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IMO1999

This document contains 6 math and geometry problems from the IMO (International Mathematical Olympiad) in 1999. Problem 1 asks to determine all finite sets of at least 3 points in the plane where the perpendicular bisector of any two points is an axis of symmetry for the set. Problem 2 involves determining the least constant C such that a given inequality holds for all non-negative real numbers x1, x2, ..., xn. Part b asks for cases when equality holds. Problem 3 asks for the smallest number of squares, N, that must be marked on an n x n board such that every unmarked or marked square is adjacent to at least one marked square. The second day

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234 views1 page

IMO1999

This document contains 6 math and geometry problems from the IMO (International Mathematical Olympiad) in 1999. Problem 1 asks to determine all finite sets of at least 3 points in the plane where the perpendicular bisector of any two points is an axis of symmetry for the set. Problem 2 involves determining the least constant C such that a given inequality holds for all non-negative real numbers x1, x2, ..., xn. Part b asks for cases when equality holds. Problem 3 asks for the smallest number of squares, N, that must be marked on an n x n board such that every unmarked or marked square is adjacent to at least one marked square. The second day

Uploaded by

drssagrawal
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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IMO QUESTIONS

IMO(40th) 1999
First Day

1. Determine all finite sets S at least three points in the plane which satisfy the following condition:
For any two distinct points A and B in S, the perpendicular bisector of the line segment AB is an
axis of symmetry for S.
2. Let n be a fixed integer, with n 2 .
4

(a) Determine the least constant C such that the inequality xi x j ( x i x j ) C xi


1 i j n
1i n
holds for all real numbers x1 , x 2 ,..., x n 0 .
(b) For this constant C, determine when equality holds.
3. Consider a n n square board, where n is fixed even positive integer. The board is divide into n 2
unit squares. We say that two different squares on the board are adjacent if they have a common
side.
N unit squares on the board are marked in such a way that every square (marked or unmarked) on
the board is adjacent to at least one marked square. Determine the smallest possible value of N.
2

Second Day

4. Determine all pairs (n, p) of positive integers such that p is prime, n not exceeds 2 p and
( p 1) n 1 is divisible by n p1 .
5. Two circles G1 and G 2 are contained inside the circle G, and are tangent to G at the distinct points
M and N respectively. G1 passes through the centre of G 2 . The line passing through the two points
of intersection of G1 and G 2 meets G at A and B. The lines MA and MB meet G1 at C and D
respectively. Prove that CD is tangent to G 2 .
6. Determine all functions f : R R such that f ( x f ( y )) f ( f ( y )) xf ( y ) f ( x) 1 for all real
numbers x, y .
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Dr. Shyam Sundar Agrawal

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