Regional Mathematics Olympiad 2 Question Paper

1. Let ABC be an acute angled scalene triangle with circumcenter O orthocenter H. If M is the midpoint of BC, then show that AO and HM intersect at the circumcircle of ABC.
2. Let n be a positive integer such that 2n + 1 and 3n + 1 are both perfect squares. Show that 5n + 3 is a composite numbers.
3. Let a, b, c > 0. If 1/a , 1/b and 1/c are in arithmetic progression, and if \(a^2 + b^2 , b^2 + c^2 , c^2 + a^2 \) are in geometric progression, prove that a=b=c.
4. Find the number of 4 digit numbers with distinct digits chosen from the set {0, 1, 2, 3, 4, 5} in which no two adjacent digits are even. 
5. Let ABCD be a convex quadrilateral. Let E, F, G, H be midpoints of AB, BC, CD, DA respectively. If AC, BD, EG, FH concur at a point O, prove that ABCD is a parallelogram.
6. Find the largest real constant \(\lambda\) such that \(\frac{\lambda abc}{a+b+c}\le (a+b)^2 + (a+b+4c)^2\) for all positive real numbers a, b and c.

I.S.I. Entrance Model Test @ Cheenta Schedule

The model test are based on the entrances of Indian Statistical Institute B.Stat and B.Math Entrance which are scheduled on 13th May, 2012.

All Cheenta Model Test will be held on Saturdays. The tests will start at 10 AM in the morning in our Kolkata center.

External Students may also appear in the tests. For registration call 09804005499, 07381944396 or mail us at helpdesk at cheenta dot com.

Model Test	Date
1	17th December, 2011
2	31st December, 2011
3	14th January, 2012
4	28th January, 2012
5	11th February, 2012
6	25th February, 2012
7	10th March, 2012
8	24th March, 2012
9	7th April, 2012
10	21st April, 2012