1. Do not open this exam until you are told to do so.

2. Print your name clearly on your answer sheet. On the "Grade" line,
write

your grade in school (9,10,11, or 12) not the grade level of your team.

3. Print clearly on the answer sheet your answer to each question. You
should

not write decimal approximations to numbers such as ,and 10!.

You should simplify your answers as much possible. For example

6/4 should be simplified to 3/2, and square roots should not appear
in the

denominator. Frequently, several equivalent expressions will be considered

correct. For example, 3/2, 1 1/2, and 1.5 could each be correct.

4. Each correct answer is worth 1 point. A blank and an incorrect answer
are each

worth 0 points. An answer which is correct, but not adequately simplified,
may

receive half of one point.

5. No books, notes, calculators or headphones may be used.

6. Most students will probably be unable to finish the exam in the 2
hours allotted.

for the most part, the hardest questions are near the end of the exam.

7. denotes a line segment and denotes the length of a line
segment. The

vertices of a polygon such as ABCD, are indicated in consecutive order
as you

move around the polygon in a clockwise manner.

8. You may keep your questions and scratch work. You might want to keep
a record

of your responses for comparison with the solutions which will be given
out after the

examination.

9. The people with the five highest scores of all people taking the
test, without regard for

grade level, will receive cash awards of $250, $175, $100, $50, and
$25 respectively.

In case of ties, the cash awards will be split. For example if two
tie for first, they each

receive ($250+$175)/2 = $212.50.

10. A plaque will be awarded to the top individual and team in each
grade level (9-12 for

individuals, 10-12 for teams). In case of a tie, duplicate awards will
be presented.

**3. **Steve has 5 red poker chips and 7 blue poker chips.
Tom has 9 red poker

chips and 3 blue poker chips. If Steve's chips are worth a total
of $69 and

Tom's are worth a total of $57, determine the value of a blue chip.

**4.** Determine the slope of the straight line shown in the figure
below.

**5.** Two sides of a triangle are 14 and 20. If the third side of
the triangle is an integer,

what is the smallest possible value for the length of the third side?

**6. **This summer 1 US dollar was worth $1.45 in Canadian money.
At that time how much US money would be equivalent to 1 Canadian dollar?
Give your answer rounded to the

nearest cent.

**7. ** What is the equation of a line that passes through the
point (1,4) and is parallel

to the line *y = - 2x + 3* ? Put your answer in *y
= mx + b* form.

**10.** The sides of a triangle are *6,7* and *x*. What
is the largest value that could

be the area of such a triangle?

**11.** A * grad* is a unit of angle measurement where
there are

angle. How many regular polygons with side length 1 have angles with an

integer number of

**13.** Sally travels to work but must only travel north or east.
Furthermore due to

construction a section of road is unusable. Determine the number of
ways Sally can

get to work?

**14.** John is given *16* coins that look identical. In this
collection there are *15* genuine

coins all of the same weight and one fake coin that is lighter than
all of the others.

He has a balance as shown in the figure below that can be used to determine
if a stack

of coins on one side is lighter than a stack of coins on the other.
Determine the fewest

number of times the scale can be used so that John can guarantee the
identification of

the fake coin.

**18.** In the figure below a square with side length *1* is
inscribed in a circle. Find the area

of the shaded region.

**21.** A class has three teachers, Mr. P, Ms. Q and Mrs. R and six
students, Ali, Bob, Cal, Dee, Emy, and Fay. How many ways can they sit
in a line of *9* chairs if we have the property that between any two
teachers there are exactly two students?

**22.** From a group of *10* students an executive board is
chosen that has one president, one vice-president, one secretary and one
treasurer. If each person can fill at most one of the positions, how many
different executive boards are possible?

**24.*** * Let *P(x) = 0* be a fifth-degree polynomial
equation with integer coefficients

that has at least one integral root. If *P(2) = 13*
and *P(10) = 5*, compute a

value of x that must satisfy *P(x) = 0*.

**25. ** Two concentric circles are such that the smaller divides
the larger into two

regions of equal area. If the radius of the smaller circle is
*3*,
compute the

length of a tangent from any point *P* on the larger circle to
the smaller

circle.

**33**. A sphere and a regular octahedron have the same surface area.
Determine the square of the ratio between the diameter of the sphere and
the distance between opposite vertices of the octahedron.

**36.** How many positive integers n are there such that n
is an exact divisor of

at least one of the numbers

*10 ^{50}, 20^{30}*?

**37.** A *2 x 3* rectangle has vertices *(0,0), (2,0), (0,3*),
and *(2,3)*.

Perform the following sequence of steps:

1. Rotate the rectangle *90*? degrees clockwise about the point
*(2,0)*.

2. Then rotate this new rectangle *90*? clockwise about the point
*(5,0)*.

(At this point the side of the rectangle that originally was on the
*x*-axis
is now parallel to the *x*-axis).

3. Then rotate the current rectangle *90*? clockwise about the
point *(7,0)*.

Find the area of the region above the *x*-axis and below the curve
traced out by the point whose initial position is *(1,1)*.

**38. ** Consider the following carnival game. You toss
a penny onto the board and you win

if the entire penny (3/4" in diameter) lies inside one of the gray
squares (side length 1").

The board is placed on a square table that has dimensions .

Assuming that the entire penny lies on the table, what is the probability
of winning?

**39**. Consider the following tiles.

A perfect tiling of a *1 x n* rectangle is an arrangement of tiles
that completely covers the entire rectangle such that no two tiles overlap
or extend beyond the boundary of the rectangle. For example when *n =
6* we could have the following tilings:

Determine the number of different tilings for a *1 x 12* rectangle.

**40.** Let *S* be a set of *n* distinct real numbers.
Let *A _{S}* be the set of numbers that occur as the
averages of

two distinct elements of