Indian National Math Olympiad, INMO 2019 Problems

This post contains problems from Indian National Mathematics Olympiad, INMO 2019. Try them and share your solution in the comments.

INMO 2019, Problem 1

Let ABC be a triangle with BAC>90. Let D be a point on the line segment BC and E be a point on the line AD such that AB is tangent to the circumcircle of triangle ACD at A and BE perpendicular to AD. Given that CA=CD and AE=CE, determine BCA in degrees

INMO 2019, Problem 2

Let A1B1C1D1E1 be a regular pentagon. For 2n≤11, let AnBnCnDnEn whose vertices are the midpoints of the sides of An−1Bn−1Cn−1Dn−1En−1. All the 5 vertices of each of the 11 pentagons are arbitrarily coloured red or blue. Prove that four points among these 55 have the same colour and form the vertices of a cyclic quadrilateral.

INMO 2019, Problem 3

Let m, n be distinct positive integers. Prove that gcd(m,n)+gcd(m+1,n+1)+gcd(m+2,n+2)≤2|mn|+1. Further, determine when equality holds.

INMO 2019, Problem 4

 Let n and M be positive integers such that M>nn−1. Prove that there are n distinct primes p1, such that pj divides M+j for 1jn.

INMO 2019, Problem 5

Let AB be a diameter of a circle Γ and let C be a point on Γ different from A and B. let D be the foot of perpendicular from C on to AB. let K be a point of the segment CD such that AC is equal to the semiperimeter of the triangle ADK. Show that the excircle of triangle ADK opposite A is tangent to Γ.

INMO 2019, Problem 6

Let \(f\) be a function defined from the set \(\{(x,y) : x,y\) are real, \(xy \neq 0\}\) to the set of all positive real number such that

(i)$f(x y, z)=f(x, z) f(y, z),$ for all $x, y \neq 0$
(ii)$\quad f(x, y z)=f(x, y) f(x, z),$ for all $x, y \neq 0$
(iii)$f(x, 1-x)=1,$ for all $x \neq 0,1$
Prove that
(a) $\quad f(x, x)=f(x,-x)=1,$ for all $x \neq 0$
(b) $f(x, y) f(y, x)=1,$ for all $x, y \neq 0$