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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