Wednesday, November 01, 2000

Puzzles and mathematics

  1. Three people decided to eat some ensaymada together. Andy brought 3 pieces, Belen 5 pieces, and Carlos none. All the ensaymada were identical. And all were eaten, each person eating an equal amount. Because Carlos had not brought any ensaymada he calculated his share (1/3 of expenses) and contributed 4 pesos. How should Andy and Belen divide the 4 pesos?

  2. The sum of an odd number of consecutive odd numbers beginning from 1 to N. Show that the sum of the first and last numbers is 2/√N.

  3. Twenty-one identical soft drink cases are to be loaded onto three carts. Seven of the cases are empty, 7 of the cases are full of softdrink bottles, and 7 are half full of soft drink bottles. The soft drink bottles are all identical, and therefore have equal weights. How can all these be loaded so that the weights are equally distributed among the three carts and without transferring any of the bottles from one case to another?

  4. A rectangular sheet of paper 30 cm x 40 cm is folded so that one corner is placed on the diagonally opposite corner as in the figure below. How long is the resulting fold?

  5. The number 30 can be expressed as the sum of one or more consecutive positive integers in these 4 ways:

    30; 6+7+8+9; 9+10+11; 4+5+6+7+8;

    Express 240 as sum of consecutive positive integers in as many ways as possible.

  6. In the figure the triangles OAB and OPQ are similar with angles A and P congruent. If OA/OQ = 3 and OB/OP = 2, then AB/PQ = ?

  7. A, B, C, D, E, F, G, H are the vertices (in order) of a regular octagon. The diagonals AD and BH cross at I. How large is angle BID?

  8. From a large cardboard circle four touching circles were cut out as shown. The two larger circles are congruent, and the two smaller circles are congruent. After cutting out the four circles, what part of the cardboard was left?

  9. Let A be a two-digit integer and let B be a two-digit integer whose digits are the same as those of A but in the reverse order. Find A so that A2-B2 is a perfect square.

  10. A small square is cut out from the corner of a large square, leaving a L-shape. Given that the side lengths of both the squares are whole numbers in centimeters, and that the L-shape has area 60 cm2, how many possible values are there for the area of the original large square?

  11. If x is a real number such that x2+2x-7>0, show that (x2+34x-71)/(x2+2x-7)≤5.

  12. Show that the product of any three consecutive integers is divisible by 3.

  13. If a and b are integers and b is odd, show that x2+2ax+2b=0 has no rational root.

  14. Determine the radius of the largest circle that can be drawn inside a quarter-circle of radius r.

  15. In the figure, points C and F are on sides AD and EG respectively. Show that that area of parallelogram ADFB is equal to that of parallelogram BCGE.


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