dan's math@home - problem of the week - archives
Problem Archives page 11
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
101 -- The Fab Five !
102 - Similar but Diff
103 - Weird Numbers
104 Dan's Prime Code
105- Ez Come, Ez Go
106- New Year Phone
107- Geometry Gems
108 - Cover The Cube
109- Based On What?
110 - Family Of Four
Problem #101 - Posted Sunday, November 19, 2000
The Fab Five ! (back to top)
A set contains five integers. When distinct elements of this set are added together,
two at a time, the complete list of different possible sums that result is:
637, 669, 794, 915, 919, 951, 1040, 1072, 1197.
a) Figure out (if possible) the original five integers in the set.
b) What are the means; of the original set, and the set of sums ?
c) Given a set of 'sums', do five such numbers always exist ?
Problem #102 - Posted Monday, November 27, 2000
Similar but Different (back to top)
Drew drew two similar triangles, both with integer sides. Two sides of one triangle
were the same as two in the other.The other (unmatching) sides differed by a prime.
a) What are the smallest triangles making this possible? (Give corresp. prime difference.)
b) What is the smallest prime for which this is possible? (Give corresponding triangles.)
Answers for a) and b) may or may not be different.
Problem #103 - Posted Friday, December 8, 2000
Weird Numbers! (back to top)
You may know a 'perfect number' is one whose proper divisors add up to the number,
such as 6: 1+2+3 = 6. In an 'abundant number' the divisors add up to more than the
number, like 12: 1+2+3+4+6 = 16 > 12. A 'weird number' is an abundant number with
no subset of divisors adding to the number. The number 12 isn't weird 'cause 1+2+3+6 = 12.
What are the first two weird numbers?
Problem #104 - Posted Sunday, December 17, 2000
Dan's Prime CodeTM (back to top)
The first 26 primes (2, 3, 5, ...) can be put in correspondence with the letters A through
Z, so the 'Prime Code' for the word CAB would be the product 5 * 2 * 3 = 30.
a) What's the Prime Code for the word BIKER ?
b) Decode this message... 913 1511191 110618.
c) What (English) word comes closest to a million?
Please submit all parts in one message. No proper nouns. Ranked by order received AND best c); results avgd.
One point penalty for each resubmission of 'improved' answers to c).
Problem #105 - Posted Monday, December 25, 2000
Easy Come, Easy Go! (back to top)
At their traditional end-of-the-millenium poker game, Clifton and Lawrence agree on
the stakes for each hand: The loser pays 1/3 of the money he has remaining, to the winner.
After a while, Lawrence gives up: "You now have exactly three times the cash I have,
you've won the last few hands, and I've lost just about four bucks!"
"But you won every hand before that," Clifton replied, "in fact you've won the same
number of hands I have!" How many hands were played, how much money did they
each start with, and how much do they have now?
Problem #106 - Posted Wednesday, January 3, 2001
New Year, New Phone! (back to top)
After my recent move, I got a new phone number. I forgot to write it down, but it had a
3-digit prefix and the rest was another 4 digits, like this: xxx-xxxx. I did remember that
the prefix, subtracted from half the square of the rest of the number, gave me the whole
phone number as a result ! What was my new number ? (Explain steps fully for best ranking!)
Problem #107 - Posted Friday, January 12, 2001
Gee, Geometry Gems ! (back to top)
a) Given that two of the three sides of a right triangle are 3 and 4, what's the shortest possible
length for the third side? b) If (6,9) and (10,3) are the coordinates of two opposite vertices of
a square, what are the coords of the other two? c) One circle has radius 5 and center at (0, 5).
A second circle has radius 12, center (12,0). Find the radius and center of a third circle which
passes through the center of the 2nd circle and both intersection pts of the first two circles.
Problem #108 - Posted Sunday, January 21, 2001
Cover The Cube ! (back to top)
The T-shape at the right can cover the six faces of a cube. How many other shapes can you find that cover the cube? Please give your answers as lists of the six squares used. The T-shape in the picture would be called {a5, b2, b3, b4, b5, c5}. Rotations and reflections (flips) don't count as different. Shapes must be connected; squares must touch along a whole edge.

Problem #109 - Posted Tuesday, January 30, 2001
Based On What?! Here's a fun rhyme I found; see if you can answer it! (back to top)
The square of nine is 121 ; I know it looks quite weird.
But still I say it's really true ; the way we figure here.
And nine times ten is 132 ; the self-same rule, you see.
So whatcha say I'd have to write for five times twenty-three?
Problem #110 - Posted Saturday, February 10, 2001
Family Of Four (There may be more than one answer.) (back to top)
"Hey, nice looking family!" said Fred, seeing the photo. "I met your younger boy today;
he said he was nine, and your wife reminded me his brother is older."
Frank agreed, saying, "It's odd about all our ages. If you total the squares of my age and
the boys' ages, you get my wife's age times the total of my age and the boys' ages."
If the ages are all whole numbers, can you figure them out?
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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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