dan's math@home - problem of the week - archives
Problem Archives page 15

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

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
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
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
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
#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
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)
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?
 \ V S C V .0 .6 .7 S .4 .0 .9 C .3 .1 .0

Problem #150 - Posted Sunday, July 28, 2002
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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Problem Archives 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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