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 index
 
121 -- 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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Problem Archive Index
 
Probs & answers . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90
Problems only . . . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90
 
Probs & answers . . . 91-100 . 101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151+
Problems only . . . . . 91-100 . 101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151+
 
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