2015 Pascal Contest
(Grade 9)

Tuesday, February 24, 2015
(in North America and South America)

Wednesday, February 25, 2015
(outside of North American and South America)

©2014 University of Waterloo

Instructions

Time: 60 minutes

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.

1. Do not open the Contest booklet until you are told to do so.
2. You may use rulers, compasses and paper for rough work.
3. Be sure that you understand the coding system for your response form. If you are not sure, ask your teacher to clarify it. All coding must be done with a pencil, preferably HB. Fill in circles completely.
4. On your response form, print your school name and city/town in the box in the upper right corner.
5. Be certain that you code your name, age, grade, and the Contest you are writing in the response form. Only those who do so can be counted as eligible students.
6. This is a multiple-choice test. Each question is followed by five possible answers marked A, B, C, D, and E. Only one of these is correct. After making your choice, fill in the appropriate circle on the response form.
7. Scoring:
   1. Each correct answer is worth 5 in Part A, 6 in Part B, and 8 in Part C.
   2. There is no penalty for an incorrect answer.
   3. Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
8. Diagrams are not drawn to scale. They are intended as aids only.
9. When your supervisor tells you to begin, you will have sixty minutes of working time.
10. You may not write more than one of the Pascal, Cayley and Fermat Contests in any given year.

Part A: Each correct answer is worth 5.

1. The value of ${\displaystyle \frac{20+15}{30-25}}$ is

   1. $1$
   2. $5$
   3. $2$
   4. $7$
   5. $0$

2. Which of the following figures is obtained when the shaded figure shown is reflected about the line segment $PQ$?

   

   1. 
   2. 
   3. 
   4. 
   5. 

3. If $8+6=n+8$, then $n$ equals

   1. $14$
   2. $22$
   3. $6$
   4. $-2$
   5. $9$

4. Which of the following numbers is greater than 0.7?

   1. $0.07$
   2. $-0.41$
   3. $0.8$
   4. $0.35$
   5. $-0.9$

5. The expression $4+\frac{3}{10}+\frac{9}{1000}$ is equal to

   1. $4.12$
   2. $4.309$
   3. $4.039$
   4. $4.012$
   5. $4.39$

6. The average age of Andras, Frances and Gerta is 22 years.

   Name	Age (Years)
   Andras	23
   Frances	24
   Gerta	?

   What is Gerta’s age?
   1. $19$
   2. $20$
   3. $21$
   4. $22$
   5. $23$

7. If $n=7$, which of the following expressions is equal to an even integer?

   1. $9n$
   2. $n+8$
   3. $n^2$
   4. $n(n-2)$
   5. $8n$

8. Jitka hiked a trail. After hiking 60% of the length of the trail, she had 8 km left to go. What is the length of the trail?

   1. $28\text{ km}$
   2. $12.8\text{ km}$
   3. $11.2\text{ km}$
   4. $13\frac{1}{3}\text{ km}$
   5. $20\text{ km}$

9. In the diagram, line segments $PQ$ and $RS$ intersect at $T$.

   

   The value of $x$ is

   1. $30$
   2. $20$
   3. $40$
   4. $50$
   5. $35$

10. The value of $\sqrt{16\times \sqrt{16}}$ is

    1. $2^1$
    2. $2^2$
    3. $2^3$
    4. $2^4$
    5. $2^5$

Part B: Each correct answer is worth 6.

11. Jim wrote the sequence of symbols $\mathrm{♡}\mathrm{♠}\mathrm{♠}\mathrm{♡}\mathrm{♢}\mathrm{♡}\mathrm{♡}\mathrm{♡}\mathrm{♢}$ a total of 50 times. How many more $\mathrm{♡}$ symbols than $\mathrm{♠}$ symbols did he write?

    1. $50$
    2. $150$
    3. $200$
    4. $250$
    5. $275$

12. What is the smallest positive integer that is a multiple of each of 3, 5, 7, and 9?

    1. $35$
    2. $105$
    3. $210$
    4. $315$
    5. $630$

13. Sixteen squares are arranged to form a region, as shown.

    

    Each square has an area of 400 ${\text{m}}^2$. Anna walks along the solid line path formed by the outer edge of the region exactly once. Aaron walks along the dotted line path formed by the inner edges of the region exactly once. In total, how far did Anna and Aaron walk?

    1. $160\text{ m}$
    2. $240\text{ m}$
    3. $320\text{ m}$
    4. $400\text{ m}$
    5. $640\text{ m}$

14. The operation $\otimes$ is defined by $a\otimes b={\displaystyle \frac{a}{b}}+{\displaystyle \frac{b}{a}}$. What is the value of $4\otimes 8$?

    1. ${\displaystyle \frac{1}{2}}$
    2. $1$
    3. ${\displaystyle \frac{5}{4}}$
    4. $2$
    5. ${\displaystyle \frac{5}{2}}$

15. At the end of the year 2000, Steve had $100 and Wayne had $10 000. At the end of each following year, Steve had twice as much money as he did at the end of the previous year and Wayne had half as much money as he did at the end of the previous year. At the end of which year did Steve have more money than Wayne for the first time?

    1. $2002$
    2. $2003$
    3. $2004$
    4. $2005$
    5. $2006$

16. Anca and Bruce left Mathville at the same time. They drove along a straight highway towards Staton.

    

    Bruce drove at 50 km/h. Anca drove at 60 km/h, but stopped along the way to rest. They both arrived at Staton at the same time. For how long did Anca stop to rest?

