**dan's math@home - problem of the week - archives****Problem Archives**page 15**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 index141 Where's the Road 142 - - Strike A Chord 143- Easy as A-B-C-D 144 - - - Click to 222 ! 145 - Up The Volume! 146 - Maxes & Means 147 - Circle the Roots! 148 - - Weird Bases !! 149 - Indep'd'ce Events 150 Hpy Bday Hermen - Problem #141 - Posted Monday, February 18, 2002
- Where's The #%@ Road ? (back to top)
- After parachuting into the woods at night, you land right next to a sign that
- says : "Road: One Mile." But it has fallen over, and there's no way to know
- what direction the road is. Describe the shortest path you might take that will
- guarantee reaching (touching) the road. (You can't see the road until you reach it!)
- What's the maximum distance you might walk on your path?

**ROAD : 1 mi**

- Problem #142 - Posted Wednesday, March 13, 2002
- Strike a Chord ! (back to top)
- The circle at right has some interesting properties:
- 1) The chords are at right angles (perpendicular),
- 2) Red chord is one inch longer than green chord,
- 3) The values of a, b, c, d are given by clues below.
- What is the area of the circle, in square inches?
- Clues for a, b, c, d : a d = b c (true in any chord situation)
- Length of a = average of b and d ; (a+b+1)(a+b-1) = bad

- Problem #143 - Posted Sunday, March 24, 2002
- Easy as A-B-C-D ! You can decide how easy these are (or aren't): (back to top)
- A is the number of real solutions to the equation (x^2 -- 9x + 19)^(2x^3 -- x^2 -- 10x) = 1
- B and C are such that the x solving the equation 3^(2 -- x) = 2^(3x -- 1) is log(base B) of C
- D is the number of ways A+B+C can be written as a sum of increasing positive integers
- What is the value of A + B * C ^ D ? . . . . . Please explain reasoning!
- Problem #144 - Posted Monday, April 8, 2002
- Click to 222 ! (not factorial, not 'Room 222') (back to top)
- You start at 0, so does k . Your goal is to get to 222 (two hundred twenty-two).
- There are three buttons you can click ; one will increase the value of
- k by 1, another decreases k by 1, the third will ADD k to your total.
- (For example the sequence [incr][ADD][incr][incr][ADD] gives 1+3=4 in 5 clicks.)
- How can you reach your goal in the fewest number of clicks?
- Ties with same number of clicks will be won by fewest [decr] then fewest [incr].

**[incr k]****[decr k]****[ADD k]**Click any button!

- Problem #145 - Posted Tuesday, April 30, 2002
- Up The Volume! (back to top)
- Mia needed to calculate the volume of a rectangular room. She multiplied the length and
- the width correctly but the width had been incorrectly written down, it was one-third
- larger than it should have been. To make up for this, she lowered the height by one-third,
- then multiplied it on. She figured this was okay since the width was equal to the height.
- She then found her volume was off by 20 cubic meters. Why was Mia wrong, and what
- was the actual volume?
- Problem #146 - Posted Wednesday, May 15, 2002
- Maxes and Means? (back to top)
- I have a list of positive integers; not necessarily distinct. The number 68 appears in the list.
- The arithmetic mean (average) of the list is 56; but if the 68 is removed, the average drops
- to 55. What is the largest number that could have appeared in my list?
- Problem #147 - Posted Tuesday, June 4, 2002
- Circle the Roots ! - - Three problems about circles, squares, and angles. (back to top)
- #1. A semicircular arched doorway is 6 ft high, 1 ft from the edge.
- What's the height of the arch 5 ft from the edge (above the floor)?
- #2. The square of the diagonal of square ABCD is 8. Point P on side BC makes the ratio
- of PC to PB equal to 3. If a circle passes through A, P, and the center of the square,
- how far is the center of the circle from the point D?
- #3. How much is the quantity : sin[arcsec(17/8) - arctan(-2/3)] ?
- Give each answer in simplest form (A *B ) / C , with B square-free, and find the product
- of these three answers, also written in this form.
- Problem #148 - Posted Saturday, June 22, 2002
- Weird Bases ! (back to top)
- Our base 10 system has columns for 1's, 10's, 100's places, etc.;
- digits can be 0, 1, ..., 9 in each. What if we use other systems?
- A. Factorial Base: Columns 1, 2, 6, 24, etc; digits in k! col. can be 0, 1, ..., k.
- B. Fibonacci Base: Columns 1, 2, 3, 5, 8, etc. Digits can only be 0 or 1 each.
- C. Square Base: Cols 1, 4, 9, 16, etc. Digits 0-3 in 1's, 0-2 in 4's, 0-1 others.
- (1) Express one million (uniquely) in factorial base.
- (2) Express one hundred in as many ways as possible for the other two systems.
- Problem #149 - Posted Thursday, July 4, 2002 (time's up on this one)
- Independence Events? (back to top)
- Three sisters : Venus , Serena , and the well-hidden little Cathy,
- play one tennis opponent per week. The chances of each sister
- beating any unrelated player are 7/10, 8/10, and 9/10, respectively.
- The probabilities of any sister beating another are given in the
- table at right (for example Venus beats Cathy 7/10 of the time).
- Each week, one pair of sisters plays each other (at random) ;
- the third plays an outsider. In a 50-match season, which
- player is likely to win the most matches, and how many
- matches would each one expect to win, on the average?
- Please explain your steps and reasoning; resubmissions carry a 1-pt penalty.

**\****V****S****C****V****.0****.6****.7****S****.4****.0****.9****C****.3****.1****.0**

- Problem #150 - Posted Sunday, July 28, 2002
- Happy Birthday Hermen ! (back to top)
- Hermen knows how old he is turning this birthday; you don't. He is as many years old as
- the largest number of divisors of any integer N less than or equal to 20,000 (twenty thousand).
- How old is Hermen turning, and what's the smallest such N? (Bonus point for giving the three
- next-smallest N's with the same no. of divisors.)
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