- problem #241 - posted
sunday, feb
19, 2006
- Local (loco)motion (back to top)
- Dan goes 49 miles in 9 hours; a mixture of running & walking.
- Can you tell how fast he runs and how fast he walks in each (i)-(v)?
- (i) Dan runs twice as fast as he walks . . . . . (ii) he runs 4 mph faster than he walks
- (iii) he runs 7 mph and he walks 3 mph . . . (iv) he runs 5 mph and he walks 4 mph
- (v) he runs 5 min, walks 1 min, repeats
- a) Which of these have unique solutions? What are they?
- b) Which of the above can not occur? Why not, exactly?
- c) Give a range of solutions, for any that are not unique. . . . Show your reasoning.
- Local (loco)motion (back to top)
- problem #242
(back to top)
- posted tuesday, march 7, 2006
- : Put a Hex around You :
- The picture shows all possible
- equiangular hexagons with integer
- sides, having perimeter 12 units
- (up to congruence).
- How many such hexagons
- are there with perimeter:
- a) 18 ; b) 19 ; c) 20 ?
- List each according to ordered sides
- (e.g. 1,1,4,1,1,4); no need for pictures!
- Show your steps and reasoning.
- posted tuesday, march 7, 2006
- problem #242
(back to top)
- problem #243 - posted
tuesday, march
28, 2006 (back
to top)
- Altitudes with Attitude! (thanks to Ed Wern for the title; I forgot one! - Dan)
- Call the altitudes of a triangle: h , k , m ;
- and the radius of its inscribed circle: r .
- Find the minimum value of the expression
- (h + k + m) / r (over all triangles).
- Which triangle achieves this minimum?
- Show your steps and reasoning.
- and the radius of its inscribed circle: r .
- Altitudes with Attitude! (thanks to Ed Wern for the title; I forgot one! - Dan)
-
- problem #244 - posted saturday, april 8, 2006
- Half More Number! (back to top)
- Find the smallest positive integer such that if the ones digit is moved (from the right) all
- the way to the left, the resulting number is exactly 50% more than the original number.
- Show your steps and reasoning.
- problem #244 - posted saturday, april 8, 2006
- problem #245 - posted
friday, april
28, 2006.
- Sphere of Influence (back to top)
- Let P1 , P2 , . . . , Pn be points on a sphere of radius 1 .Prove that the sum of the squares of
- the distances (in R3) between all pairs of points is at most n^2, and state conditions for this sum
- to be equal to n^2. Show your steps and reasoning.
- Sphere of Influence (back to top)
-
- problem #246 - posted saturday, may 20, 2006.
- Beating the Cycle! (back to top)
- Suppose we have nine cards, numbered 1-9 (or Ace, 2, . . . , 9) separated into three piles
- X, Y, and Z, of three cards each,such that : pile X beats pile Y, Y beats Z, and Z beats X.
- We say "pile X beats pile Y" if a random card from X has a bigger number than a random
- card from Y, most of the time. How is it possible to arrange the cards into piles like this?
- Show your steps and reasoning.
- problem #246 - posted saturday, may 20, 2006.
- problem #247 - posted
monday, may
29, 2006
- I Am the Bugman! (back to top)
- "Bugs" can have six legs (insects), eight legs (spiders), or ten legs (decipedes).
- (a) Is it possible to find a set of bugs that has any even number of legs > 4?
- (b) How many sets of bugs have 100 legs total?
- (c) How many bug collections are possible with 100 bugs and 800 legs?
- List or describe b) and c), giving the number of insects, spiders, and decipedes.
- Show your steps and reasoning.
- I Am the Bugman! (back to top)
-
- problem #248 - posted
tuesday, july
4, 2006
- How Many Runners? (back to top)
- A 'number' of runners entered a recent road race.
- These are two true statements regarding the 'number':
- a) It has 3 distinct digits. . . b) If you add 99 the number reverses.
- Here are four other statements regarding the 'number':
- 1) It is divisible by the sum of its digits . . . 2) It is not prime. . .
- 3) It has only one common digit with the product of its digits. . .
4) The sum of the first and last digit is one more than the middle digit- If I told you which of these 'other' statement(s) were false, you'd be
- able to determine the number. How many runners entered the race?
- Show your steps and reasoning.
- How Many Runners? (back to top)
that's me (middle)
- problem #249 - posted
saturday, july
22, 2006
- Eat a Peach (a Mile) (back
to top)
- A truck driver has 3000 peaches in Ashland and wants to move them to
- Blythe, 1000 miles away by freeway. The truck can hold up to 1000 peaches,
- but the driver has an addiction: if there are any peaches in the truck, the
- driver will eat them at a rate of one peach per mile. Give a plan to deliver
- the most peaches possible to Blythe. (Hint: There is safe fruit storage at all
- points along the way.) Show your steps and reasoning.
- A truck driver has 3000 peaches in Ashland and wants to move them to
the cargo 
and the fuel
- problem #250 - posted
thursday, august 10,
2006
- Pack 'Em All In ! (back to top)
- Shirley needs to ship off 250 small rectangular boxes that each measure 1" by 1" by 4" (inches).
- (a) What are the dimensions and surface area of the rectangular packing box of smallest
- surface area, if there can be no empty space in the (big) box? (b) Does this answer change
- if we allow empty space in the box? If so, give the dimensions and area of this smallest-area box.
- Show your steps and reasoning.
- Pack 'Em All In ! (back to top)
