M CUBE: MatheMatics by Maheshwari

IMPORTANT DATES, ELIGIBILITY CRITERIA, EXAM PATTERN, APPLICATION PROCESS..... Kishore Vaigyanik Protsahan Yojana (KVPY) is a national-level scholarship program funded by the Department of Science and Technology to provide scholarship to meritorious students up to pre-PhD level. The exam is conducted every year by the prestigious Indian Institute of Science (IISc). The scholarship program offered by … Continue reading KVPY- 2019: NOTIFICATION…

M CUBE: MatheMatics by Maheshwari

Theorem: If two circles touch one another, the centres and the point of contact are in one straight line.

Let R denote the set of all real numbers. Find all functions f : R → R satisfying the condition f(x + y) = f(x)f(y)f(xy) for all x, y in R

All possible 6 digit numbers, in each of which the digits occur in non-increasing order (from left to right, eg: 877550) are written as a sequence in increasing order. Find the 2005-th number in this sequence.

KVPY- 2019: NOTIFICATION…

Theorem: The tangent at any point of a circle is perpendicular to the radius drawn to the point of contact.

Let M be the mid-point of side BC of a triangle ABC. Let the median AM intersect the in-circle of ABC at K and L, K being nearer to A than L. If AK=KL=LM, prove that the sides of the triangle ABC are in the ratio 5:10:13 in the same order.

How many 6- digit numbers are there such that: (a) The digits of each number are all from the set {1,2,3,4,5} (b) Any digit that appears in the number appears at least twice? [Example 225252 is an admissible number while 222133 is not]

Let a,b,c be three natural numbers such that a<b<c and gcd(c-a,c-b)=1. Suppose there exists an integer d such that a+d,b+d,c+d form the sides of a right-angled triangle. Prove that there exists integer l, m such that c+d=l^2+m^2

Let ABC be a triangle and D be the mid-point of side BC. Suppose ∠DAB = ∠BCA and ∠DAC = 15 . Show that ∠ADC is obtuse. Further, if O is the circumcentre of ADC, prove that triangle AOD is equilateral.

Show that the equation: x^2+y^2+z^2=(x-y)(y-z)(z-x) has infinitely many solutions in integers x,y,z.