Fryer Contest 2015年真题

1. A company builds cylinders. Its Model

A cylinder has radius r = 10 cm and height

h = 16 cm.

(a) What is the volume in cm3 of a Model A cylinder?

(b) The company also builds a Model B cylinder having a radius of 8 cm. Each Model B cylinder has the same volume as each Model A cylinder. What is the height in cm of a Model B cylinder?

called Box A, that holds six Model A cylinders.

The cylinders are placed into the box vertically

and tightly packed, as shown. Determine the

volume in cm3 of Box A.

(d) The company makes another rectangular box, called Box B, that holds six Model B cylinders. The cylinders are placed into the box vertically and tightly packed, just as was shown in part (c). State whether the volume of Box B is less than, greater than, or equal to, the volume of Box A.

2. In Canada, a quarter is worth $0.25, a dime is worth $0.10, and a nickel is worth $0.05.

(a) Susan has 3 quarters, 18 dimes and 25 nickels. What is the total value of Susan’s coins?

(b) Allen has equal numbers of dimes and nickels, and no other coins. His coins have a total value of $1.50. How many nickels does Allen have?

3. A formula for the sum of the first n positive integers is . For example, to calculate the sum of the first 4 positive integers, we evaluate .

(a) What is the sum of the first 200 positive integers,

1 + 2 + 3 + · · · + 198 + 199 + 200 ?

(b) Calculate the sum of the 50 consecutive integers beginning at 151, that is,

151 + 152 + 153 + · · · + 198 + 199 + 200 .

(c) Starting with the sum of the first 1000 positive integers, 1+2+3+· · ·+999+1000, every third integer is removed to create the new sum

1 + 2 + 4 + 5 + 7 + 8 + 10 + 11 + · · · + 998 + 1000 .

Calculate this new sum.

4. The token • is placed on a hexagonal grid, as shown. At each step, the token can be moved to an adjacent hexagon in one of the three directions .

(The token can never be moved in any of the three directions,.)

(a) What is the minimum number of steps required to get the token to the hexagon labelled A?

(b) With justification, determine the maximum number of steps that can be taken so that the token ends at A.

(c) Using exactly 5 steps, the token can end at the hexagon labelled C in exactly 20 different ways. Using exactly 5 steps, the token can end at n different hexagons in at least 20 different ways. Determine, with justification, the value of n.