2015 Euclid Contest

Wednesday, April 15, 2015
(in North America and South America)

Thursday, April 16, 2015
(outside of North American and South America)

©2015 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 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 be inserted into your answer booklet. Write your name, school name, and question number on 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. What is value of ${\displaystyle \frac{{10}^2-9^2}{10+9}}$?

   2. If ${\displaystyle \frac{x+1}{x+4}}=4$, what is the value of $3x+8$?

   3. If $f(x)=2x-1$, determine the value of $(f(3){)}^2+2(f(3))+1$.

2. 1. If $\sqrt{a}+\sqrt{a}=20$, what is the value of $a$?

   2. Two circles have the same centre. The radius of the smaller circle is 1. The area of the region between the circles is equal to the area of the smaller circle.

      

      What is the radius of the larger circle?

   3. There were 30 students in Dr. Brown’s class. The average mark of the students in the class was 80. After two students dropped the class, the average mark of the remaining students was 82. Determine the average mark of the two students who dropped the class.

3. 1. In the diagram, $BD=4$ and point $C$ is the midpoint of $BD$. If point $A$ is placed so that $\mathrm{△}ABC$ is equilateral, what is the length of $AD$?

      

   2. $\mathrm{△}MNP$ has vertices $M(1,4)$, $N(5,3)$, and $P(5,c)$. Determine the sum of the two values of $c$ for which the area of $\mathrm{△}MNP$ is 14.

4. 1. What are the $x$-intercepts and the $y$-intercept of the graph with equation $y=(x-1)(x-2)(x-3)-(x-2)(x-3)(x-4)$?

   2. The graphs of the equations $y=x^3-x^2+3x-4$ and $y=a{x}^2-x-4$ intersect at exactly two points. Determine all possible values of $a$.

5. 1. In the diagram, $\mathrm{\angle }CAB={90}^{\circ }$. Point $D$ lies on $AB$ and point $E$ lies on $AC$ so that $AB=AC=DE$, $DB=9$, and $EC=8$.

      

      Determine the length of $DE$.

   2. Ellie has two lists, each consisting of 6 consecutive positive integers. The smallest integer in the first list is $a$, the smallest integer in the second list is $b$, and $a<b$. She makes a third list which consists of the 36 integers formed by multiplying each number from the first list with each number from the second list. (This third list may include some repeated numbers.) If

      • the integer 49 appears in the third list,

      • there is no number in the third list that is a multiple of 64, and

      • there is at least one number in the third list that is larger than 75,

      determine all possible pairs $(a,b)$.

6. 1. A circular disc is divided into 36 sectors. A number is written in each sector. When three consecutive sectors contain $a$, $b$ and $c$ in that order, then $b=ac$. If the number 2 is placed in one of the sectors and the number 3 is placed in one of the adjacent sectors, as shown, what is the sum of the 36 numbers on the disc?

      

   2. Determine all values of $x$ for which $0<{\displaystyle \frac{x^2-11}{x+1}}<7$ .

7. 1. In the diagram, $ACDF$ is a rectangle with $AC=200$ and $CD=50$. Point $B$ lies close to $A$ on $AC$, and point $E$ lies close to $F$ on $FD$. Also, $\mathrm{△}FBD$ and $\mathrm{△}AEC$ are congruent triangles which are right-angled at $B$ and $E$, respectively.

      

      What is the area of the shaded region?

   2. The numbers $a_1,a_2,a_3,\dots$ form an arithmetic sequence with $a_1\ne a_2$. The three numbers $a_1,a_2,a_6$ form a geometric sequence in that order. Determine all possible positive integers $k$ for which the three numbers $a_1,a_4,a_k$ also form a geometric sequence in that order.

      (An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, $3,5,7,9$ are the first four terms of an arithmetic sequence.

      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.)

8. 1. For some positive integers $k$, the parabola with equation $y={\displaystyle \frac{x^2}{k}}-5$ intersects the circle with equation $x^2+y^2=25$ at exactly three distinct points $A$, $B$ and $C$. Determine all such positive integers $k$ for which the area of $\mathrm{△}ABC$ is an integer.

   2. In the diagram, $\mathrm{△}XYZ$ is isosceles with $XY=XZ=a$ and $YZ=b$ where $b<2a$. A larger circle of radius $R$ is inscribed in the triangle (that is, the circle is drawn so that it touches all three sides of the triangle). A smaller circle of radius $r$ is drawn so that it touches $XY$, $XZ$ and the larger circle.

      

      Determine an expression for ${\displaystyle \frac{R}{r}}$ in terms of $a$ and $b$.

9. Consider the following system of equations in which all logarithms have base 10: $$\begin{array}{rl}(\log x)(\log y)-3\log 5y-\log 8x& =a\\ (\log y)(\log z)-4\log 5y-\log 16z& =b\\ (\log z)(\log x)-4\log 8x-3\log 625z& =c\end{array}$$

   1. If $a=-4$, $b=4$, and $c=-18$, solve the system of equations.

   2. Determine all triples $(a,b,c)$ of real numbers for which the system of equations has an infinite number of solutions $(x,y,z)$.

10. For each positive integer $n\ge 1$, let $C_n$ be the set containing the $n$ smallest positive integers; that is, $C_n=\{1,2,\dots ,n-1,n\}$. For example, $C_4=\{1,2,3,4\}$. We call a set, $F$, of subsets of $C_n$ a Furoni family of $C_n$ if no element of $F$ is a subset of another element of $F$.

    1. Consider $A=\{\{1,2\},\{1,3\},\{1,4\}\}$. Note that $A$ is a Furoni family of $C_4$. Determine the two Furoni families of $C_4$ that contain all of the elements of $A$ and to which no other subsets of $C_4$ can be added to form a new (larger) Furoni family.

    2. Suppose that $n$ is a positive integer and that $F$ is a Furoni family of $C_n$. For each non-negative integer $k$, define $a_k$ to be the number of elements of $F$ that contain exactly $k$ integers. Prove that $${\displaystyle \frac{a_0}{{\displaystyle (\genfrac{}{}{0ex}{}{n}{0})}}}+{\displaystyle \frac{a_1}{{\displaystyle (\genfrac{}{}{0ex}{}{n}{1})}}}+{\displaystyle \frac{a_2}{{\displaystyle (\genfrac{}{}{0ex}{}{n}{2})}}}+\cdots +{\displaystyle \frac{a_{n-1}}{{\displaystyle (\genfrac{}{}{0ex}{}{n}{n-1})}}}+{\displaystyle \frac{a_n}{{\displaystyle (\genfrac{}{}{0ex}{}{n}{n})}}}\le 1$$ (The sum on the left side includes $n+1$ terms.)

       (Note: If $n$ is a positive integer and $k$ is an integer with $0\le k\le n$, then ${\displaystyle (\genfrac{}{}{0ex}{}{n}{k})={\displaystyle \frac{n!}{k!(n-k)!}}}$ is the number of subsets of $C_n$ that contain exactly $k$ integers, where $0!=1$ and, if $m$ is a positive integer, $m!$ represents the product of the integers from $1$ to $m$, inclusive.)

    3. For each positive integer $n$, determine, with proof, the number of elements in the largest Furoni family of $C_n$ (that is, the number of elements in the Furoni family that contains the maximum possible number of subsets of $C_n$).

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