Last weekend was incredibly busy, as I, along with many others wrote two competitive exams: the AMTI Inter NMTC, Second Level, and the KVPY
First, the AMTI Second Level
This is the paper:
a) If and are positive reals such that , prove that
b) The angles of a triangle are in arithmetical progression. The altitudes of this triangle are also in arithmetical progression. Show that the triangle is equilateral.
is an acute angle triangle in which the three sides are unequal. are respectively the midpoints of the sides . The perpendicular bisectors of the sides intersect at respectively. cut at inside the triangle. Show that are concyclic.
is a set of all positive integers, such that for any two distinct members of , and , is also a member of . Find and describe all such finite sets.
a) Find all real number triples which satisfy
Ab) Show that all the numbers to can be written in a line, but not in a circle, so that the sum of any two adjacent numbers is a perfect square.
A certain number is the product of three different prime factors, the sum of whose squares is . There are number (including unity) which are less than the number and prime to it. The sum of all its divisors (including unity and the number) is . Find the number.
A rectangular parallelepiped is given, such that its intersection with a plane is a regular hexagon. Prove that the rectangular parallelepiped is a cube
A triangle is given. The midpoints of the sides and are and respectively. The in centre of the is . The lines and meet the sides at respectively. If the areas of and are equal, find the measure of the angle
If and are relatively prime integers for some natural numbers , find the greatest common divisor of and
I would like to know how other Brilliant people who qualified did in their levels
The other exam was the KVPY
I found the exam rather tough. How did you do?
After checking the answer key, I am getting around 45. So, unfortunately, I will not be qualifying.
But how did you do? Post a comment about your KVPY experience or AMTI experience.