**dan's math@home - problem of the week - archives****Problem Archives**page 8**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 index71- Reciprocal Sums 72- Valentines 4 All! 73 - Divisible Dates? 74 - Four Centuries ! 75 - Work/Study Pts. 76 - The Triple Kiss! 77- Polynomial Road 78 RunForYourLife! 79 Open & Shut Case 80 Try Our Products - Problem #71 - Posted Friday, February 4, 2000
- Reciprocal Sums (back to top)
- If n is a natural number, the reciprocal of n, 1/n, is called a 'unit fraction.'
- Historically, numbers were expressed as sums of unit fractions, like
- 4/7 = 1/2 + 1/14 or 1 = 1/2 + 1/3 + 1/6.
- Write each of these as sums of distinct unit fractions (with denoms at least 2; read rules below):
**a) 3/23 . . . b) 14/15 . . . c) 7/11 . . . d) 11/7**- RULES: The winner will be the one using the smallest total number of fractions. In case of a tie, the winner is the
- one with the smallest sum of all denominators. You can use the same fraction for more than one of the numbers.
- Problem #72
Valentines 4 All! **In Miss Lizzy's . . . . .4th grade class, all****the kids gave each of the other kids a valentine.****The girls received a total of 798 valentines and****the boys got 684. How many kids were in****Miss Lizzy's class,how many girls****and how many boys?**Explain your reasoning carefully.

- Problem #73 - Posted Sunday, February 20, 2000
- Divisible Dates? (back to top)
- The posting date (Feb 20) of this problem can be written as 2|20|2000.
- Notice the 2 goes into the 20 (without remainder) and the 20 goes into 2000.
- a) What are all the days this year that have this "divisible date" (dd) property, how many
- dd's are there, and what percentage of the days of 2000 do they amount to (nearest 1/10 %)?
- b) In France they would name Feb 20th as 20|2|2000, not a "dd." But October 5 would be a French dd.
- How many French dd's are there this year? c) How many days are either dd, French dd, or both?
- Problem #74 - Posted Tuesday, February 29, 2000
- Four Centuries! (back to top)
- The posting date of this problem (Feb 29) is the 'leap day' in the only leap year that ends in 00 for the next 400 years. This is because of the rule of "a leap year every 4 years, except for every 100 years, except for every 400 years."
- a) Prove that February 29, 2400 will be on the same day of the week as Feb 29, 2000. (This says the days of the week follow a 400-year cycle, not 2800 years.)
- b) How many times, during the 400-year period from Jan 1, 2000 to Dec 31, 2399, is the "first of the month" on a Saturday, as was 1/1/2000? (There are 4800 months.)
- c) Which day (or days) of the week have the most "first days of the month," and how many "firsts" are on that day in a four-century period? Answer any or all the parts you can; please include your methods and reasoning.
- Problem #75 - Posted Wednesday, March 8, 2000
- Work/Study Points (back to top)
- You want to win a scholarship by compiling the most "work/study points."
- You get 3 points for each semester-unit of classes you take, and 4 points for
- each hour of work (per week) on your job.
- . . (1) You have to take at least 7 units,
- . . (2) Triple your units, plus your work-hours, is at most 75,
- . . (3) The sum of the squares of your units and your work-hours is at most 625.
- How should you plan your schedule (x units and y work-hours) to maximize your work/study points?
- Explain your strategy carefully, it's the essay question for your scholarship!
- Problem #76 - Posted Saturday, March 18, 2000
- The Triple Kiss (back to top)
- The term "kissing circles" means mutually tangent.
- If the circles at the right have radii 3, 4, and 5 cm,
- what is the area of the (green) shaded region between
- the three circles? Explain your procedure carefully;
- give your answer to the nearest thousandth of a sq cm.

- Problem #77 - Posted Monday, March 27, 2000
- Pacific Polynomial Road (back to top)
- In Problem 59, we tried to minimize the length of a zig-zag path that went through 5 cities.
- Now let's build a smooth curving road through 4 cities:
**L**os Angeles (3,4),**N**ewport Beach (5,1),**P**asadena (4,5),**S**anta Monica (2,3).- a) What's the smallest degree polynomial y=f(x) that will pass through all four cities? . . .
- b) What is the exact equation of this polynomial? (Hint: use fractions not decimals)
- c) Would this road go through
**C**hatsworth at (1,6)? (Actual approximate SoCal-ordinates!) - Problem #78 - Posted Wednesday, April 5, 2000
- Run For Your Life! (back to top)

- One day Dan was out bicycling. After entering a one-lane tunnel and riding one-fourth of the way through
- it, he glanced back over his shoulder and saw a truck approaching the tunnel entrance at 80 miles per hour!
- Doing the quick mental math, Dan realized that if he accelerated immediately to his top speed, he could just
- escape with his life, whichever direction he rode. What is Dan's top biking speed?
- Problem #79 - Posted Tuesday, April 18, 2000
- Open and Shut Case! (back to top)
- At the Metric Academy there are 100 students, and 100 lockers numbered 1-100. One day the lockers are all closed, and student #1 opens all of them. Then student #2 shuts every other locker, starting with #2, then #4, etc. Student #3 changes the state of every third one, starting with #3: opens it if it's shut, shuts it if it's open. Student #4 changes the state of every 4th locker, etc. This continues until the hundredth student changes the state of locker #100. Exactly which lockers are now open, and which are closed, after all the students have open and shut them as described? Show your steps and reasons.

- Problem #80 - Posted Sunday, April 30, 2000
- Try Our Products! (back to top)
- Answer all of these; they increase in difficulty.
- a) What is the maximum possible product of two whole numbers whose sum is 10? What are the numbers?
- b) What's the maximum product of any number of whole numbers whose sum is 10? What are they?
- c) What's the maximum product if the sum of the set of whole numbers is 20? What are the numbers?
- d) What's the maximum product of any number of whole numbers whose sum is 100? What are they?
- e) What's the maximum product of any whole number of real numbers whose sum is 100? What are they?
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