Select Page

## Golden Ratio and Right Triangles – when geometry meets number theory

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...

## I.S.I Entrance Solution – locus of a moving point

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; ...

## I.S.I. Entrance Solution Sequence of isosceles triangles -2018 Problem 6

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...

## 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 +...