dan's math@home - problem of the week - archives
 
 
Problem Archives page 26
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 . 151-160 . 161-170 . 171-180 . 181-190
191-200 . 201-210 . 211-220 . 221-230 . 231-240 . 241-250 . 251-260 . 261+ index
 
251 - Nice / Dice / Twice
252 - - Nearly Multiples!
253 -- Powers Five Apart
254 Vote Four Pumpkins
255 - The Round Robins
256 - Get A Clue (or three)
257 -- nEver Say Never ?
258 -- The Math Contest
259- Ah, Now We Know
260 - White,Black,What?
 

.problem #251 - posted friday, august 25, 2006
> Nice Dice Twice < (back to top)
a) Using a pair of standard dice (each labeled 1, 2, 3, 4, 5, 6), what
are the probabilities of rolling each total from 2 through 12 ?
b) Is it possible to relabel the dice with positive integers and get
the very same table of probabilities, for the same totals ?
> > If so, show how . . . if not, prove it can't be done. < <
("relabel" means use different lists of numbers, not just permute the 1 thru 6)
Show your steps and reasoning.

 
problem #252 - posted sunday, september 10, 2006
next-to-last problem of the 2005-06 contest ...
Nearly Multiples ! (back to top)
a) Prove that the square of any odd number is always 1 more than a multiple of 8.
b) Prove that the square of any odd prime > 3 is always 1 more than a multiple of 24.
c) If p is prime and is 1 more than a multiple of 4, prove there's always a perfect square
that's 1 less than a multiple of p. Show your steps and reasoning. "Numbers, primes" are positive
integers. You may quote famous theorems. Partial answers accepted.
 
problem #253 - posted wednesday, september 27, 2006
 
Powers Five Apart last problem of the 2005-06 contest. (back to top)
Find pairs of positive integers (a, b) such that
and prove you have found all of them. Show your steps and reasoning.
 
problem #254 - posted monday, october 23, 2006
Vote Four Pumpkins ! first problem of the 2006-07 contest, my tenth season! (back to top)
Three pumpkins are sitting on my front porch. They are spherical, of radii 4 ft, 3 ft,
and 2 ft, and they are mutually tangent. I put a fourth pumpkin of radius 1 ft on top,
in the hollow . . .How high off the porch is the top of the last pumpkin?
Show your steps and reasoning.
 
problem #255 - posted monday, november 13, 2006
The Round Robins second problem of 2006-07 contest . . . (back to top)
A 'round robin' tournament (RRT) is one in which each player (or team, named A, B, C, ...)
plays every other one once. i) Give clear examples of how to schedule an RRT for:
4 players, 6 players, 8 players, and (if possible) 10 players, where (for part i) all players
play once per day (think tennis). ii) Are tournaments possible with 3, 5, 7, 9 players?
Show your steps and reasoning.
 
problem #256 - posted monday, november 27, 2006
Get A Clue (or three) . . . (back to top)
"A" is a perfect square, and it is also the product of four consecutive odd integers; find it
and prove it's unique. "B" is the product of three primes, whose squares add up to 2331.
There are 7560 numbers less than B and relatively prime to B, and the sum of all positive
divisors of B is 10560. . . . find these numbers, A and B. Show your steps and reasoning.

problem #257 - posted thursday, december 14, 2006
? nEver Say Never ? ...we assume n is an integer... (back to top)
a) Prove that n^4 + 2n^3 + 2n^2 + 2n + 1 is never a square for n > 0
b) Prove that 1 + 1/2 + 1/3 + . . . + 1/n is never an integer for n > 1
Show your steps and reasoning.
 
problem #258 - posted wednesday, december 27, 2006
The Math Contest . . . (back to top)
Twenty-one girls and twenty-one boys entered a math contest.
(1) Each contestant solved at most six problems, and: (2) For each pair of a girl and
a boy, there was at least one problem that was solved by both the girl and the boy.
Prove that there was a problem that was solved by at least three girls and at least
three boys. Show your steps and reasoning.
 
problem #259 - posted monday, january 15, 2007
Ah, Now We Know ! (back to top)
Two smart mathematicians (Ben and Jen) are told that a rectangle with integer
sides L and W has been drawn, having perimeter less than 200, where L > W > 1 .
Ben is told the area, Jen is told the perimeter. They now say:
Ben: "I can't determine the dimensions." Jen: "I knew that."
Ben responds, "Now I can determine them." Jen: "So can I."
Given these are true statements, what are the length and the width?
Show your steps and reasoning.
 
problem #260 - posted wednesday, january 31, 2007
White, Black, What ? (back to top)
Three bags contain 6 marbles each; all white or black: one bag has 5 white and
1 black, another has 4 white and 2 black, the third has 3 white and 3 black. One
white marble is drawn from one bag and one black marble is drawn from another
(it is not known which bags). What is the probability of drawing a white marble from
the remaining bag? Show your steps and reasoning.
 
 
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YOU CAN ALWAYS FIND ME AT dansmath.com - Dan the Man Bach - 2006 A.D.
 
 
Problem Archives Index
 
Probs & answers . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100 . 101-110
Problems only . . . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100 . 101-110
Probs & answers . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180 . 181-190
Problems only . . . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180 . 181-190
Probs & answers . 191-200 . 201-210 . 211-220 . 221-230 . 231-240 . 241-250 . 251-260 . 261+
Problems only . . . . 191-200 . 201-210 . 211-220 . 221-230 . 231-240 . 241-250 . 251-260 . 261+
 
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