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New Zealand Maths Olympiad Committee Camp 2009 Preparation Problems

The document contains 8 math problems for preparation for the 2009 New Zealand Maths Olympiad. The problems cover topics like geometry, number theory, sequences, sets, and inequalities. The goal is to help students prepare and practice solving complex math problems.

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Destroyer74
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68 views1 page

New Zealand Maths Olympiad Committee Camp 2009 Preparation Problems

The document contains 8 math problems for preparation for the 2009 New Zealand Maths Olympiad. The problems cover topics like geometry, number theory, sequences, sets, and inequalities. The goal is to help students prepare and practice solving complex math problems.

Uploaded by

Destroyer74
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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New Zealand

Maths Olympiad Committee


Camp 2009
Preparation Problems

1. The diagonals AC and BD of a convex quadrilateral ABCD are perpendicular. Through the midpoints
of AB and AD the lines perpendicular to the corresponding opposite sides CD and BC are drawn. Prove
that these lines intersect at a point on AC.
2. Prove that for any positive integer n,√there exists an integer k which can be expressed as a sum of two
squares and such that n ≤ k ≤ n + 3 4 n.
3. Let x1 , x2 , . . . , xn be a decreasing sequence of positive real numbers. Prove that:

n
! 12 n
x
√k .
X X
x2k ≤
k=1 k=1
k

4. A finite set of squares, all the same size, are drawn in the plane so that each square has a side which is
parallel to a given line. Among any 2009 of the squares, there are at least two that intersect. Prove that
the set of squares can be divided into no more than 4015 subsets so that all the squares in each subset
have a point in common.
5. Let ABCD be a quadrilateral inscribed in a semicircle with diameter AB. The lines AC and BD intersect
at E and AD and BC at F . The line EF intersects the semicircle at G, and AB at H. Prove that E is
the midpoint of GH if and only if G is the midpoint of F H.
6. Is it possible for the product of five consecutive positive integers to be a perfect square?
7. A collection of points in the plane are connected by arcs (possibly more than one arc connects any two
given points, and there may well be points which are not connected together by an arc). There are
2008 × 2009 arcs. It is possible to label the arcs with the numbers from 1 to 2008 (repetitions allowed) in
such a way that all the labels of arcs ending at any given point are different from one another. Show that
it is also possible to do this in such a way that each label occurs exactly 2009 times.
8. Suppose that a, b, and c are positive real numbers satisfying

a2 + b2 + c2 = 1.

Prove that:
1 1 1 2(a3 + b3 + c3 )
+ + ≥ 3 + .
a2 b2 c2 abc

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