Now that summer is starting, the team is helping kids on break from the local high school stay sharp for next year’s contests. Here are some of the practice problems they use.
You have an unlimited supply of stamps in each denomination from eight cents to the number of cents equal to the answer to last week’s Duplicate Division problem. However, there is an unusual rule when you apply postage to a package: the number of stamps of a given denomination must be the denomination of a different kind of stamp on the package, and the denomination of every stamp must be the number of stamps of some different denomination. For example, the postage on a package could consist of nine eight-cent stamps and eight nine-cent stamps
What is the maximum number of cents of postage you can apply to a single package?
Figure P is a spiral that starts at the point (1,0) and spirals outward clockwise making eight revolutions around the origin to reach the point (10,0). Figure R is a different spiral that starts at (1,0) and spirals outward counterclockwise making seven revolutions around the origin to reach (10,0).
Not counting either (1,0) or (10,0), what is the maximum number of times that figures P and R can intersect?
Solution to week 38
Phenomenal Packing. As you can see in the diagram, Ariana’s boxes have sizes (in decimeters) 1, 1, 2, 2, 2, 3, 3, 4, 6, 6, and 7.
The answer to Duplicate Division will be revealed in two weeks, when the relay is over.
Week 38: Phenomenal Packing & Duplicate Division, solutions to Hat Trick, Possible Parabolas, Equal Laterals, & Isocircules
Week 37: Equal Laterals & Isocircules, solution to Circle Crossing
Week 36: Possible Parabolas & Circle Crossing, solution to Heart Trick
Week 35: Hat Trick & Heart Trick, solutions to Test Test, Simultaneous Remainders, Different Dice, & No-Dispute Knights
Week 34: Different Dice & No-Dispute Knights, solution to Subtle Sequences
Week 33: Simultaneous Remainders & Subtle Sequences, solution to Late Show
Week 32: Test Test & Late Show, solutions to Talking Midpoints, Split Vote, Dream Sequence, & Alien Arithmetic
Links to all of the puzzles and solutions are on the Complete Varsity Math page.
Come back next week for answers and more puzzles.