Problem #121 - Posted Monday, July 2, 2001

Gear Numbers ! (back to top)

In the picture, assume the gears have 15, 16, and 17 teeth from left to right.

Number each gear clockwise, 0-14 0-15, 0-16, with (0, 0, 0) at the top at noon.

Assuming the gears take exactly one second per tooth, answer these:

a) At what exact time will the gears first say (0, 0, 0) again?

b) What's the first time the gears will say (1, 2, 3) ?

c) What do the numbers say at 12:34 p.m.?

Problem #123 - Posted Monday, July 23, 2001

Tour de Ants ! (back to top)

Here are n ants who encounter some 'forks in the road.' A positive (whole) number

of ants crawl along each path. RULES: (1) The net ant flow around a loop is zero,

(2) The same number of ants go into a node as out. a) Find the smallest number, n,

of ants that can do this. b) What nine values {a, b, . . . , i} will 'go with this flow'?

Problem #124 - Posted Wednesday, August 1, 2001

Lattice Midpoint ? (Problem thanks to Mark Jaeger) (back to top)

Below (in red) is a statement that may or may not be true. As we always say at Ohio State in the Arnold Ross Program, Prove or Disprove, and Salvage If Possible:

 

"Given any nine lattice points in 3-space (each coordinate (a,b,c) is an integer), there is at least one pair of points whose midpoint is also a lattice point."

 

Prove the statement is true or come up with a counterexample. If it's false, see if you can

recover something that's true. Show reasoning, keep resubmissions to a minimum.

Problem #127 - Posted Thursday, September 6, 2001 first problem of 2001-02 contest!

 

Hexagonal Floor Tiles (back to top)

 

The hexagonal tiles on my large bathroom floor are 2 inches across from

the middle of one side to the middle of the opposite side. They are separated

by regions of white cement that are all 1/8 inch wide. What percentage of

the floor covering is cement?

 

Answer exactly and also (if necessary) to the nearest 1/100%.

Show reasoning, keep resubmissions to a minimum.