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Indian National Math Olympiad 2013


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

Problem 1

Let \Gamma_{1} and \Gamma_{2} be two circles touching each other externally at R. Let l_{1} be a line which is tangent to \Gamma_{2} at P and passing through the center O_{1} of \Gamma_{1}. Similarly, let l_{2} be a line which is tangent to \Gamma_{2} at Q and passing through the center O_{2} of \Gamma_{2}. Suppose l_{1} and l_{2} are not parallel and interesct at K . If K P=K Q, prove that the triangle P Q R is equilateral.

Solution

Problem 2

Find all positive integers m, n, and primes p \geq 5 such that
m(4m2+m+12)=3(pn−1)

Problem 3

Let a, b, c, d be positive integers such that a \geq b \geq c \geq d. Prove that the equation x^{4}-a x^{3}-b x^{2}-c x-d=0 has no integer solution.

Solution

Problem 4

Let n be a positive integer. Call a nonempty subset S of {1,2, \ldots, n} good if the arithmetic mean of the elements of S is also an integer. Further let t_{n} denote the number of good subsets of {1,2, \ldots, n} . Prove that t_{n} and n are both odd or both even.

Solution

Problem 5

In an acute triangle A B C, O is the circumcenter, H is the orthocenter and G is the centroid. Let O D be perpendicular to B C and H E be perpendicular to C A, with D on B C and E on CA. Let F be the midpoint of A B. Suppose the areas of triangles ODC,HEA and GFB are equal. Find all the possible values of \widehat{C}.

Problem 6

Let a, b, c, x, y, z be positive real numbers such that a+b+c=x+y+z and a b c=x y z . Further, suppose that a \leq x<y<z \leq c and a<b<c . Prove that a=x, b=y and c=z .


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

Problem 1

Let \Gamma_{1} and \Gamma_{2} be two circles touching each other externally at R. Let l_{1} be a line which is tangent to \Gamma_{2} at P and passing through the center O_{1} of \Gamma_{1}. Similarly, let l_{2} be a line which is tangent to \Gamma_{2} at Q and passing through the center O_{2} of \Gamma_{2}. Suppose l_{1} and l_{2} are not parallel and interesct at K . If K P=K Q, prove that the triangle P Q R is equilateral.

Solution

Problem 2

Find all positive integers m, n, and primes p \geq 5 such that
m(4m2+m+12)=3(pn−1)

Problem 3

Let a, b, c, d be positive integers such that a \geq b \geq c \geq d. Prove that the equation x^{4}-a x^{3}-b x^{2}-c x-d=0 has no integer solution.

Solution

Problem 4

Let n be a positive integer. Call a nonempty subset S of {1,2, \ldots, n} good if the arithmetic mean of the elements of S is also an integer. Further let t_{n} denote the number of good subsets of {1,2, \ldots, n} . Prove that t_{n} and n are both odd or both even.

Solution

Problem 5

In an acute triangle A B C, O is the circumcenter, H is the orthocenter and G is the centroid. Let O D be perpendicular to B C and H E be perpendicular to C A, with D on B C and E on CA. Let F be the midpoint of A B. Suppose the areas of triangles ODC,HEA and GFB are equal. Find all the possible values of \widehat{C}.

Problem 6

Let a, b, c, x, y, z be positive real numbers such that a+b+c=x+y+z and a b c=x y z . Further, suppose that a \leq x<y<z \leq c and a<b<c . Prove that a=x, b=y and c=z .

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