INMO 2012 | Problems

This post contains problems from Indian National Mathematics Olympiad, INMO 2012. Try them and share your solution in the comments.

Problem 1

Let ABCD be a quadrilateral inscribed in a circle. Suppose AB = $\sqrt {2 + \sqrt {2} }$ and AB subtends 1350 at the center of the circle. Find the maximum possible area of ABCD.

Problem 2

Let $p_1<p_2< p_3< p_4$ and $q_1<q_2<q_3<q_4$ be two sets of prime numbers such that $p_4 - p_1 = 8$ and $q_4 - q_1 = 8$ . Suppose $p_1>5$ and $q_1>5$ . Prove that 30 divides $p_1 - q_1$ .

Problem 3

Define a sequence $<fn(x)>$ n∈N of functions as $f_0(x )=1, f_1(x )=x$ , $(f_n(x))^2 - 1 = f_{n-1} (x) f_{n+1} (x)$ , for $n \ge 1$ . Prove that each $f_n(x )$ is a polynomial with integer coefficients.

Problem 4

Let ABC be a triangle. An interior point P of ABC is said to be good if we can find exactly 27 rays emanating from P intersecting the sides of the triangle ABC such that the triangle is divided by these rays into 27 smaller triangles of equal area. Determine the number of good points for a given triangle ABC.

Problem 5

Let ABC be an acute angled triangle. Let D, E, F be points on BC, CA, AB such that AD is the median, BE is the internal bisector and CF is the altitude. Suppose that angle FDE = angle C and angle DEF = angle A and angle EFD = angle B. Show that ABC is equilateral.

Problem 6

Let $f :Z \mapsto Z$ be a function satisfying $f(0) \neq 0$ , $f(1)=0$ and

1. $f(xy) + f(x)f(y) = f(x) + f(y)$ ,
2. for all x , simultaneously.
  1. Find the set of all possible values of the function f.
  2. If $f(10) \neq 0$ and $f(2) = 0$ , find the set of all integers n such that $f(n) \neq 0$ .

Some Useful Links:

RMO 1990 Problems

INMO 2018 Problem 6 - Video

Let ABC be a triangle. An interior point P of ABC is said to be good if we can find exactly 27 rays emanating from P intersecting the sides of the triangle ABC such that the triangle is divided by these rays into 27 smaller triangles of equal area. Determine the number of good points for a given triangle ABC.

Problem 5

Let ABC be an acute angled triangle. Let D, E, F be points on BC, CA, AB such that AD is the median, BE is the internal bisector and CF is the altitude. Suppose that angle FDE = angle C and angle DEF = angle A and angle EFD = angle B. Show that ABC is equilateral.

Problem 6

Let $f :Z \mapsto Z$ be a function satisfying $f(0) \neq 0$ , $f(1)=0$ and

1. $f(xy) + f(x)f(y) = f(x) + f(y)$ ,
2. for all x , simultaneously.
  1. Find the set of all possible values of the function f.
  2. If $f(10) \neq 0$ and $f(2) = 0$ , find the set of all integers n such that $f(n) \neq 0$ .