dan's math@home - problem of the week - archives
Problem Archives page 7

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

61 - The Right Sticks
62 - Eat Lots of Fruit
63 - Sqrs into Primes
64 A Prime Complex
65 -- Goat or Sheep ?
67 - Into 2 Thousand
68 -- Powerful Digits
69 -- The Do-It-Alls!
70 -- "Can" Be Done!

Problem #61 - Posted Sunday, October 24, 1999
Stella sees six sticks, of lengths 2, 3, 4, 5, 6, and 7 inches.
She arranges them into two separate triangles (each with positive area).
a) How can she maximize the sum of the two areas?
b) How can she minimize the sum of the two areas?
c) What are these extreme area sums (to 3 decimal places)?
Winner is first with the best answers. Supply steps and any formulas used.

Problem #62 - Posted Thursday, Novenber 4, 1999
Cafe de la Peche offers three fruit bowls:
Bowl A has 2 apples and one banana;
Bowl B has 4 apples, 2 bananas, and 3 pears;
Bowl C has 2 apples, 1 banana, and 3 pears.
Your doctor tells you to eat exactly:16 apples, 8 bananas, and 6 pears per day.
How many of each type of bowl can you buy so there's no fruit left over?
Find all possible answers. The numbers of bowls must be integers; show equations and steps.

Problem #63 - Posted Saturday, November 13, 1999
Some primes can be expressed as the sum of two squares, as in 13 = 2^2 + 3^2;
some can't: 7 =/= a^2 + b^2. Other primes can be c^2 + 2 e^2, like 11 = 3^2 + 2 * 1^2;
still others are f^2 + 3 g^2 , like 31 = 2^2 + 3 * 3^2. Find all primes less than 100
that can each be written in all three ways: p = a^2 + b^2 = c^2 + 2 e^2 = f^2 + 3 g^2,
and show how it's done with each one (as above with 13, 11, and 31).
The numbers you're squaring must be integers; show your thinking!

Problem #64 - Posted Wednesday, November 24, 1999
One of our contestants asked me, "What is it with you and primes?" Well, I admit to a certain 'infinity' towards primes, and not just real ones. Let's call m + ni an 'iprime' if it can't be factored as (a+bi)(c+di), with a,b,c,d integers (not counting "units" 1, -1, i, -i as factors; the symbol i represents the imaginary square root of -1, so that i^2 = -1. For example, 2 is not an iprime: (1+i)(1-i) = 1^2-i^2 = 1+1 = 2. Neither is 7+6i, which is (2+i)(4+i). However, 2+i itself is an iprime, and so is 3, because they can't be factored into 'smaller' parts.) What are all iprimes with positive (integer) real part at most 7 and (integer) imaginary part between -7 and 7 (inclusive)?

Problem #65 - Posted Monday, December 6, 1999
My friend Charlie sometimes gets confused; 80% of the time he identifies a sheep as a sheep, and 80% of the time he calls a goat a goat. (Otherwise he thinks it's the other animal.) In his area, 85% of the animals are sheep and the rest are goats. Charlie sees a random animal and says it's a goat. What's the probability that he's right?

Problem #66 - Posted Thursday, December 16, 1999
 Holiday Cookies? (back to top) You decide to bake just two cookies for your holiday party. One is circular, the other is square. The diameter D of the circle and the side S of the square (in inches) are both (positive) integers less than 100. a) What choice of D and S give cookies of closest area, which cookie has more area, and by how much? b) If these closest areas were equal, what would be the implied value of Pi? Include a brief summary of your method and/or steps involved.
Problem #67 - Posted Tuesday, December 28, 1999
 Into Two Thousand! (back to top) The number 2000 has 20 divisors, as shown in "Dan's Divisor Chart" at the right. a) What are all (natural) numbers less than 2000 that have at least that many (20) divisors? b) How many divisors does each of those have? c) Which of them has (have) the most divisors? Find as many as you can; partial answers get partial credit. Explain your thinking or give a procedure.

Problem #68 - Posted Thursday, January 6, 2000 (we lived!)
a) Which digits (0-9) can be the last digit of a power of 2? Which occur infinitely often?
b) Which pairs of digits (00-99) can be the last two digits of a power of 2?
. .Which ones occur infinitely often? (Consider 1 as 01, etc.)
c) What's the last digit of 2^34 ? What are the last two digits of 2^345 ?
Explain your thinking and give a procedure.

Problem #69 - Posted Sunday, January 16, 2000
There's a job opening at dansmath.com for a Webmaster. There are three desirable skills:
Writing, Design, Programming. I received 45 applications; 80% have at least one of these skills.
Twenty know design, 25 can write well, and 21 have programming ability. There are 7 with
writing and design skills only, 9 with just writing and programming, and six who can design
and program but can't write. I will only interview those with all three skills.
How many interviews will there be?

Problem #70 - Posted Wednesday, January 26, 2000
 "Can" Be Done! (back to top) At a carnival game, you see nine paint cans stacked and numbered as shown. You get three throws, and you must knock down one (and only one) can per throw. Your first throw scores the number on that can, the second one counts twice the number on the can, and the third shot counts triple its number. To win a prize you must score exactly 50 points; no more, no less. Describe precisely how you can win the prize.
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YOU CAN ALWAYS FIND ME AT dansmath.com - Dan the Man Bach - 3*23*29 A.D.

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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