TITA A vessel contains 80 litres of pure milk. In each operation, exactly x litres of the liquid in the vessel is removed and replaced by x litres of water, where x is the same positive integer in every operation. After exactly 4 operations, the volume of milk

MCQ Let a, b, c be positive real numbers satisfying a+b+c=12 and abc=64. The maximum possible value of a is:

TITA Let N be the smallest positive integer such that N leaves remainder 3 when divided by 7, remainder 5 when divided by 11, and is divisible by 13. If M is the next positive integer (after N) satisfying all three conditions, find the value of M-N.

MCQ In the coordinate plane, A=(0,0) and B=(10,0). A point P=(x,y) with y>0 is chosen such that the area of triangle APB is exactly 20 square units. Over all such points P, the quantity PA+PB attains its minimum value at a unique point P^*. The value of PA PB

[Algebra -- Symmetric polynomials / Optimisation, Very Hard, MCQ]\ Let a, b, c be non-negative real numbers satisfying a + b + c = 6 and ab + bc + ca = 9. The maximum possible value of abc is:

[Geometry -- Triangle / Incircle, Very Hard, TITA]\ In triangle ABC, AB = 14, BC = 13 and CA = 15. Let M be the midpoint of side BC, and let P be the point at which the incircle of triangle ABC touches side AB. The value of 4 MP^2 is 2cm0.4pt.

[Mensuration -- Cone & inscribed spheres, Very Hard, TITA]\ A right circular cone has its apex at the top and opens downward, resting on its circular base. The half-vertex angle of the cone (i.e.\ the angle between its axis and any slant edge) satisfies = 35.

A vessel contains 80 litres of pure milk. In each operation, some quantity of the mixture is removed and replaced with the same quantity of water. In the first operation, x litres are removed; in the second, 2x litres are removed; in the third, 3x litres are r

Let a and b be positive integers such that the quadratic equation [ x^2 - ax + b = 0 ] has two distinct positive integer roots, and the quadratic equation [ x^2 - bx + a = 0 ] has real roots. The number of ordered pairs (a, b) satisfying both conditions is

Let k be the largest positive integer such that 5^k divides [ 1! 2! 3! 50!. ] Find k. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%