Problems for Mathematics Olympiads

Dr. Nilotpal Ghosh

PROBLEMS

1. Prove that if p and q are positive integers (p ³ q), the number of ways the numbers 1 to p + q can be arranged in 2 rows (left-aligned) with p numbers on top row and q numbers on the bottom row and the numbers in ascending order from left to right (for rows) and top to bottom (for columns) is _p+qC_p - _p+qC_p+1 .

2. In a large country with thousands of cities, the following game is played. At exactly the same moment, one airplane takes off from an airport in each city and lands at the nearest airport. Assume that the distances between the airports are all distinct. Show that there is no airport at which more than 5 planes land.

3. If there are 100 points on a plane, no three of them on the same line, show that at most 70% of the ₁₀₀C₃ triangles formed with three of these points as vertices are acute-angled.

4. Let n be a positive integer. Consider a regular polygon with n sides inscribed in a circle of radius a. If P is a point on the circle, show that the sum of squares of the distances of P from the vertices equals 2na².

5. Let n be an odd positive integer. Consider a regular polygon with n sides inscribed in a circle. If the vertices are numbered from 1 to n and P is a point on the circle on the arc between the first and the n-th vertex, show that the sum of the distances of P from the odd numbered vertices equals the sum of its distances from the even numbered vertices.

6. If four points are chosen at random on a sphere, what is the probability that the center of the sphere lies inside the tetrahedron with the four points as vertices?

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For solutions call Dr. Nilotpal Ghosh (410) 455-4281
or email : Nilotpal Ghosh

Last Update : May 1, 2002