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Problems > High School Math > 2004

Rush Hour

Two people stand back to back next to the rails in a small railway station. As the head of the express train that passes the station reaches them, they start to walk parallel to the rails. As the tail of the train reaches each of them, they stop, having walked 30m and 40m respectively.

 

If they both walked with identical, constant speed and the train kept its speed as well, can you tell how long the train was?

 

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Row, Row, Row

Two boats on the opposite shores of a river start moving towards each other. When they pass each other they are 750 yards from one
shoreline. They each continue to the opposite shore, immediately turn around and start back. When they meet again they are 250 yards from the other shoreline. Each boat maintains a constant speed throughout. How wide is the river?

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Prove Composite

Oh, no, not another number theory problem?!

 

Suppose a²+ab+b²-c²-cd-d²=0, where a, b, c, d are positive integers.

Then I think a+b+c+d is a composite number. It is true? Prove it!

 

 

Note: Composite means "not prime." A composite number is a number that has factors other than one and itself.

 

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Circle Game

Show that you cannot cover a circular disk with two circular disks of smaller diameter.

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Integer Solutions Wanted

Find all solutions in nonnegative integers to x4 + (x+1)4 = y2 + (y+1)2

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How Many Divisors?

The number of divisors of 55n3 are 55 (Including 1 and the number itself). How many divisors does 7n7 have?

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Polynomial Puzzle

Given that f(x) is a polynomial of degree EIGHT such that

f(m) = 1/m for m = 1,2,3,4,5,...,9


Find f(10) 

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What's the Point?

Let P be a point inside a square S so that the distances from P to the four vertices, in order, are 7, 35, 49, and x. What is x?

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