Fryer Contest 2018年真题

1.       Sandy’s Fruit Market sells cherries,

plums and blueberries. For each type

of fruit, the price of one box is shown

in the table.

(a) On Monday, Shane visited Sandy’s Fruit Market. He bought 4 boxes of cherries, 3 boxes of plums, and 2 boxes of blueberries. How much did Shane pay in total?

(b) On Wednesday, Shane bought 2 boxes of plums. He bought some boxes of cherries, no blueberries, and spent $22.00 in total. How many boxes of cherries did he buy?

(c) On Saturday, Shane bought twice as many boxes of plums as boxes of cherries. He also bought 3 boxes of blueberries. How many boxes of cherries did Shane buy if he gave the cashier $100.00 and received $14.50 in change?

2. In the diagrams shown, ABCD

represents a rectangular fifield.

There are three flflagpoles: M on

BC, P on AD, and Q on CD. Paul

runs along the path A → D → C →

M → A. Tyler runs along the path

A → P → Q → C → B → A.

(a) What is the length of MA?

(b) What is the total distance that Tyler runs?

(c) Paul and Tyler start running at the same

time. Tyler runs at a speed of 145 m/min.

Paul runs at a constant speed and finishes

1 minute after Tyler. Determine Paul’s speed, in m/min.

3. (a) A line has equation y = 2x - 6. What is its x-intercept and what is its y-intercept?

(b) A line has equation y = kx -  6, where k ≠ 0. What is its

x-intercept? Express your answer in terms of k.

(c) A triangle is formed by the positive x-axis, the negative

y-axis, and the line with equation y = kx -  6, where k > 0. The area of this triangle is 6. What is the value of k?

(d) A triangle is formed by the positive x-axis, the line with equation y = mx -  m2, and the line with equation y = 2mx - m2 . Determine all values of m > 0 for which the area of the triangle is .

4.       A Bauman number is a positive integer each of whose digits is 1 or 2. Each Bauman number consists of blocks of digits. Each block contains at least one digit and includes all of the consecutive equal digits. For example, 2222111112111 is a

13-digit Bauman number consisting of four blocks: a block of four 2s, then a block of five 1s, then a block of one 2, then a block of three 1s; 2222222 is a 7-digit Bauman number consisting of a single block of seven 2s.

(a) How many 3-digit Bauman numbers are there?

(b) How many 10-digit Bauman numbers consist of fewer than three blocks?

(c) Determine the number of Bauman numbers that consist of at most three blocks and have the property that the sum of the digits is 7.

(d) Some Bauman numbers include a block of exactly 2018 2s. Determine the number of 4037-digit Bauman numbers that include at least one block of exactly 2018 2s.