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This post contains problems from Indian National Mathematics Olympiad, INMO 2013. Try them and share your solution in the comments.

**Problem 1**

Let and be two circles touching each other externally at . Let be a line which is tangent to at and passing through the center of . Similarly, let be a line which is tangent to at and passing through the center of . Suppose and are not parallel and interesct at If prove that the triangle is equilateral.

**Problem 2**

Find all positive integers and primes such that

m(4m2+m+12)=3(pn−1)

**Problem 3**

Let be positive integers such that . Prove that the equation has no integer solution.

**Problem 4**

Let be a positive integer. Call a nonempty subset of good if the arithmetic mean of the elements of is also an integer. Further let denote the number of good subsets of Prove that and are both odd or both even.

**Problem 5**

In an acute triangle is the circumcenter, is the orthocenter and is the centroid. Let be perpendicular to and be perpendicular to with on and on CA. Let be the midpoint of . Suppose the areas of triangles ODC,HEA and GFB are equal. Find all the possible values of .

**Problem 6**

Let be positive real numbers such that and Further, suppose that and Prove that and

This post contains problems from Indian National Mathematics Olympiad, INMO 2013. Try them and share your solution in the comments.

**Problem 1**

Let and be two circles touching each other externally at . Let be a line which is tangent to at and passing through the center of . Similarly, let be a line which is tangent to at and passing through the center of . Suppose and are not parallel and interesct at If prove that the triangle is equilateral.

**Problem 2**

Find all positive integers and primes such that

m(4m2+m+12)=3(pn−1)

**Problem 3**

Let be positive integers such that . Prove that the equation has no integer solution.

**Problem 4**

Let be a positive integer. Call a nonempty subset of good if the arithmetic mean of the elements of is also an integer. Further let denote the number of good subsets of Prove that and are both odd or both even.

**Problem 5**

In an acute triangle is the circumcenter, is the orthocenter and is the centroid. Let be perpendicular to and be perpendicular to with on and on CA. Let be the midpoint of . Suppose the areas of triangles ODC,HEA and GFB are equal. Find all the possible values of .

**Problem 6**

Let be positive real numbers such that and Further, suppose that and Prove that and

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