2025 Euclid Contest
Wednesday, April 2, 2025
(in North America and South America)
Thursday, April 3, 2025
(outside of North American and South America)
Wednesday, April 2, 2025
(in North America and South America)
Thursday, April 3, 2025
(outside of North American and South America)
©2025 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 \(4(x-2) =
2(x-4)\), what is the value of \(x\)?
If \(2x =
9\), what is the value of \(2^{6x-23}\)?
Determine the coordinates of the point of
intersection of the lines with equations \(y =
3x + 7\) and \(y = 7x +
3\).
There is one positive integer \(k\) for which \(3
< \sqrt{k^2+4} < 4\). What is this positive integer \(k\)?
What is the sum of the \(20\) smallest odd positive
integers?
In the diagram, \(\triangle BDF\) is equilateral and \(\angle FAB = \angle BCD = \angle DEF =
90\degree\). Also, \(AB = 16\),
\(BC = 8\), \(DE = 5\), and \(FA = 13\). Determine the perimeter of
hexagon \(ABCDEF\).
Suppose that \(p+q+r=18\) and \(p+q = 5\) and \(q
+ r = 9\). What is the value of \(q\)?
The line with equation \(6x + y = 24\) has its \(x\)-intercept at point \(P\) and its \(y\)-intercept at point \(Q\). What is an equation of the parabola
whose \(y\)-intercept is at \(Q\) and whose only \(x\)-intercept is at \(P\)?
Suppose that \(\dfrac{1}{2w} = \dfrac{1}{3y} =
\dfrac{1}{4z}\) and \(\dfrac{1}{2w} +
\dfrac{1}{3y} + \dfrac{1}{4z} = \dfrac{1}{24}\). Determine the
value of \(w+y+z\).
Terry’s bicycle has a larger front wheel
with radius \(15\) cm and a smaller
rear wheel with radius \(9\) cm, as
shown.
Terry ties a ribbon to the top of each wheel, and then starts to ride forward. Terry travels \(d\) cm forward and stops. Both ribbons are again at the top of the wheels. What is the integer closest to the smallest possible value of \(d\) with \(d>0\)?
In the diagram, \(ABCD\) is a rectangle with \(AB = 24\) and \(AD = 18\). Also, \(E\) is on \(BC\) with \(EC =
6\). If segments \(DE\) and
\(AC\) intersect at \(F\), determine the length of \(CF\).
Alice has a lock whose combination
consists of three integers \(a\), \(b\), \(c\)
which need to be entered in that order. The three integers satisfy the
following:
each of \(a\), \(b\) and \(c\) is between \(1\) and \(40\), inclusive;
\(a\), \(b\) and \(c\) are all different;
\(b\) is less than \(a\), and \(b\) is less than \(c\); and
one of the integers is \(20\) and another of the integers is \(30\).
How many possible combinations satisfy these conditions?
For some angles \(\theta\), the three numbers \(2 - 2\cos\theta\), \(1 + \sin\theta\), \(2 + 2\cos\theta\) form a geometric sequence
in that order. Determine all possible exact values of \(\cos\theta\).
(A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant. For example, \(3\), \(6\), \(12\) is a geometric sequence with three terms.)
Twelve points are equally spaced around
the entire circumference of a circle. In how many ways can three of
these points be chosen so that the triangle that they form has at least
two sides of equal length?
In the diagram, quadrilateral \(ABCD\) has \(\angle ABC = \angle CDA = 90\degree\),
\(AB=56\), \(BC=33\), \(CD=39\), and \(DA=52\). Point \(P\) is on \(AD\) so that \(BP\) is perpendicular to \(AD\). Determine the exact length of \(BP\).
The functions \(f\) and \(g\) are defined by the tables of values
below:
| \(x\) | \(f(x)\) |
|---|---|
| \(1\) | \(5\) |
| \(2\) | \(3\) |
| \(3\) | \(4\) |
| \(4\) | \(1\) |
| \(5\) | \(2\) |
| \(x\) | \(g(x)\) |
|---|---|
| \(1\) | \(3\) |
| \(2\) | \(1\) |
| \(3\) | \(4\) |
| \(4\) | \(5\) |
| \(5\) | \(2\) |
The functions \(f^{-1}\) and \(g^{-1}\) are the inverse functions of \(f\) and \(g\), respectively. If \(f^{-1}(g^{-1}(a)) = 3\), what is the value of \(a\)?
Determine all pairs \((x,y)\) of real numbers that satisfy the
following system of equations: \[\begin{align*}
x^2 - 8xy + 16y^2 & = 0 \\
(\log_{10}x)^2 + 2(\log_{10}x)(\log_{10}y) + (\log_{10}y)^2 & =
4\end{align*}\]
Leilei, Jerome and Farzad write a test
independently. The probability that Leilei passes the test and Jerome
fails the test is \(\frac{3}{20}\). The
probability that Jerome passes and Farzad fails is \(\frac{1}{4}\). The probability that Leilei
and Farzad both pass is \(\frac{2}{5}\). Determine the probability
that at least one of Leilei, Jerome and Farzad fails the test.
The integer \(7447\) is a palindrome because it reads the
same forwards and backwards. Suppose that positive integers \(m\) and \(n\) are palindromes between \(1001\) and \(9999\), inclusive, with \(m > n\). Determine the number of pairs
\((m,n)\) for which the difference
\(m-n\) is a multiple of \(35\).
Suppose that \(p(x) = qx^3 - rx^2 - sx +
t\) for some positive integers \(q<r<s<t\) which form an arithmetic
sequence.
Show that \(x = 1\) is a root of \(p(x)\).
Suppose that the average of \(q\), \(r\), \(s\), \(t\) is \(19\) and that \(p(x)\) has three rational roots. Determine the roots of \(p(x)\).
Prove that, for every positive integer \(n > 3\), there are at least two arithmetic sequences of positive integers \(q<r<s<t\) with common difference \(2n\) for which \(p(x)\) has three rational roots.
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant, called the common difference. For example, \(3\), \(5\), \(7\), \(9\) are the first four terms of an arithmetic sequence.)
An equilateral triangle is formed using \(n\) rows of coins. There is \(1\) coin in the first row, \(2\) coins in the second row, \(3\) coins in the third row, and so on, up
to \(n\) coins in the \(n\)th row. Initially, all of the coins show
heads (H). Carley plays a game in which, on each turn, she chooses three
mutually adjacent coins and flips these three coins over. To win the
game, all of the coins must be showing tails (T) after a sequence of
turns. An example game with \(4\) rows
of coins after a sequence of two turns is shown.
Below (a), (b) and (c), you will find instructions about how to refer to these turns in your solutions.
If there are \(3\) rows of coins, give a sequence of \(4\) turns that results in a win.
Suppose that there are \(4\) rows of coins. Determine whether or not there is a sequence of turns that results in a win.
Determine all values of \(n\) for which it is possible to win the game starting with \(n\) rows of coins.
Note: For a triangle with \(4\) rows of coins, there are \(9\) possibilities for the set of three coins that Carley can flip on a given turn. These \(9\) possibilities are shown as shaded triangles below:
   
Participants should use the names for these moves shown inside the 9 shaded triangles when answering (b). Participants should adapt this naming convention in a suitable way when answering parts (a) and (c).
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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