Mathematics Olympiad : 2005 Pennsylvania State University Hazleton
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Muhammad Yusuf
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Mathematics Olympiad
Pennsylvania State University Hazleton
Fall 2005
First Round
Problem 1
A train was moving in the same direction for 5 ½ hours. It is known that the train covered exactly 100 km over any 1hour time interval. Is it true that the train was moving at a constant speed? Is it true that its average speed was 100 km/h?
Problem 2
Solve the equation : √2 . (sin x + cos x) = tan2005 x + cot2005 x .
Problem 3
Let a1 = 32005 , and let an+1 be the sum of digits of an , n ≥ 1. Find a 5 .
Problem 4
Show that there exists such a positive integer N that the fractional part of N . √2005 is less than 2005−2005 .
Problem 5
Let f (x) be continuous on the interval [0, 2005] such that f (0) = f (2005) . Show that the graph of f (x) has a chord of length 1 parallel to the x − axis.
Problem 6
The vertices of a triangle lie on a circle of radius 2005. Is it possible to put this triangle inside a circle of radius 2004?
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