**dan's math@home - problem of the week - archives****Problem Archives**page 5**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 index41- Car AND Train! 42 - Co-Perfect Nos. 43 -"Divisorly" Nos. 44 - The Lying Truth 45 - Not Sold at 7-11 46- Round the Corner 47 - Roll That Prime! 48 - Cut The Cheese! 49 - Not So Peachy! 50 - Bouncing Ball... - Problem #41 - Posted Tuesday, March 16, 1999
- Car AND Train!! (back to top)
- Every day, Ellie takes the commuter train and arrives at the station 8:30 AM, where she's
- immediately picked up by a car and driven to work.
- One day she takes the early train, arrives at the station at 7:00 AM, and begins to walk
- towards work. The car picks her up along the way and she gets to work 10 minutes
- earlier than usual. When did Ellie meet the car on this day? (Explain fully.)
- Problem #42 - Posted Sunday, March 28, 1999
- Co-Perfect Numbers (back to top)
- A perfect number is one whose positive proper divisors add up to the
- number itself. (For example: 6 = 1 + 2 + 3 ; 28 = 1 + 2 + 4 + 7 + 14 ; the next is 496.)
- Can you find a pair of numbers, so that the proper divisors of each
- number add up to the other one? (Give a complete proof.)
- Problem #43 - Posted Friday, April 9, 1999
- Divisorly Numbers (back to top)
- A 'divisor' of a natural number is one that goes in with no remainder.
- (For example,18 has 6 divisors: 1, 2, 3, 6, 9, 18.)
- a) What's the smallest number with at least 14 divisors?
- b) What's the smallest number with exactly 14 divisors?
- c) Find the smallest number with exactly 100 divisors.
- (Hint? See the solution for last week's problem in the archives.)
- Problem #44 - Posted Saturday, April 24, 1999
- The Lying Truth (back to top)
- You missed math class yesterday and you want to know if there's a test today.
- There are two students outside the classroom; one always lies and one always tells
- the truth, but you don't know which is which. What single yes-or-no question can
- you ask (one of them) so that no matter who answers, you'll know if there's a test?
- Problem #45 - Posted Tuesday, May 4, 1999
- Not Sold at 7-11! (back to top)
- The local convenience store sells a gallon of milk for $7 and a box of cereal for $11. If you bought 3 milks and 2 cereals you'd spend $43, but there's no way to spend exactly, say, $20. Starting at $1, how many whole numbers (such as $20) can't equal the "total" of some milks and cereals, and what are they?
- It's ok to buy all milk or all cereal. (Hint: After a certain point, all totals are possible!)
- Problem #46 - Posted Saturday, May 15, 1999
Round the Corner! (back to top)

a) There's a big circle (7 feet across) painted in the corner of a room, and a little circle crammed in that just touches the big circle. What's the diameter of the little circle?

b) Do the same problem using spheres: a big ball (of diam 7 ft) just touches the three walls, and a little ball touches three walls and the big ball. What's the diameter of the little ball?

Give answers involving fractions or radicals, not decimal approximations.

- Problem #47 - Posted Monday, May 24, 1999
Roll That Prime! (back to top)

Three standard fair dice are thrown. What's the probability that the total of the three is a prime number?

- Problem #48 - Posted Saturday, June 5, 1999
- Cut The Cheese! (back to top)
- Take a big cube of cheese (12" x 12" x 12"), and cut through it with a knife. Only cut through the corners and midpoints shown, and look at the cross sections you get. a) What are all regular polygons you can get, and what are their areas and perimeters? b) Allowing non-regular (but planar) polygons, which has the most area, and what is its area? c) Which planar polygon has the most perimeter, and what is that longest perimeter? d) Which polygon has the largest area- to- perimeter ratio, and what is that ratio?

- Problem #49 - Posted Friday, June 18, 1999
- Not So Peachy! (back to top)
As a fuzzy fruit farmer, I know how many peaches people produce. For fun I figured that last year my peach trees averaged as many peaches per tree as there were trees. But this year I reckoned the average was only 97 ppt (peaches per tree); that's 3500 fewer total peaches with the same number of trees. How many peach trees do I farm? Explain your solution! - Problem #50 - Posted Tuesday, June 29, 1999
- The Bouncing Ball . . . (back to top)
- A basketball is dropped straight down from the rim height of 10 feet. Each bounce, it rises to the same
- percentage (p%) of its previous height. The ball finally comes to rest after traveling a total of 70 (vertical) feet.
- What is the mystery "rebound" percentage p ?
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