgupta

Let, \( a \geq b \geq c > 0 \) be real numbers such that for all natural number n, there exist triangles of side lengths \( a^n,b^n,c^n \) Prove that the triangles are isosceles. Hint 1 - Triangular Inequality"Hint"Hint If a, b, c are sides of a... by Ashani Dasgupta

Let \(a, b, c\) are natural numbers such that \(a^{2}+b^{2}=c^{2}\) and \(c-b=1\). Prove that (i) a is odd. (ii) b is divisible by 4 (iii) \( a^{b}+b^{a} \) is divisible by c Hint 1 - Isolate aHint 2 - Eliminate cHint 3 - A bit of Modular Arithmetic Notice that \( a^2... by Ashani Dasgupta

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... by Ashani Dasgupta

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 +...