2023 Galois Contest
(Grade 10)
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.Jigsaw puzzles often have pieces that are arranged in a rectangular grid of rows and columns, where every cell in the grid represents one piece. The grid has two types of pieces: edge pieces which form the outer edge of the grid; and middle pieces which form the inside of the grid. In the example shown, there are 7 rows and 8 columns, and the middle pieces are shaded.
How many pieces, in total, does a grid
with 12 rows and 15 columns have?
How many middle pieces does a grid with
\(6\) rows and \(4\) columns have?
If a grid has \(14\) middle pieces, then it either has
\(s\) edge pieces or it has \(t\) edge pieces. Determine the values of
\(s\) and \(t\).
A grid with \(5\) rows and \(c\) columns has the same number of edge
pieces as middle pieces. Determine the value of \(c\).
In an Ing sequence, the first term is a positive integer and each term after the first is determined in the following way:
if a term, \(x\), is odd, the next term is \(x+3\), and
if a term, \(x\), is even, the next term is \(x+4\).
For example, if the first term in an Ing sequence is 13, then the second term is 16, and the third term is 20.
If the first term in an Ing sequence is
7, what is the fifth term in the sequence?
If the fifth term in an Ing sequence is
62, what are the two possibilities for the first term?
If the first term in an Ing sequence is
49, determine the terms appearing in the sequence whose values are
greater than 318 and less than 330.
The number 18 appears somewhere in an Ing
sequence after the first term. If the first term is the positive integer
\(n\), determine all possible values of
\(n\).
The shaded triangle shown is bounded by
the \(x\)-axis, the line \(y=x\), and the line \(x=a\), where \(a>0\).
If the area of this triangle is 32, what is the value of \(a\)?
A triangle is bounded by the \(x\)-axis, the line \(y=2x\), and the line \(x=10\). Diego draws the vertical line \(x=4\). This line divides the original
triangle into a trapezoid, which is shaded, and a new unshaded triangle,
as shown.
What is the area of the shaded trapezoid?
A triangle is bounded by the \(x\)-axis, the line \(y=3x\), and the line \(x=21\). Alicia draws the vertical line
\(x=c\), where \(0<c<21\). This line divides the
original triangle into a trapezoid and a new triangle. If the area of
the trapezoid is 8 times the area of the new triangle, determine the
value of \(c\).
A triangle is bounded by the \(x\)-axis, the line \(y=4x\), and the line \(x=1\). Ahmed draws his first vertical line
at \(x=p\), where \(0<p<1\). This line divides the area
of the original triangle in half. Ahmed then draws a second vertical
line at \(x=q\), where \(0<q<p\). This line divides the area
of the triangle bounded by the \(x\)-axis, the line \(y=4x\), and the line \(x=p\) in half. Ahmed continues this process
of drawing vertical lines at decreasing values of \(x\) so that each such line divides the area
of the previous triangle in half. If the 12th vertical line that he
draws is at \(x=k\), determine the
value of \(k\).
When people gather for a meeting, each person shakes hands with all, some or none of the other people, and never with the same person twice. When a handshake occurs between two people, this is counted as one handshake.
At a meeting of 5 people, Amrita shook
hands with exactly 1 person, Bin and Carlos each shook hands with
exactly 2 people, Dennis shook hands with exactly 3 people, and Eloise
did not shake hands with anyone. How many handshakes took
place?
At a meeting of 9 people, each
participant said that they shook hands with exactly 3 people. Explain
why this is not possible.
At a meeting of 7 people, at least one
handshake occurred within each group of 3 people. Determine the minimum
possible number of handshakes, \(m\),
that took place at this meeting. A complete solution must include the
value of \(m\), an explanation of why
the given conditions can be satisfied with some specific set of \(m\) handshakes, and an explanation of why
fewer than \(m\) handshakes does not
satisfy the given conditions.
Thank you for writing the Galois 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