    1. $40\text{ minutes}$
    2. $10\text{ minutes}$
    3. $67\text{ minutes}$
    4. $33\text{ minutes}$
    5. $27\text{ minutes}$

17. In the diagram, six identical circles just touch the edges of rectangle $PQRS$ and each circle just touches the adjacent circles. The centres $T,V,W,Y$ of four of these circles form a smaller rectangle $TVWY$, as shown.

    

    The centres $U$ and $X$ lie on this rectangle. If the perimeter of $TVWY$ is 60, what is the area of $PQRS$?

    1. $600$
    2. $900$
    3. $400$
    4. $1200$
    5. $1000$

18. In a magic square, the numbers in each row, the numbers in each column, and the numbers on each diagonal have the same sum.

    $\begin{array}{|ccc|}\hline a& 13& b\\ 19& c& 11\\ 12& d& 16\\ \hline\end{array}$

    In the magic square shown, the sum $a+b+c$ equals

    1. $49$
    2. $54$
    3. $47$
    4. $50$
    5. $46$

19. Krystyna has some raisins. She gives one-third of her raisins to Mike. She then eats 4 raisins, after which she gives one-half of her remaining raisins to Anna. If Krystyna then has 16 raisins left, how many raisins did she have to begin?

    1. $42$
    2. $54$
    3. $60$
    4. $84$
    5. $108$

20. André has an unlimited supply of $1 coins, $2 coins, and $5 bills. Using only these coins and bills and not necessarily using some of each kind, in how many different ways can he form exactly $10?

    1. $10$
    2. $9$
    3. $8$
    4. $7$
    5. $6$

Part C: Each correct answer is worth 8.

21. Each diagram shows a triangle, labelled with its area.

    

    What is the correct ordering of the areas of these triangles?

    1. $m<n<p$
    2. $p<n<m$
    3. $n<m<p$
    4. $n<p<m$
    5. $p<m<n$

22. The chart shown gives the cost of installing carpet in four rectangular rooms of various sizes. The cost per square metre of installing carpet is always the same.

    		Width (metres)
    		$10$	$y$
    Length (metres)	$15$	$\mathrm{\$}397.50$	$\mathrm{\$}675.75$
    $x$	$\mathrm{\$}742.00$	$z$

    What is the value of $z$?

    1. $331.25$
    2. $463.75$
    3. $1815.25$
    4. $476.00$
    5. $1261.40$

23. How many triples $(a,b,c)$ of positive integers satisfy the conditions $6ab=c^2$ and $a<b<c\le 35$?

    1. $10$
    2. $8$
    3. $6$
    4. $7$
    5. $9$

24. Paula, Quinn, Rufus, and Sarah are suspects in a crime. The police found links between exactly four pairs of suspects: Paula and Quinn, Quinn and Rufus, Rufus and Paula, and Quinn and Sarah. These links can be shown in a diagram by drawing a point to represent each suspect and a line or curve joining two points whenever the two corresponding suspects are linked. An example of a drawing that represents this information is:

    

    Ali, Bob, Cai, Dee, Eve, and Fay are suspects in a second crime. The police found links between exactly eight pairs of suspects: Ali and Bob, Bob and Cai, Cai and Dee, Dee and Eve, Eve and Fay, Fay and Ali, Ali and Dee, and Bob and Eve. For how many of the following drawings can the six dots be labelled with the names of the six suspects so that each of the eight links given is represented by a line or curve in that drawing?

    

    Hide/Reveal Description of Diagram for Question 24

    1. In the first drawing, six dots lie along a straight horizontal line with line segments connected each dot to the next dot in the line. Also, the first (leftmost) dot is connected by a curve to the sixth (rightmost) dot and the second dot is connected by a curve to the fifth dot.
    2. In the second drawing, four dots are connected by lines forming a square. An additional dot is inside the square and is connected by lines to two opposite vertices of the square. A final dot is connected by a line to the dot inside the square and to one of the other two vertices of the square.
    3. In the third drawing, six dots are arranged into a circle with each dot connected by a curve to the two dots on either side of it in the circle. Also, in a group of three dots in a row, the outer two dots are connected by a line. Similarly, in the group of remaining three dots, the outer two dots are connected by a line.
    4. In the fourth drawing, six dots are arranged into two rows of three dots. The first dot in the first row is connected by a line to the first and second dots in the second row. The second dot in the first row is connected by a line to all three dots in the second row. The third dot in the first row is connected by a line to the second and third dots in the second row and connected by a curve to the first dot in the second row.
    5. In the fifth drawing, three dots are connected by lines forming a triangle. Another dot is at the midpoint of the base of the triangle and a line connects this dot to the opposite vertex of the triangle. Also, two more dots are on the base, one to the left of the midpoint and one to the right of the midpoint. These two final dots are connected by a curve.

    ---

    1. $4$
    2. $2$
    3. $1$
    4. $3$
    5. $5$

25. The first four rows of a table with columns $V$, $W$, $X$, $Y$, and $Z$ are shown.

    $V$	$W$	$X$	$Y$	$Z$
    $1$	$3$	$4$	$6$	$8$
    $2$	$5$	$7$	$11$	$15$
    $9$	$19$	$28$	$46$	$64$
    $10$	$21$	$31$	$51$	$71$

    For each row, whenever integer $n$ appears in column $V$, column $W$ contains the integer $2n+1$, column $X$ contains $3n+1$, column $Y$ contains $5n+1$, and column $Z$ contains $7n+1$. For every row after the first, the number in column $V$ is the smallest positive integer that does not yet appear in any previous row. The integer 2731 appears in column $W$. The complete list of columns in which 2731 appears is

    1. $W$
    2. $W\text{, }X\text{, }Y\text{, and }Z$
    3. $W\text{, }X\text{ and }Z$
    4. $W\text{, }Y\text{ and }Z$
    5. $W\text{ and }Z$

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