2023 Euclid Contest

Tuesday, April 4, 2023
(in North America and South America)

Wednesday, April 5, 2023
(outside of North American and South America)

©2023 University of Waterloo

Instructions

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:

1. SHORT ANSWER parts indicated by
   • worth 2 or 3 marks each
   • full marks are given for a correct answer which is placed in the box
   • part marks are awarded if relevant work is shown in the space provided
2. FULL SOLUTION parts indicated by
   • worth the remainder of the 10 marks for the question
   • must be written in the appropriate location in the answer booklet
   • marks awarded for completeness, clarity, and style of presentation
   • a correct solution poorly presented will not earn full marks

WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED.

• Extra paper for your finished solutions supplied by your supervising teacher must beinserted into your answer booklet. Write your name, school name, and question numberon any inserted pages.
• Express answers as simplified exact numbers except where otherwise indicated. For example, $\pi +1$ and $1-\sqrt{2}$ are simplified exact numbers.

Questions

1. 1. The average of $n$, $2n$, $3n$, $4n$, $5n$ is 18. What is the value of $n$?

   2. Suppose that $2x+y=5$ and $x+2y=7$. What is the average of $x$ and $y$?

   3. The average of $t^2$, $2t$ and $3$ is 9. If $t<0$, determine the value of $t$.

2. 1. If $Q(5,3)$ is the midpoint of the line segment with endpoints $P(1,p)$ and $R(r,5)$, what are the values of $p$ and $r$?

   2. A line with slope 3 and another line with slope $-1$ intersect at $P(3,6)$. What is the distance between the $x$-intercepts of the two lines?

   3. For some value of $t$, the line with equation $y=tx+t$ is perpendicular to the line with equation $y=2x+7$. Determine the point of intersection of these two lines.

3. 1. The positive divisors of 6 are 1, 2, 3, and 6. What is the sum of the positive divisors of 64?

   2. Fionn wrote 4 consecutive integers on a whiteboard. Lexi came along and erased one of the integers. Fionn noticed that the sum of the remaining integers was 847. What integer did Lexi erase?

   3. An arithmetic sequence with 7 terms has first term $d^2$ and common difference $d$. The sum of the 7 terms in the sequence is 756. Determine all possible values of $d$.

      (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.)

4. 1. Liang and Edmundo paint at different but constant rates. Liang can paint a room in 3 hours if she works alone. Edmundo can paint the same room in 4 hours if he works alone. Liang works alone for 2 hours and then stops. Edmundo finishes painting the room. How many minutes will Edmundo need to finish painting the room?

   2. On January 1, 2021, an investment had a value of $400.
      From January 1, 2021 to January 1, 2022, the value of the investment increased by $A$% from its value on January 1, 2021 for some $A>0$.
      From January 1, 2022 to January 1, 2023, the value of the investment decreased by $A$% from its value on January 1, 2022.
      On January 1, 2023, the value of the investment was $391.
      Determine all possible values of $A$.

5. 1. Suppose that $f(x)=x^2+(2n-1)x+(n^2-22)$ for some integer $n$. What is the smallest positive integer $n$ for which $f(x)$ has no real roots?

   2. In the diagram, $\mathrm{△}PQR$ has $PQ=a$, $QR=b$, $PR=21$, and $\mathrm{\angle }PQR=60\mathrm{°}$. Also, $\mathrm{△}STU$ has $ST=a$, $TU=b$, $\mathrm{\angle }TSU=30\mathrm{°}$, and $\sin (\mathrm{\angle }TUS)=\frac{4}{5}$. Determine the values of $a$ and $b$.

      

6. 1. A triangle of area $770{\text{ cm}}^2$ is divided into 11 regions of equal height by 10 lines that are all parallel to the base of the triangle. Starting from the top of the triangle, every other region is shaded, as shown.

      

      What is the total area of the shaded regions?

   2. A square lattice of 16 points is constructed such that the horizontal and vertical distances between adjacent points are all exactly 1 unit. Each of four pairs of points are connected by a line segment, as shown.

      

      Hide/Reveal Description of Diagram for 6(b).

      Sixteen points arranged into 4 rows of 4. Four line segments connect pairs of points as follows:

      • The first line segment joins the first point in the top row to the last point in the row below.

      • The second line segment is parallel to the first and joins the first point in the second row to the last point in the row below.

      • A third line segment joins the second point in the first row to the first point in the last row. This line meets the first line at D and the second line at C.

      • A fourth line segment is parallel to the third and joins the third point in the first row to the second point in the last row. This line meets the first line at A and the second line at B.

