2023 Fryer Contest
(Grade 9)
Wednesday, April 5, 2023
(in North America and South America)
Thursday, April 6, 2023
(outside of North American and South America)
Wednesday, April 5, 2023
(in North America and South America)
Thursday, April 6, 2023
(outside of North American and South America)
©2023 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.
, place your answer in the appropriate box in the answer booklet and show your work., provide a well-organized solution in the answer booklet. Use mathematical statements and words to explain all of the steps of your solution. Work out some details in rough on a separate piece of paper before writing your finished solution.Useful Fact:
It may be helpful to know that the sum of the \(n\) integers from 1 to \(n\) equals \(\frac{1}{2}n(n+1)\);
that is, \(1+2+3+\cdots+
(n-1)+n=\frac{1}{2}n(n+1)\).
At Monday’s practice, Lonnie sprints 200 m a total of 24 times. At Tuesday’s practice, he sprints 240 m a total of 20 times. On both days, he rests for 30 s between each consecutive pair of sprints. Lonnie sprints at a constant speed of 8 m/s.
On Monday, how many times does Lonnie
take the 30 s rest between consecutive pairs of sprints?
On Monday, determine Lonnie’s total
practice time. That is, determine the total number of seconds between
the start of his first sprint and the end of his last sprint, including
the rests.
Determine how many fewer seconds
Tuesday’s practice will take compared to Monday’s practice.
Consider the following arrangement of positive integers.
| \(1\) | ||||
| \(2\) | \(4\) | |||
| \(5\) | \(7\) | \(9\) | ||
| \(10\) | \(12\) | \(14\) | \(16\) | |
| \(\vdots\) |
The 1st row includes the odd integer \(1\), and the 2nd row includes the two even integers \(2\) and \(4\). For \(k\geq2\), the \(k\)th row
begins with the integer that is one more than the last integer in the previous row,
includes, in increasing order, \(k\) consecutive odd integers when \(k\) is odd, and
includes, in increasing order, \(k\) consecutive even integers when \(k\) is even.
A useful fact about this arrangement is that the integer in the \(k\)th row and \(k\)th position (that is, the last position in the \(k\)th row) is \(k^2\). For example, \(4^2=16\) and 16 is the integer in the 4th position of the 4th row.
What is the average of the integers in
the 5th row?
Which row has the integer 145 in the 1st
position?
Determine the row and the position in
which the integer \(1598\)
appears.
The average of the integers in row \(r\) is 241. Determine the value of \(r\).
A positive integer is divisible by 3 exactly when the sum of its
digits is divisible by 3.
A positive integer is divisible by 4 exactly when the positive integer
formed by its last two digits is divisible by 4. For example:
\(3816\) is divisible by \(3\), since \(3+8+1+6=18\) and \(18\) is divisible by \(3\);
\(3817\) is not divisible by \(3\), since \(3+8+1+7=19\) and \(19\) is not divisible by \(3\);
\(3816\) is divisible by \(4\), since 16 is divisible by \(4\);
\(3817\) is not divisible by \(4\), since 17 is not divisible by \(4\).
In each part that follows, \(A\), \(B\) and \(C\) are non-zero digits (1, 2, 3, 4, 5, 6, 7, 8, or 9), and not necessarily distinct.
The five-digit positive integer \(4\,B\,5B\,2\) is divisible by \(3\). What are the possible values of the
non-zero digit \(B\)?
The five-digit positive integer \(ABABA\) is divisible by \(4\) and not divisible by 3. Determine the
number of different pairs of non-zero digits \(A\) and \(B\) that are possible.
A positive integer, \(t\), is equal to the product of the
four-digit positive integer \(A\,CA\,2\)
and the three-digit positive integer \(BAC\); that is, \(t=A\,CA\,2\times BAC\). If \(t\) is divisible by 15 and not divisible by
12, determine the number of different triples of non-zero digits \(A\), \(B\), \(C\)
that are possible.
A lab has \(50\) computers numbered \(1\) through \(50\). Each pair of computers is connected to each other by a cord. The cords are coloured according to the following rules.
If the numbers of the two computers are both even or both odd, then the cord connecting them is red.
Otherwise, the cord connecting them is blue.
A route is a sequence of cords along which data can travel to get from one computer to another computer within the lab. For example, data could travel the route from Computer 5 to Computer 12 directly, or the route from Computer 5 to Computer 15 to Computer 12.
There is a route connecting Computer 1 to
Computer \(n\) using only red cords. If
\(n\neq 1\), how many possible values
are there for \(n\)?
Show that for every pair of distinct
computers, Computer \(A\) and Computer
\(B\), there is always a route between
them that uses only blue cords.
Each red cord and blue cord is removed
and randomly replaced with either a green cord or a yellow cord. Dani
notices that there is no route that connects Computer 1 to Computer 50
that uses only green cords. Show that there is always a route between
Computer 13 and Computer 14 that uses only yellow cords.
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.
Visit our website cemc.uwaterloo.ca to find
Visit our website cemc.uwaterloo.ca to