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 =$latex \sqrt {2 + \sqrt {2} } $ and AB subtends 1350 at the center of the circle. Find the maximum possible area of ABCD.

Problem 2

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

Problem 3

Define a sequence $latex <fn(x)> $ n∈N of functions as $latex f_0(x )=1, f_1(x )=x $, $latex (f_n(x))^2 - 1 = f_{n-1} (x) f_{n+1} (x) $, for $latex n \ge 1 $ . Prove that each $latex 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 $latex f :Z \mapsto Z $be a function satisfying $latex f(0) \neq 0 $ , $latex f(1)=0 $ and

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

Some Useful Links:

RMO 1990 Problems

INMO 2018 Problem 6 - Video