      ---

      The intersections of these line segments are the vertices of square $ABCD$. Determine the area of square $ABCD$.

7. 1. A bag contains 3 red marbles and 6 blue marbles. Akshan removes one marble at a time until the bag is empty. Each marble that they remove is chosen randomly from the remaining marbles. Given that the first marble that Akshan removes is red and the third marble that they remove is blue, what is the probability that the last two marbles that Akshan removes are both blue?

   2. Determine the number of quadruples of positive integers $(a,b,c,d)$ with $a<b<c<d$ that satisfy both of the following system of equations: $$\begin{array}{rl}ac+ad+bc+bd& =2023\\ a+b+c+d& =296\end{array}$$

8. 1. Suppose that $\mathrm{△}ABC$ is right-angled at $B$ and has $AB=n(n+1)$ and $AC=(n+1)(n+4)$, where $n$ is a positive integer. Determine the number of positive integers $n<100000$ for which the length of side $BC$ is also an integer.

   2. Determine all real values of $x$ for which $$\sqrt{{\log }_{2}x\cdot {\log }_2(4x)+1}+\sqrt{{\log }_{2}x\cdot {\log }_2\left({\displaystyle \frac{x}{64}}\right)+9}=4$$

9. At the Canadian Eatery with Multiple Configurations, there are round tables, around which chairs are placed. When a table has $n$ chairs around it for some integer $n\ge 3$, the chairs are labelled 1, 2, 3, $\dots$, $n-1$, $n$ in order around the table. A table is considered full if no more people can be seated without having two people sit in neighbouring chairs. For example, when $n=6$, full tables occur when people are seated in chairs labelled $\{1,4\}$ or $\{2,5\}$ or $\{3,6\}$ or $\{1,3,5\}$ or $\{2,4,6\}$. Thus, there are 5 different full tables when $n=6$.

   

   1. Determine all ways in which people can be seated around a table with 8 chairs so that the table is full, in each case giving the labels on the chairs in which people are sitting.

   2. A full table with $6k+5$ chairs, for some positive integer $k$, has $t$ people seated in its chairs. Determine, in terms of $k$, the number of possible values of $t$.

   3. Determine the number of different full tables when $n=19$.

10. For every real number $x$, define $\lfloor x\rfloor$ to be equal to the greatest integer less than or equal to $x$. (We call this the “floor” of $x$.) For example, $\lfloor 4.2\rfloor =4$, $\lfloor 5.7\rfloor =5$, $\lfloor -3.4\rfloor =-4$, $\lfloor 0.4\rfloor =0$, and $\lfloor 2\rfloor =2$.

    1. Determine the integer equal to $\lfloor {\displaystyle \frac{1}{3}}\rfloor +\lfloor {\displaystyle \frac{2}{3}}\rfloor +\lfloor {\displaystyle \frac{3}{3}}\rfloor +\cdots +\lfloor {\displaystyle \frac{59}{3}}\rfloor +\lfloor {\displaystyle \frac{60}{3}}\rfloor$.
       (The sum has 60 terms.)

    2. Determine a polynomial $p(x)$ so that for every positive integer $m>4$, $$\lfloor p(m)\rfloor =\lfloor {\displaystyle \frac{1}{3}}\rfloor +\lfloor {\displaystyle \frac{2}{3}}\rfloor +\lfloor {\displaystyle \frac{3}{3}}\rfloor +\cdots +\lfloor {\displaystyle \frac{m-2}{3}}\rfloor +\lfloor {\displaystyle \frac{m-1}{3}}\rfloor$$ (The sum has $m-1$ terms.)

       A polynomial $f(x)$ is an algebraic expression of the form $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$ for some integer $n\ge 0$ and for some real numbers $a_n,a_{n-1},\dots ,a_1,a_0$.
       

    3. For each integer $n\ge 1$, define $f(n)$ to be equal to an infinite sum: $$f(n)=\lfloor \frac{n}{1^2+1}\rfloor +\lfloor \frac{2n}{2^2+1}\rfloor +\lfloor \frac{3n}{3^2+1}\rfloor +\lfloor \frac{4n}{4^2+1}\rfloor +\lfloor \frac{5n}{5^2+1}\rfloor +\cdots$$ (The sum contains the terms $\lfloor {\displaystyle \frac{kn}{k^2+1}}\rfloor$ for all positive integers $k$, and no other terms.)

       Suppose $f(t+1)-f(t)=2$ for some odd positive integer $t$. Prove that $t$ is a prime number.

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