2024 Euclid Contest
Wednesday, April 3, 2024
(in North America and South America)
Thursday, April 4, 2024
(outside of North American and South America)
Wednesday, April 3, 2024
(in North America and South America)
Thursday, April 4, 2024
(outside of North American and South America)
©2024 University of Waterloo
Time: \(2\frac{1}{2}\) hours
Number of Questions: 10
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. If \(x=2\), what is the value of \(\dfrac{x^4 + 3x^2}{x^2}\)Â ?
In the diagram, \(\triangle ABC\) is right-angled at \(B\). Also, \(AB=10\), \(BC=t-1\), and \(AC=t+1\). What is the value of \(t\)?
Suppose that \(\dfrac{2}{y} + \dfrac{3}{2y} = 14\).
Determine the value of \(y\).
In a sequence with six terms, each term
after the second is the sum of the previous two terms. If the fourth
term is \(13\) and the sixth term is
\(36\), what is the first
term?
For some real number \(r \neq 0\), the sequence \(5r\), \(5r^2\), \(5r^3\) has the property that the second
term plus the third term equals the square of the first term. What is
the value of \(r\)?
Jimmy wrote four tests last week. The
average of his marks on the first, second and third tests was \(65\). The average of his marks on the
second, third and fourth tests was \(80\). His mark on the fourth test was \(2\) times his mark on the first test.
Determine his mark on the fourth test.
The graph of the equation \(y = r(x-3)(x-r)\) intersects the \(y\)-axis at \((0,48)\). What are the two possible values
of \(r\)?
A bicycle costs \(\$B\) before taxes. If the sales tax were
\(13\%\), Annemiek would pay a total
that is \(\$24\) higher than if the
sales tax were \(5\%\). What is the
value of \(B\)?
The function \(f\) has the following three properties:
\(f(1) = 3\).
\(f(2n) = (f(n))^2\) for all positive integers \(n\).
\(f(2m+1) = 3f(2m)\) for all positive integers \(m\).
Determine the value of \(f(2)+f(3)+f(4)\).
In the diagram, \(AB\) is perpendicular to \(CD\) (with \(B\) on \(CD\)), \(CP\) is perpendicular to \(AD\) (with \(P\) on \(AD\)), and \(N\) is the point of intersection of \(AB\) and \(CP\).
Also, \(\angle ADB = 45\degree\), \(AB=12\), and \(CB=6\). What is the area of \(\triangle APN\)?
In the diagram, the line with equation
\(y=-3x+6\) crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). Suppose that \(m>0\) and that the line with equation
\(y = mx + 1\) crosses the \(y\)-axis at \(D\) and intersects the line with equation
\(y = -3x+6\) at the point \(C\).
If \(O\) is the origin and the area of \(\triangle ACD\) is \(\frac{1}{2}\) of the area of \(\triangle ABO\), determine the coordinates of \(C\).
In the diagram, rectangle \(ABCD\) is divided into four smaller
rectangles by the lines \(PQ\) and
\(RS\), which intersect at \(X\).
The areas of these smaller rectangles are, in some order, \(2\), \(6\), \(3\), and \(a\). What are the three possible values of \(a\)?
Suppose that the parabola with equation
\(y = x^2 - 4tx + 5t^2 - 6t\) has two
distinct \(x\)-intercepts. Determine
the value of \(t\) for which the
distance between these \(x\)-intercepts
is as large as possible.
There are \(M\) integers between \(10\,000\) and \(100\,000\) that are multiples of \(21\) and whose units (ones) digit is \(1\). What is the value of \(M\)?
There are \(N\) students who attend Strickland S.S.,
where \(500 < N < 600\). Among
these \(N\) students, \(\frac{2}{5}\) are in the physics club and
\(\frac{1}{4}\) are in the math club.
In the physics club, there are \(2\)
times as many students who are not in the math club as there are
students who are in the math club. Determine the number of students who
are not in either club.
