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# 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 = and AB subtends 1350 at the center of the circle. Find the maximum possible area of ABCD.

Problem 2

Let and be two sets of prime numbers such that and . Suppose and . Prove that 30 divides .

Problem 3

Define a sequence n∈N of functions as , , for . Prove that each 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 be a function satisfying , and

1. ,
2. for all x , simultaneously.
1. Find the set of all possible values of the function f.
2. If and , find the set of all integers n such that .

RMO 1990 Problems

INMO 2018 Problem 6 - Video

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 = and AB subtends 1350 at the center of the circle. Find the maximum possible area of ABCD.

Problem 2

Let and be two sets of prime numbers such that and . Suppose and . Prove that 30 divides .

Problem 3

Define a sequence n∈N of functions as , , for . Prove that each 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 be a function satisfying , and

1. ,
2. for all x , simultaneously.
1. Find the set of all possible values of the function f.
2. If and , find the set of all integers n such that .

RMO 1990 Problems

INMO 2018 Problem 6 - Video

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