dan's math@home - problem of the week - archives
 
 
Problem Archives page 22
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 . 151-160 . 161-170 . 171-180
181-190 . 191-200 . 201-210 . 211-220 . 221-230 . 231-240 . 241-250 . 251+ index
 
211-Cogito Log-o-Sum
212 - C h e c k , P l e a s e
213 - Island Hopping ...
214 - The Four Means !
215 - Put A Hex On It !
216 - Band in Boston ?
217- Baskets of Apples
218- Holy Sphere+circle
219 -- See Double Free
220 - - Cube in a Cone!
 
 
Problem #211 - Posted Wednesday, September 1, 2004
Next-to-last problem of 2003/04 = New season starts with Problem 213
Cogito Log-o-Sum ! "I think about logs, so I exist !" (back to top)
Find the smallest value of the sum S; prove your answer is as small as possible.

S = loga(bc) + logb(ac) + logc(ab)

 
Problem #212 - Posted Monday, September 13, 2004
= Last problem of 2003/04 = New season starts with Problem 213 =
Check, Please ! (back to top)
At the bank Hank cashed a check but the teller gave him cents
instead of dollars, and dollars instead of cents. After buying a 5-
cent piece of gum, Hank had exactly twice as much as his original
check. How much was the check? Show reasoning clearly.
 
 
Problem #213 - Posted Thursday, September 23, 2004
= First problem of 2004/05 = Beginning of my eighth year! =
Island Hopping (back to top)
Three nearby islands: A, B, C, are isolated from the rest of the world.
Every year, 5% of the people of island A move to island B, 5% from
A to C, 15% from B to A, 10% from B to C, 10% from C to A, and
5% from C to B. In the long run, what fraction(s) of the population
will live on each island? Show reasoning clearly.
 

Problem #214 - Posted Monday, October 4, 2004
The Four Means ! (back to top)
Given two positive real numbers a and b ,here are four ways of
computing the "mean":
am = arithmetic mean = (a + b) / 2
gm = geometric mean = [a b]
hm = harmonic mean = 2 / (1/a + 1/b)
rms = root mean square =[(a^2 + b^2) / 2]
a) Rank these from smallest to largest, proving your result. If one type
is not always less than another, say when it's equal and when it's more.
b) Use the diagram to compare at least three of the means using the
lengths of segments. Assume E is at the center of the semicircle.
Show reasoning clearly.
Which to use, and how?

 
Problem #215 - Posted Saturday, October 23, 2004
Put A Hex On It ! (back to top)
Find the side length and area of the largest regular hexagon that
can be inscribed in a square of side 1 meter. Prove your answer.
Show reasoning clearly. Simple gif attachments ok, or just describe it!
 
Problem #216 - Posted Saturday, November 6, 2004
Band in Boston ? (back to top)
Red Sox, Patriots, John Kerry, lots to celebrate ! A marching band goes up
Main Street in single file. Then one band member sits down, and the rest of
them march on in twos. One more sits and the rest march in threes, and so on,
until one sits and the rest march in rows of ten. What is the smallest positive
number of members that could have started this big band march on Boston?
Show reasoning clearly. (I've been told there's no Main St. in Boston; sorry Sox fans!)
 

 

Problem #217 - Posted Thursday, December 2, 2004
Baskets of Apples ! (back to top)
I walked by a holiday display, and noticed there were nine straw baskets,
each one having a whole number of apples, at most 9, and possibly none.
Then I also noticed the mean number of apples was 4, the median was 4,
and the mode was 2. Is this possible, and if so, how many solutions are there?
(not counting the positions of the baskets) Show reasoning clearly.

 

 
Problem #218 - Posted Friday, December 17, 2004
Holy Sphere (and circle) (answer both) (back to top)
 
a) A hole is drilled through a wooden sphere (top). If the
hole is 6 cm long, find the volume of wood that remains.
 
b) A chord of a circle is tangent to an inner, concentric
circle (bot). If the chord is 6 cm long, find the area of the ring.
 
Show reasoning clearly.
 
Problem #219 - Posted Monday, December 27, 2004
See Double Free (back to top)
Call a set of numbers "double-free" if there is no pair {m, 2m} in the set.
For example, {1, 3, 4, 5} is double-free. Find the size of the largest double-free
subset of {1, 2, 3, . . . , 2^n} for each n = 3, 4, 5, 6, 7, and 8;
give actual maximal sets for n = 3, 4, 5. Show reasoning clearly.
 
Problem #220 - Posted Sunday, January 9, 2005
Cube in a Cone (back to top)
A cone has a circular base of radius 1 ft, and vertex at height 3 ft directly
above the center of the circle. A cube has four vertices on the base and
four on the cone's lateral side. a) What length is a side of the cube?
b) What cone height (r = 1) makes the cube side 1 ft? Show reasoning clearly.

 
 
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YOU CAN ALWAYS FIND ME AT dansmath.com - Dan the Man Bach - 2005 A.D.
 
 
Problem Archives Index
 
Probs & answers . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100
Problems only . . . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100
Probs & answers . 101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180
Problems only . . . 101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180
Probs & answers . 181-190 . 191-200 . 201-210 . 211-220 . 221-230 . 231-240 . 241-250 . 251+
Problems only . . . . 181-190 . 191-200 . 201-210 . 211-220 . 221-230 . 231-240 . 241-250 . 251+
 
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