**dan's math@home - problem of the week - archives****Problem Archives**page 6**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 index51- The Age-Old Lie 52 - Determine - It ! 53 - Chickens'n'Eggs 54-The Three Boards 55 - The Millionaire! 56 - Subtrac.Diamond 57 - Close to the Line 58 - - Y 2 Digits - - ? 59- - - Bicycle Shorts 60-Odd Combinations - Problem #51 - Posted Wednesday, July 14, 1999
- The Age-Old Lie (back to top)
- A reporter on New Year's Eve 1993 wanted to know, from Pat and Chris, how old they were, but felt (correctly, it turns out) that one would lie. So the reporter asked them both, "Write down your age now, your age at the end of next year, add these together, then multiply the result by 5," quickly followed by: "now add the last digit of the year you were born." They had no time to fake that last digit; Pat answered 281, while Chris announced 229. Who was lying, and what were their real ages at the time?
- Problem #52 - Posted Saturday, July 24, 1999
- Determine-It! (back to top)
- Find the determinant of the 2 x 2 matrix at the right, where the values are:
**A**is the absolute value of the difference of two primes, such that the sum of the cubes of the two primes is 6244;**B**is the largest even number that can't be written as the sum of two odd composite numbers;**C**is the smallest integer satisfying: C is divisible by 11, C+1 is divisible by 12, C+2 is a multiple of 13, and C is at least 14; and**D**is the largest odd divisor of 10! (10 factorial).Explain your answers as clearly as you can!

**A****B****C****D**- Problem #53 - Posted Tuesday, August 3, 1999
- Chickens 'n' Eggs (back to top)
- I reckon a chicken and a half can lay an egg and a half in a day and a half. Reckon these:
- a) How many eggs can ten chickens lay in ten days?
- b) How many days does it take ten chickens to lay ten eggs?
- c) How many chickens does it take to lay ten eggs in ten days?
- Problem #54 - Posted Saturday, August 14, 1999
- The Three Boards (back to top)
- Craig was remodeling a house and found three perfectly square boards (not pictured).
- He noticed that one of them was five square feet bigger than the smallest, and was
- the same amount smaller than the biggest. To his surprise, each of the three boards
- measured a whole number of inches on a side! How big were the three boards?
- Problem #55 - Posted Friday, August 27, 1999
- The Millionaire! (back to top)
- Eddie, the eccentric rich guy, wants to give away one MILLION dollars.
- Eddie has two quirks: (1) he gives each person either $1, $7, $49, or some power
- of seven, and (2) he won't give more than six people the same size gift.
- How many people received money, and exactly how did he distribute his fortune?
- Problem #56 - Posted Sunday, September 5, 1999
`First problem of the 1999-2000 contest!`- Subtraction Diamond (back to top)
- Put four different whole numbers from 1 to 49, in the corners, then subtract the smaller from the larger and put the answers in between. Keep doing this until all four numbers become equal. How many steps can you make it last? The demo is "(14, 30, 18, 37) lasts 3 steps." The winner is the contestant with the first correct longest-lasting entry.

- Problem #57 - Posted Tuesday, September 14, 1999
- Close to the Line (back to top)
- The picture at the right shows a line of slope -/2 (square root of 2), through the origin, that passes "close" to the point (2,3). This means that the slope 3/2 is close to -/2. We define "m/n is really close to -/2" to mean that its distance from -/2 is less than 1/n^2. (In our example, |3/2 - -/2| := 1.500-1.414 := 0.086 < 1/4, so 3/2 is "really close.") What are all fractions m/n, with 0<m+n<75, that are really close to -/2? m and n have to be integers. The winner is the first entry with the largest difference of (correct fractions) -- (incorrect fractions).

- Problem #58 - Posted Friday, September 24, 1999
- Y 2 Digits? (back to top)
- Why not? Figure out all of these whole numbers that you can; each one has two digits.
- A's double exceeds its half by 99. B is twice the product of its digits. C is thrice the sum of its digits. Half of D exceeds its third by the sum of its digits. E is increased by 20% if its digits are reversed. F can be squared by sandwiching in two more digits between its original two. G differs from its reverse by twice the product of its digits. The product of H's digits is twice their sum. And turn I upside down and you'll increase it by 12.The winner will be the first entry with the largest difference of (correct answers) -- (incorrect answers) that's sent in before the deadline.
- Problem #59 - Posted Monday, October 4, 1999
- Bicycle Shorts (back to top)
- 'Bikeman' wants to tour five cities in one day: Anamoose, Brindle, Catfish, Danville, & Easton; located at (1,0), (2,3), (-2,4), (5,-1), and (-3,-2), respectively. He wants to see all cities (not nec. in that order) and travels from town to town in straight lines. What is the shortest total distance possible, where should he start, and what is the sequence of cities he visits? (The winner is the first entry with the shortest distance submitted before the deadline. Exact answer or at least to four decimal places.)
- Problem #60 - Posted Wednesday, October 13, 1999
- Odd Combinations! (back to top)
- You have one each of the numbers 1, 3, 5, 7, 9. Using just basic operations
- +, -, *, /, and ( ), 'stitch together' as many answers from 1 to 10 as you can.
- (Example: 1 - (5 - 3) + (9 + 7) = 15 , but that's not between 1 and 10.) The winner will be
- the first entry with the highest score, figured by (correct answers) - (incorrect answers).
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