**dan's math@home - problem of the week - archives****Problem Archives**page 1**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 index1- The Most Divisors 2 - Trains versus Fly! 3 - Unit fractions 1/n 4 - The Census Taker 5 - Square root probs 6- Buying house nos. 7 - Name that pattern! 8 - Clink wine glasses 9 - Strange Powers !! 10- Odd & Abundant? - Problem #1
- The Most Divisors (back to top)
- A
**divisor**of a positive whole number n is a whole number that divides evenly into n - (with no remainder).What integer from 1 through 1000 has the most divisors?
- To win you must prove it and explain your method.
- Problem #2 - Posted Friday, November 28, 1997
- Trains versus Fly (back to top)
- Two trains are 2 miles apart and are traveling towards each other on the same
- track, each train going 30 mph. A fly going 60 mph starts at the nose of one train,
- flies toward the other train, and upon reaching the second train immediately turns
- around and flies back towards the first train. The fly buzzes back and forth until all
- three collide. How far did the fly fly?
- Problem #3 - Posted Sunday, December 7, 1997
- Unit Fraction Problems (back to top)
- A
**unit fraction**is of the form 1 / n (where n = whole no. >= 1.) - a) Starting with 1 / 1 , then 1 / 2 , 1 / 3 , etc., how many unit fractions
- does it take to add up to more than pi ?
- b) Express 3 / 23 as the sum of two unit fractions; 3 / 23 = 1 / a + 1 / b .
- c) Write 5 / 4 as the sum of distinct unit fractions. Fine print . . .
- The winner in part c) will be the one using the smallest number of fractions.
- In case of a tie, the winner is the one with the "smallest biggest denominator."
- Problem #4 - Posted Thursday, December 18, 1997
- The Census Taker Problem (back to top)
- A census-taker rings Mr. Simpson's bell and asks how many children he has.
- "Three daughters," he replies.
- "And how old are they, in whole numbers?" asks the census-taker.
- "Well, I'll tell you this: the product of their ages is 72, and the sum of their ages is my house number."
- "But that isn't enough information!" complains the census taker.
- "Okay, my oldest daughter (in years) likes chocolate milk," replies Mr. Simpson.
- With that, the census-taker nods and writes down the three ages.
- How old are the Simpson girls, and how did the census-taker figure it out?
- Include a full explanation!
- Problem #5 - Posted Monday, December 29, 1997
- A Tree-o of Square Root Problems (back to top)
- Let
**Sqrt(x) or -/x**denote the (positive) square root of x, - as in
**Sqrt(100) = -/100 = 10**. - Also
**x ^ 2**will mean x squared, as in**10 ^ 2 =**10 * 10 =**100**. **1)**. . If:**Sqrt(m) + Sqrt(n) = 13**, and**m and n differ by 65**,- what is the
**largest possible value of m**? **2)**. . Notice that the equation**x^2**-**3 = 0**has a**solution x = Sqrt(3)**.- Find a polynomial equation
in
**x**, with integer coefficients, having **x = Sqrt(3) + Sqrt(5)**as a**solution**.**3)**. . What is a really good**fraction**approximation for**Sqrt(17)**,- and why? Generalize your answer if possible to Sqrt(n^2 + 1).
- Problem #6 - Posted Wednesday, January 7, 1998
- New Year, More House Numbers! (back to top)
- The people living on Sesame Street all decide to buy new house numbers,
- so they line up at the store in order of their addresses: 1, 2, 3, . . . .
- If the store has 100 of each digit, what is the first address that
- won't be able to buy its house numbers?
- Problem #7 - Posted Friday, January 23, 1998
- Patterns and Sequences (back to top)
- a) 2, 3, 5, 8, 13, _?_
- b) 2, 3, 5, 7, 11, _?_
- c) 3, 3, 5, 4, 4, 3, 5, 5, 4, _?_
- d) 1, 3, 7, 15, 31, _?_
- e) 1, 4, 27, 256, 3125, _?_
- f) 1, 2, 6, 24, 120, 720, _?_
- g) 1, 2, 6, 30, 210, _?_
- What number comes next in each sequence? Give reasons!
- Problem #8 - Posted Friday, January 30, 1998
- Clinking Wine Glasses
- When I have wine with a few people and we clink glasses and say "salud", I can always
- tell if everyone has "clinked" with everyone else, because I know math! Let's assume
- each person clinks each other person exactly once. If there are 2 people, there is one
- "clink." If there are 3 people, there are 3 clinks.
- 8a. How many clinks are there for 4, 5, 6, . . . 10 people?
- 8b. How many people were there if I heard 903 clinks?
- 8c. What is the formula for the quantity : c(n) = number of clinks for a group of n people ?
- Problem #9 - Posted Sunday, February 15, 1998
- Strange Powers
- The expression b ^ n means b to the power n.
- 9a) If 3 ^ a = 4 and 4 ^ b = 8, what is 9 ^ (a - b) ?
- 9b) Can you find numbers a =/= b such that a ^ b = b ^ a ?
- 9c) If a $ b means a ^ b - b ^ a , what is 4 $ 6 ?
- 9d) Is 2 $ (3 $ 4) the same as (2 $ 3) $ 4 ?
- Problem #10 -
**Posted Saturday, February 28, 1998** - Odd and Abundant?
- A natural number is
**abundant**if its proper divisors (not including itself) add up to more than - the number. (12 is abundant because the divisors of 12 are 1, 2, 3, 4, 6, and 12, and 1 + 2 + 3 + 4 + 6 = 16 > 12.)
- Note 6 = 1 + 2 + 3 ; 6 is
called
**perfect**; 15 is**deficient**(not abundant or perfect): 1 + 3 + 5 = 9 < 15. - What is the smallest odd abundant number? Prove your answer and spell out your thinking.
- Hint: Numbers like 12 = 2 * 2 * 3 . that have lots of small prime factors, tend to be abundant.
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