Thursday, April 16, 2015
(in North America and South America)
Friday, April 17, 2015
(outside of North American and South America)
©2015 University of Waterloo
Time: 75 minutes
Number of Questions: 4
Each question is worth 10 marks.
Calculating devices are allowed, provided that they do not have any of the following features: (i) internet access, (ii) the ability to communicate with other devices, (iii) information previously stored by students (such as formulas, programs, notes, etc.), (iv) a computer algebra system, (v) dynamic geometry software.
Parts of each question can be of two types:
WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED.
A company builds cylinders. Its Model A cylinder has radius \(r=10\) cm and height \(h=16\) cm.
Volume of a Cylinder: \(V=\pi r^2h\)
What is the volume in cm\(^3\) of a Model A cylinder?
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?
The company makes a rectangular box, 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 cm\(^3\) of Box A.
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.
In Canada, a quarter is worth $0.25, a dime is worth $0.10, and a nickel is worth $0.05.
Susan has 3 quarters, 18 dimes and 25 nickels. What is the total value of Susan’s coins?
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?
Elise has $10.65 in quarters and dimes. If Elise has \(x\) quarters and \(2x+3\) dimes, what is the value of \(x\)?
A formula for the sum of the first \(n\) positive integers is \(1+2+3+\cdots+n=\dfrac{n(n+1)}{2}\).
For example, to calculate the sum of the first 4 positive integers, we evaluate \(1+2+3+4=\dfrac{4(4+1)}{2}=10\).
What is the sum of the first 200 positive integers, \[1+2+3+\cdots+198+199+200~?\]
Calculate the sum of the 50 consecutive integers beginning at 151, that is, \[151+152+153+\cdots+198+199+200~.\]
Starting with the sum of the first 1000 positive integers, \(1+2+3+\cdots+999+1000\), every third integer is removed to create the new sum \[1+2+4+5+7+8+10+11+\cdots+998+1000~.\] Calculate this new sum.
The token \(\bullet\) 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 \(\nwarrow, \uparrow, \nearrow\).
(The token can never be moved in any of the three directions, \(\swarrow, \downarrow, \searrow\).)
What is the minimum number of steps required to get the token to the hexagon labelled \(A\)?
With justification, determine the maximum number of steps that can be taken so that the token ends at \(A\).
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 \(n\).
Thank you for writing the Fryer Contest!
Encourage your teacher to register you for the Canadian Intermediate Mathematics Contest or the Canadian Senior Mathematics Contest, which will be written in November.
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