 Select Page ## An isosceles triangle,on Trigonometry, I.S.I Entrance 2016, Solution to Subjective problem no. 6

Understand the problem Let (a,b,c) be the sides of a triangle and (A,B,C) be the angles opposite to those sides respectively. If ( sin (A-B)=frac{a}{a+b}sin Acos B-frac{b}{a+b} cos A sin B), then prove that the triangle is isosceles. Source of the problem I.S.I.... ## Four Points on a Circle, ISI Entrance 2017, Subjective Problem no 2

Understand the problem  Consider a circle of radius 6 as given in the diagram below. Let (B,C,D) and (E) be points on the circle such that (BD) and (CE), when extended, intersect at (A). If (AD) and (AE) have length 5 and 4 respectively, and (DBC) is a right angle,... ## The Product of Digits, ISI Entrance 2017, Subjective Solution to problem – 5.

Understand the problem   Let (g : mathbb{N} to mathbb{N} ) with ( g(n) ) being the product of digits of (n).        (a) Prove that ( g(n)le n) for all ( n in mathbb{N} ) .        (b) Find all (n in mathbb{N} ) , for which ( n^2-12n+36=g(n) ). Source of the problem... ## Three Primes, ISI Entrance 2017, Subjective solution to Problem 6.

Understand the problem Let (p_1,p_2,p_3)  be primes with (p_2neq p_3), such that (4+p_1p_2) and (4+p_1p_3) are perfect squares. Find all possible values of (p_1,p_2,p_3).    Start with hints Hint 0Hint 1Hint 2Hint 3Hint 4 Do you really need a hint? Try it first!... ## System of n equations of Real Analysis , I.S.I Entrance 2008, Solution to Subjective Problem No. 9.

Understand the problem  For (nge3 ), determine all real solutions of the system of (n) equations :                                                (x_1+x_2+cdots+x_{n-1}=frac{1}{x_n})                                                                ...