Fryer Contest 2020年真题

1. Annie (A), Bogdan (B), Cao (C),

Daniel (D), and Emily (E) are running

for student government positions.

Their campaign posters are printed by

different printing companies. The graph

to the right plots each student’s total

price of their first batch of posters versus

the number of posters in their first batch.

(a) Daniel pays $16.00 for 10 posters which is a price of $1.60 per poster. What does Cao pay per poster?

(b) Which two students are paying the same price per poster?

(c) In Daniel’s second batch, he needs to print 40 posters. He can pay $60.00 to print these 40 posters at his local library. To spend less money on the second batch, should Daniel print these 40 posters at the library, or should he continue to pay

the same price per poster using the company that printed his first batch?

(d) Annie’s printing company charged $10.00 for her first 5 posters. Her company will charge her a lower price per poster for additional posters. Emily’s printing company charges a fixed price per poster for any number of posters. By the end of the campaign, Annie and Emily each have printed 25 posters and have spent the same amount of money. What price is Annie charged per additional poster?

2. In the diagram, rectangle JKLM is drawn

with its vertices on the sides of ∆PQR so

that PJ = PK = 5 m, JQ = KR = 50 m,

KL = 40 m, and QR = 66 m, as shown.

(a) What is the length of LR?

(b) What is the length of ML?

(d) Determine the fraction of the area of ∆PQR that is covered by rectangle JKLM.

3. In a Dlin sequence, the first term is a positive integer and each term after the first is calculated by adding 1 to the previous term in the sequence, then doubling the result. For

example, the first seven terms of the Dlin sequence with first term 4 are:

4, 10, 22, 46, 94, 190, 382

(a) The 5th term in a Dlin sequence is 142. What are the 4th

and 6th terms in the sequence?

(b) Determine all possible first terms which give a Dlin sequence that includes 1406.

(c) Which possible first terms from 10 to 19 inclusive produce a Dlin sequence in which all terms after the first have the same ones (units) digit?

(d) Determine the number of positive integers between 1 and 2020, inclusive, that can be the third term in a Dlin sequence.

4. An m × n grid has m rows and n columns. Each cell in the grid is coloured either red (R) or blue (B). For example, a 1

× 2 grid can be coloured in 4 different ways, as shown below.

(a) How many different ways can a 5 × 1 grid be coloured so that exactly 3 cells are red and 2 cells are blue?

(b) Carrie writes down all possible colourings of a 1 × 13 grid. Looking at her first 1 × 13 grid, Carrie counts the number of cells coloured red and the number of cells coloured blue, and begins a list by writing down the maximum of these two numbers. Carrie continues this process and writes down this maximum for each of the 1×13 grids that she has coloured. What is the smallest number in Carrie’s list?

(c) Determine the smallest value of n so that no matter how a 3 ×n grid is coloured, it must have at least two columns that are coloured in an identical way.

(d) Consider the following statement:

In a 5 × 41 grid, we can always find 3 rows and 3 columns such that the 9 cells located at the intersections of these 3 rows and 3 columns are all the same colour.

Determine whether the statement above is true or false and justify your answer.