Arun and Bella run around a circular
track, starting from diametrically opposite points. Arun runs clockwise
around the track and Bella runs counterclockwise. Arun and Bella run at
constant, but different, speeds. They meet for the first time after Arun
has run \(100\) m. They meet for the
second time after Bella runs \(150\) m
past their first meeting point. What is the length of the
track?
Determine all angles \(\theta\) with \(0\degree \leq \theta \leq 360\degree\) for
which \(4^{1 + \cos^3\theta} =
2^{2-\cos\theta}\cdot 8^{\cos^2\theta}\).
In the diagram, the circle with centre
\(A\) has radius \(4\), and \(A\) lies on the circle which has centre
\(B\) and radius \(3\). The line passing through \(A\) and \(B\) lies along the diameter of each circle
and is perpendicular to \(MN\) at \(T\). Also, \(MN\) is tangent to the larger circle, and
\(MC\) and \(NC\) are each tangent to both circles at
points \(D\), \(E\), \(F\), and \(G\), as shown. Determine the area of \(\triangle MNC\).
Determine all triples \((x,y,z)\) of real numbers that are
solutions to the following system of equations: \[\begin{align*}
\log_9 x + \log_9 y + \log_3 z & = 2\\
\log_{16} x + \log_4 y + \log_{16} z & = 1 \\
\log_5 x + \log_{25} y + \log_{25} z & =
0\end{align*}\]
An ant walks along the \(x\)-axis by taking a sequence of steps of
length \(1\). Some, all or none of
these steps are in the positive \(x\)-direction; some, all or none of these
steps are in the negative \(x\)-direction. The ant begins at \(x=0\), takes a total of \(n\) steps, and ends at \(x=d\). For each such sequence, let \(c\) be the number of times that the ant
changes direction.
Determine the number of different sequences of steps for which \(n=9\) and \(d=5\).
Suppose that \(n=9\) and \(d=3\). Determine the number of sequences for which \(c\) is even.
Determine the number of pairs \((d,n)\) of integers with \(1 \leq n \leq 2024\) and \(d \geq 0\) for which \(c\) is even for exactly half of the sequences of \(n\) steps that end at \(x = d\).
Suppose that \(s\) and \(t\) are real numbers with \(0 < s \leq 1\) and \(0 < t \leq 1\). Points \(A(-1,0)\), \(B(0,4)\) and \(C(1,0)\) form \(\triangle ABC\). Points \(S(s,0)\) and \(T(-t,0)\) lie on \(AC\). Point \(P\) lies on \(AB\) and point \(Q\) lies on \(BC\), with neither \(P\) nor \(Q\) at a vertex of \(\triangle ABC\). Line segments \(SP\) and \(TQ\) intersect at \(X\) and partition \(\triangle ABC\) into four regions. For some
such pairs \((s,t)\) of real numbers
and points \(P\) and \(Q\), the line segments \(SP\) and \(TQ\) in fact partition \(\triangle ABC\) into four regions of equal
area. We call such a pair \((s,t)\) a
balancing pair.
Suppose that \((s,t)\) is a balancing pair with \(s=1\) and that line segments \(SP\) and \(TQ\) partition \(\triangle ABC\) into four regions of equal area. Determine the coordinates of \(P\).
Prove that there exist real numbers \(d\), \(e\), \(f\), and \(g\) for which all balancing pairs \((s,t)\) satisfy an equation of the form \[s^2 + t^2 = dst + es + ft + g\] and determine the values of \(d\), \(e\), \(f\), and \(g\).
Determine an infinite family of distinct pairs of rational numbers \((s,t)\) with \(0 < s \leq t \leq 1\) that satisfy the equation in (b).
Thank you for writing the Euclid Contest!
If you are graduating from secondary school, good luck in your future endeavours! If you will be returning to secondary school next year, encourage your teacher to register you for the Canadian Senior Mathematics Contest, which will be written in November.
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