Fryer Contest 2019年真题
1. (a) A rectangle with dimenions 7 by 8 is shown in Figure A. What is the perimeter of this figure?
(b) A 3 by 1 rectangle is removed from one corner of a 7 by 8 rectangle, as shown in Figure B. What is the perimeter of this figure?
(c) A 4 by 2 rectangle is removed from one corner of a k + 4 by k + 2 rectangle, as shown in Figure C. Suppose that the perimeter of Figure C is 56. Determine the value of the integer k.
(d) Four 4 by 7 rectangles are removed from the corners of a square having side length 8n + 1, as shown in Figure D. Determine the largest integer n for which the perimeter of Figure D is less than 1000.
2. Rope is fed into a machine at a constant rate of 2 metres per second. The machine can be set to cut off one piece of rope every t seconds for various values of t. For example, if
the machine is set to make one cut every 5 seconds, then 12 pieces of rope are cut off in 1 minute.
(a) If the machine is set to make one cut every 8 seconds, how many pieces of rope are cut off in 10 minutes?
(b) If the machine is set to make one cut every 3 seconds, what is the length of each piece of rope that is cut off?
(c) If each piece of rope that is cut off is 30 m long, determine the number of cuts per minute that the machine is set to make.
(d) If the machine is set to make 16 cuts per minute, determine the length of each piece of rope that is cut off.
3. Tania lists the positive integers, in order, leaving out the integers that are multiples of 5.
Her resulting list is
1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, . . . .
(a) How many integers has Tania listed just before she leaves out the 6th multiple of 5?
(b) If the kth integer in Tania’s list is 2019, determine the value of k.
(c) Determine the sum of the first 200 integers in Tania’s list.
4. A Shonk sequence is a sequence of positive integers in which
• each term after the fifirst is greater than the previous term, and
• the product of all terms is a perfect square.
For example: 2, 6, 27 is a Shonk sequence since 6 > 2 and 27 > 6 and 2 × 6 × 27 = 324 = 182.
(a) If 12, x, 24 is a Shonk sequence, what is the value of x?
(b) If 28, y, z, 65 is a Shonk sequence, what are the values of y and z?
(c) Determine the length of the longest Shonk sequence, each of whose terms is an integer between 1 and 12, inclusive. This means that your solution should include an example of a sequence of this longest length, as well as justification as to why no longer sequence is possible.
(d) A sequence of four terms a, b, c, d is called a super-duper-Shonkolistic sequence (SDSS) exactly when each of a, b, c, d and a, b, c and b, c, d is a Shonk sequence.
Determine the number of pairs (m, n) such that m, 1176, n,
