Fryer Contest 2021年真题

1. A company sells rectangular business cards. Each card has dimensions 5 cm × 9 cm. Cards are printed on a page and then the page is cut to produce the individual cards.

(a) What is the area of each business card in cm2?

(b) Several business cards are printed without overlapping on a single 20 cm × 27 cm page. If the entire page is used with no waste, how many business cards are printed?

l The portrait page layout is printed so that every card is positioned with its 5 cm edges parallel to the 19 cm edges of the page.

l The landscape page layout is printed so that every card is positioned with its 5 cm edges parallel to the 29 cm edges of the page.

Which of these two page layouts allows the greatest number of business cards from a single page?

2. Franklin and Giizhig travel from their school to their own homes each day. The school is located at O(0, 0). Franklin’s home is at F(240, 100) and Giizhig’s home is at G(240, 180). The straight paths from their school to each of their homes are shown on the graph. (Throughout this problem, all coordinates represent lengths in metres.)

(a) What is the distance, in metres, along the straight path from the school to Franklin’s home?

(b) On Monday, Franklin walks at a constant speed of 80 m/min. How many minutes does it take Franklin to walk from school straight to his home?

(c) On Tuesday, Franklin and Giizhig leave school at the same time. Franklin walks at 80 m/min straight to his own home and then immediately turns and walks straight toward Giizhig’s home. Giizhig walks at g m/min straight to her own home and then immediately turns and walks straight toward Franklin’s home. If they meet exactly halfway between their homes, what is the value of g?

3. Given a list of six numbers, the Reverse Operation, Rn, reverses the order of the first n numbers in the list, where n is an integer and 2 ≤ n ≤ 6. For example, if the list is 1, 4, 6, 2, 3, 5, then after performing R4 the list becomes 2, 6, 4, 1, 3, 5.

(a) R3 is performed on the list 5, 2, 3, 1, 4, 6. What is the new list?

(b) A Reverse Operation is performed on the list 1, 2, 3, 4, 5, 6. A second Reverse Operation is performed on the resulting list to give the final list 3, 4, 2, 1, 5, 6. Which two Reverse Operations were performed and in what order were they performed?

(c) Suppose that m is the minimum number of Reverse Operations that need to be performed, in order, on the list 1, 2, 3, 4, 5, 6 so that 3 ends up in the last position (that is, the list takes the form , , , , , 3). The value of m can be determined by answering (i) and (ii), below.

(i) Find m Reverse Operations and show that after performing them, the desired result is achieved (that 3 ends up in the last position).

(ii) Explain why performing fewer than m Reverse Operations can never achieve the desired result.

(d) Determine the minimum number of Reverse Operations that need to be performed, in order, on the list 1, 2, 3, 4, 5, 6 so that the last number in the list is 4 and the second last number in the list is 5 (that is, the list takes the form , , , , 5, 4).

4. An SF path starts at S, follows along the edges of the squares, never visits any vertex more than once, and finishes at F. An example of an SF path is shown. (A vertex is a point where two or more of the squares’ edges meet.)