dan's math@home - problem of the week - archives
 
 
Problem Archives page 18
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+ . prob index
 
171 - Drowning Child!
172 - To Catch A Thief
173- Three Equal Areas
174 Pigs Pkeets Profits
175 Fruit Season Again
176 - Jack - and - Diane
177 - Cover That Table!
178 - - The Cat-Census
179 Around The World
180 Cover That Board?
 
 
Problem #171 - Posted Saturday, May 3, 2003
Help ! Drowning Child ! . (back to top)
At Newport Beach, CA, the long, straight shoreline separates the clear
water (blue) from the sand (tan). Mitch, the lifeguard, is at M, 80 meters
from the water, and sees a drowning child C, 120 meters from the sand.
The points A and B are 280 meters apart along the shoreline. Mitch runs
only 4 m/sec but he can swim 8 m/sec. At what spot on the shoreline AB
should Mitch aim in order to reach the child at C as soon as possible?
Compute the minimum time to the nearest tenth of a second, compare it to the times
for the paths: MC, MAC, MBC. Show steps and reasoning.
Can Mitch save the child?
 
 
Problem #172 - Posted Monday, May 19, 2003
~ To Catch A Thief ~ . (back to top)
There are five men married to five women in a big five-bedroom house near me. One of the men
is a thief. The other men are: a poor man, a rich man, a baker, and a doctor. The women don't work.
One member of each couple always tells the truth and the other one always lies. In four of the five
couples, it's the husband who lies; the exception is the poor man, who tells the truth. All ten people
have names that can designate men or women. In these statements, nobody mentions the name of their own spouse:
 Bobby:  1. Pat is married to Terry.
 Elly:  2. Kim is married to the rich man.
 Freddie:  3. Elly is married to Ronnie.
 Jerry:  4. Elly is married to Willie. 5. Freddie is the baker.
 Kim:  6. Jerry is not married to Terry.
 Lou:  7. Pat is not the poor man. 8. Bobby is married to Jerry.
 Pat:  9. Freddie is married to Ronnie. 10. Willie is not the doctor.
 Ronnie:  11. Jerry is the rich man or is a woman. 12. Bobby is married to Kim.
 Terry:  13. Freddie is married to Lou. 14. Elly is a woman.
 Willie:  15. Kim is the thief. 16. Freddy tells the truth.
Decide who is married to whom, which are men, which are women, and
the profession of each man. Who was the thief?
 
Problem #173 - Posted Monday, June 2, 2003
Three Equal Areas . (back to top)
A point P is placed inside a triangle so that segments
from this point to the vertices X, Y, and Z divide the
big triangle into three smaller triangles of equal area.
Decide which of these statements is true, and prove it.
a) P is the center of the inscribed circle of the triangle
b) P is the center of the triangle's circumscribed circle
c) The three angles meeting at P are each 120 degrees
d) P is the intersection of the altitudes of the triangle
e) P is the intersection of the medians of the triangle
f) P may not exist, or be unique, or be determinable
Bonus: Point a is the "incenter." What are geometric names
(if any) for the other four points described in b, c, d, and e?
Three equal areas?
 
 
Problem #174 - Posted Saturday, June 14, 2003
Pigs, Parakeets, and Profits . (back to top)
The Pet Shop Boys bought a certain number of guinea pigs, and half that many pairs of
parakeets, to sell in their shop. On every pet sold, there was a ten percent markup on what
the Boys paid for it. They paid twice as much for each pig as for each parakeet. The price
in dollars for a guinea pig was equal to half of the second perfect number (the one after six).
 
After all but seven of the pets had been sold, the Boys found they had taken in the same
amount of money they'd paid out for the pets. Their profit was the combined retail value
of the remaining pets. a) What was their profit? b) How many of each type of pet had
been sold? c) Hadn't been sold? Show steps and reasoning.
 
 
Problem #175 - Posted Friday, June 27, 2003
Fruit Season Again ! . (back to top)
What could be better than fresh fruit season? Yumm! Doctors say to get plenty of
carbohydrates and protein each week, but not too much fat or too many calories !

Fruit

 Protein grams

Fat grams

Carbo grams

Calories

Apple

0.6

0.8

24.1

106

Banana

5.0

6.0

89.0

100

Cantaloupe

1.5

0.0

23.5

50

Peach

 0.5

0.1

7.6

33
You've been ordered to eat, each week, a total of at least 230g of protein, at least
4400g of carbos, and at most 260g fruit fat, while minimizing your total calories.
How much of each fruit should you have weekly, if: a) you're required to eat
whole numbers of fruits? b) you can cut up the fruit and eat any amounts?
Show steps and reasoning.
 
 
Problem #176 - Posted Tuesday, July 8, 2003
"Jack and Diane" . (back to top)
They've been immortalized in song, but how old are they? When Jack was twice as old as
Diane was when Jack was twice as old as Diane was when Jack was half as old as he is now,
Diane was half as old as Jack was when Diane was a year older than half as old as Jack is now.
Assuming their ages are whole numbers of full years, a) How old are Jack & Diane if they're both in their
twenties? b) What are all possible solutions if their ages are under 100? Show steps and reasoning.
 
 
Problem #177 - Posted Wednesday, July 16, 2003
Cover That Table! . (back to top)
I was eating in a restaurant recently where the tables were squares
1.5 meters on a side. They were putting down paper tablecloths
for the next people, using four 1 meter squares for each table.
"That seems wasteful," I told them. "I bet you could cover a table
using only three uncut squares of paper." a) Was I right about three? Show/explain how to do it. b) What's the largest square table that
can be covered with three 1-meter squares? Answer is independent of a).
Show steps and reasoning.
graphics dan bach
 
 
Problem #178 - Posted Thursday, July 24, 2003
The Cat-Census (This problem comes to us in limerick form!) . (back to top)
"They've counted the cats in Llanfair,
Which number a third of a square.
If a quarter were gone,
Just a cube would stay on.
How many, at least, must be there?" (Show steps & reasoning.)
 
 
Problem #179 - Posted Wednesday, August 6, 2003
Around The World ! . (back to top)
A group of airplanes is based on a small island. Each plane holds just enough
fuel to make it halfway around the world. Any amount of fuel can be transferred
from one plane to another in flight but the only source of fuel is on the island.
What is the smallest number of planes that will ensure the flight of one plane all
the way around the world (on a great circle), and how is it done?
Assume the planes have the same constant ground speed and rate of fuel consumption, that
no time is lost refueling in the air or on the ground, that all planes return safely to base island,
and can land nowhere else. Show steps and reasoning. One point penalty for resubmissions.
One plane makes it
around the world.
 
 
Problem #180 - Posted Sunday, August 17, 2003
Cover That Board ? . (back to top)
Can you cover the board at the right
with just copies of the T-shaped tile?
Show how to
do it, or else
prove that it
can't be done.
No cutting, overlapping, or hanging T's off the edge!
(You can refer to columns A,B,... , rows 1, 2, ...
if you want.) Show steps and reasoning.
Either show the board can
or can't be covered by T's.
 
 
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YOU CAN ALWAYS FIND ME AT dansmath.com - Dan the Man Bach - 2003 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
Problems only . . . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100
Probs & answers . 101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180
Problems only . . . 101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180
Probs & answers . 181-190 . 191-200 . 201-210 . 211-220 . 221-230 . 231+
Problems only . . . . 181-190 . 191-200 . 201-210 . 211-220 . 221-230 . 231+
 
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