02 Oct State Level Mathematics Competition-2013-Category-IV : Assam Academy of Mathematics
01 September 2013
(Class XI and XII)
Marks: 10 X 10 = 100
Time: 1.30 pm to 4.30 pm
Answer the following ten questions
- Prove that has no solution in positive integers.
(Hint: Suppose on the contrary Also use )
- Find all non negative integers a, b, c, d, n that satisfy
(Hint: Put n=0 and use )
- Let ABCD be convex quadrilateral. Suppose that the lines AB and CD intersect at E and the lines AD and BC intersect at F. Prove that the following statements are equivalent
(i) a circle is inscribed in ABCD
(Hint: Inscribe a circle in the quadrilateral ABCD to touch the quadrilateral at K, L, M and N.)
- A and B are two points situated on the same side of a line XY. Find the position of a point M on the line such that the sum AM+MB is minimal.
(Hint: Suppose B´ is the reflection of B across the line XY. M is the point of intersection of AB´ and XY.)
- Let a, b, c be non-zero real numbers such that and
Prove that is p is a prime the is an irrational number.
- Let a, b, c be positive numbers such that Prove that
(Hint: Use inequality in )
- Find all functions such that
for all rationals x, y.
The sum of two integers is 52 and their L.C.M. is 168. Find the numbers.
- Show that there does not exist a function which satisfy
(b) for all in
- If be n real numbers such that
Prove that in a if and only if
- The number of class of 27 pupils each goes swimming on some of the days from Monday to Friday in a certain week. If each pupil goes atleast twice, show that there must be two pupils who go swimming on exactly the same days.
Let ABC be an acute angled triangle; AD be the bisector of with D on BC; BE be the altitude from B on AC. Show that