**dan's math@home - problem of the week - archives****Problem Archives**page 13**Problems Only.**For answers & winners click here.**1-10****. 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100****101-110****. 111-120 . 121-130 . 131-140 . 141-150 . 151+ .**prob index121 -- Gear Numbers 122 Power 2b Diffrnt 123 -- Tour de Ants ! 124 Lattice Midpoint 125- Goats for Sheep 126 -- Longest Train! 127- Hex Floor Tiles 128 - Two or 3 Kids? 129 Grand Exponents 130 - Triang-U-Later! - 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 #122 - Posted Thursday, July 12, 2001
- Power To Be Different ! (back to top)
- From the set {0, 1, 2, 3, 4, 5, 6, 7}, pick three sequences of four distinct numbers (a, b, c, d)
- so that placing parentheses
in the expression
**a ^ b ^ c ^ d**in all possible ways yields: - i) the fewest number of distinct values . . . ii) the greatest number of distinct values . . .
- iii) the greatest number of prime values. Note: m^0 = 1 if m =/= 0; 0^p = 0 if p =/= 0; 0^0 is undefined.
- There may be more than one sequence with the same number of values. Explain reasoning, minimize resubmissions.
- 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 #125 - Posted Sunday, August 12, 2001
- Goats for Sheep! (back to top)
- Marlee and Charlie decide to sell all their sheep and go into goat herding(!) They get
- as many dollars per sheep as they had sheep, and they buy as many $10 goats as they
- can with the money. This leaves them with a few dollars, with which they buy a rabbit.
- They now have an even number of animals which they split evenly; Charlie has all goats.
- How much money should he give to Marlee to even up the value of their parts?
- Problem #126 - Posted Saturday, August 25, 2001 ... last problem of the 2000-01 contest!
- The Longest Train (Clues for values of n, m, and p are given below.) (back to top)
- The world's longest train is n miles long, and takes m minutes to pass a certain point.
- A train robber on horseback can ride alongside from the rear of the moving train to
- the front and back to the rear in p minutes. How fast is the horse?
- Clues: 2 < n < m < p ; n+m+p = 40 ; n is prime ; p is a mult of 10 ; m is a mult of n.
- 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.

- Problem #128 - Posted Saturday, September 15, 2001
- Two or Three Kids... (back to top)
- "It's been four years since I saw you," said Martha. "How old are your two kids now?"
- "Gee, it has been a while; I have three kids now!" replied Suzanne. "If you multiply
- their (integer) ages it's 2/3 of a gross, but if you add them up you get your present age."
- Martha said, "That still doesn't pin it down for me!" Suzanne winked, "Yes it does;
- think about it!" How old are the three kids now, and why?
- Problem #129 - Posted Tuesday, September 25, 2001
- Grand Exponents! (back to top)
- Consider these two expressions raised to the same power:
- (a) (1 + x^2 -- x^3)^1000 . vs . (b) (1 -- x^2 + x^3)^1000
- When expanded, which one has the larger coefficient on the x^24 term, and why?
- Problem #130 - Posted Thursday, October 4, 2001
- Triang-U-Later ! (back to top)
- I drew four triangles with integer sides. Two were right, and two were not. The right ones had area
- numerically equal to their perimeter, and the others' areas were two-thirds of their perimeters.
- What were the sides, perimeters, and areas of my four triangles? (Hint: One of the triangles was rather thin!)
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