dan's math@home - problem of the week - archives
 
 
Problem Archives page 27
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 . 251-260 . 261-270 . 271+ . index
 
261 - - Here's To Toast !
262 -- Five Running Ten
263 - Cosine My Loan ?
264 Cubes&Sqrs 2gether
265 - Four Mini-Pollies!
266- Those Factorialoots
267- Squareful Numbers
268 Nine Tangled Wires
269 AnotherBrick inWall
270-SpheresFillingSpace
 
problem #261 - posted wednesday, february 21, 2007
Here's To Toast ! (back to top)
My old toaster can hold at most two slices of bread and toast them on one side
at a time. I want my bread toasted on both sides and buttered on one side.
It takes 3 seconds to put each slice into the toaster, 30 sec to toast one side of
1 or 2 slices, 3 sec to reverse a slice, and 12 sec to butter one (that side must
already be toasted). Assume only one of these operations can be done at one time.
In how short a time can I toast and butter three slices? Show your steps and reasoning.
 

 

problem #262 - posted tuesday, march 6, 2007
Five..... Running Ten ! (back to top)
There were five runners in this year's Waterfront 10, a ten-mile "out-and-back" race, with
a turn-around at the halfway point. Each runner ran their whole race at a constant speed.
Annie reached the 3-mile mark 6 min after the winner, but before at least 1 other runner.
Ben got to the turnaround a full 7 minutes after the third-place runner did.
Carla ran each of the 10 miles of the race in 12 seconds less than 8 minutes.
Dan passed the winner 42 min into the race, then at least 1 other runner before turning around.
Emo had run 5 5/9 miles when he passed the last runner, and had met runners in both
directions already. What was the order of finish, and what was each runner's total ten-mile
time, in minutes? Show your steps and reasoning.
 

 

problem #263 - posted sunday, april 1, 2007

 

Cosine My Loan ? (back to top) Here are a couple of cute trig problems for you; no foolin'!
a) Prove: cos(pi/7) - cos(2pi/7) + cos(3pi/7) = 1/2.
b) Find all real solutions to this transcendental equation:
cos^2 (x) + cos^2 (2x) + cos^2 (3x) = 1.
Show your steps and reasoning.
 
 
problem #264 - posted sunday, april 22, 2007
Cubes & Squares Together (back to top)
Suppose we add a perfect square to a perfect cube, and the result is the next perfect cube.
a) Prove that the square root of the square must be the sum of two consecutive squares
b) Find the first two all-positive examples of this occurrence.
[Illegal trivial example: 0^3 +1^2 = 1^3; 1 = 0^2+1^2.] Show your steps and reasoning.
 
problem #265 - posted wednesday, may 23, 2007

 

Four Mini-Pollies ! (back to top)
The 'minimal polynomial' of a number z is the f(x) of smallest degree with integer coefficients
having z as a root. (For example the min poly for z = \/ 3 is f(x) = x^2 - 3.)
Find min polys for these "algebraic" numbers: a) z = \/ 7 + \/ 11 . .b) z = 2 \/ 3 + 6 i
c) z = sqrt[5] + cuberoot[2] . d) z = tan(18 deg). Show steps & reasoning.
 
 
problem #266 - posted friday, august 31, 2007
Those Factorialoots ! (back to top)
a) Which number is larger, S or T ? (Both a numerical comparison and calculator-free proof)
b) Which is larger, S/99 or T/100 ? (Give the numerical comparison, and a proof if you can)
c) Figure out the limit, as n --> oo, of
(oo is infinity. Try for numerical value and exact proof.) Show steps and reasoning.
 
problem #267 - posted thursday, september 27, 2007
Next-to-last problem of this season! 11th season starts with Problem 269.
Square-ful Numbers (back to top)
A natural number n is 'squareful' if, whenever a prime goes into n, the square
of that prime also goes into n.
a) Find all squareful numbers less than or = to 300.
b) Prove that n is squareful if and only if n is the product of a square and a cube.
c) Prove there are an infinite number of pairs of consecutive squareful numbers, like 8'n'9.
Show your steps and reasoning.
 
problem #268 - posted wednesday, october 10, 2007
Last problem of this season! Eleventh annual contest starts with Problem 269.
Nine Tangled Wires /\|\|/\|\ (back to top)
I was working on the circuits in my house, and found a bundle of nine identical wires,
going into the wall and up to the attic. In the attic were the nine wires going down to
the main floor. Suppose I can twist any number of wires together at either end, and I
have a circuit tester that lights when a loop is complete. How can I match up all nine
pairs of wire ends, using the smallest number of trips up to (and down from) the attic?
Assume the wires are insulated except for a few bare cm at the ends.
Wires cannot be pulled thru the wall or pipe. Show your steps and reasoning.
.
problem #269 - posted tuesday, october 23, 2007
First problem of the season, my eleventh annual contest! %;-} (back to top)
Another Brick In The Wall
 
Figure out the values of the 15 bricks
marked A, B, ..., N, O. Each value is the
sum of the two below it, like F = A + B.
 
Some values are given, L is 5 more than J.
 
Show steps and reasoning.
problem #270 - posted sunday, november 11, 2007
Second problem of my eleventh contest!
Spheres Filling Space (and vice-versa) (back to top)
There are three common unit-sphere packings in 3D space, listed in increasing density:
i) Stacked square-grid layers, ii) Stacked hexagonal layers, iii) Indented hexagonal layers.
a) Find the exact density of each packing, then round to 0.1%.
b) Fill in the main spaces for each packing i), ii), iii), with small spheres that are twice
as dense as the unit spheres.Find the size of the small spheres in each case, and then
rerank the total mass densities from lowest to highest, giving both the exact values
and the approximations to 0.1%. Show steps and reasoning.
 
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YOU CAN ALWAYS FIND ME AT dansmath.com - Dan the Man Bach - 2007 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 . 101-110
Problems only . . . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100 . 101-110
Probs & ans . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180 . 181-190 . 191-200
Probs only . . . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180 . 181-190 . 191-200
Problems & answers . . . . 201-210 . 211-220 . 221-230 . 231-240 . 241-250 . 251-260 . 261-270 . 271+
Problems only . . . . . . . . . . 201-210 . 211-220 . 221-230 . 231-240 . 241-250 . 251-260 . 261-270 . 271+
 
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