42nd International Mathematical Olympiad
Washington, DC, United States of America
July 89, 2001
Each problem is worth seven points.
Let ABC be an acute-angled triangle with circumcentre O . Let P on BC be the foot of the altitude from A
Wednesday, July 7, 2010
Problem 1. Determine all functions f : R R such that the equality
f x y = f (x) f (y )
holds for all x, y R. (Here z denotes the greatest integer less than or equal to z .)
Problem 2. Let I be the incentre
Wednesday, July 15, 2009
Problem 1. Let n be a positive integer and let a1 , . . . , ak (k 2) be distinct integers in the set
cfw_1, . . . , n such that n divides ai (ai+1 1) for i = 1, . . . , k 1. Prove that n does not divide ak
49th INTERNATIONAL MATHEMATICAL OLYMPIAD
MADRID (SPAIN), JULY 10-22, 2008
Wednesday, July 16, 2008
Problem 1. An acute-angled triangle ABC has orthocentre H . The circle passing through H with
centre the midpoint of BC intersects
July 25, 2007
Problem 1. Real numbers a1 , a2 , . . . , an are given. For each i (1 i n) dene
di = maxcfw_aj : 1 j i mincfw_aj : i j n
d = maxcfw_di : 1 i n.
(a) Prove that, for any real numbers x1 x2 xn ,
maxcfw_|xi ai | : 1 i n .
12 July 2006
Problem 1. Let ABC be a triangle with incentre I . A point P in the interior of the
P BA + P CA = P BC + P CB.
Show that AP AI , and that equality holds if and only if P = I .
Problem 2. Let P be a re
46rd IMO 2005
Problem 1. Six points are chosen on the sides of an equilateral triangle
ABC : A1 , A2 on BC , B1 , B2 on CA and C1 , C2 on AB , such that they are
the vertices of a convex hexagon A1 A2 B1 B2 C1 C2 with equal side lengths.
Prove that the li
45rd IMO 2004
Problem 1. Let ABC be an acute-angled triangle with AB = AC . The
circle with diameter BC intersects the sides AB and AC at M and N
respectively. Denote by O the midpoint of the side BC . The bisectors of
the angles B AC and M ON intersect a
44th IMO 2003
Problem 1. S is the set cfw_1, 2, 3, . . . , 1000000. Show that for any subset A
of S with 101 elements we can nd 100 distinct elements xi of S , such that
the sets cfw_a + xi |a A are all pairwise disjoint.
Problem 2. Find all pairs (m, n)
43rd IMO 2002
Problem 1. S is the set of all (h, k ) with h, k non-negative integers such
that h + k < n. Each element of S is colored red or blue, so that if (h, k )
is red and h h, k k , then (h , k ) is also red. A type 1 subset of S has
n blue element
QUESTION BANK ON RESEARCH METHODOLOGY
Q1. What do you mean by research? Explain its significance in modern times.
Q2. Explain difference between research method and research methodology
Q3. A research scholar has to work as a judge an