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MATHEMATICS CONTEST 2003 SOLUTIONS

PART I: 30 Minutes; NO CALCULATORS

Section A. Each correct answer is worth 1 point.

1. Find the sum of the first six prime numbers.
Solution : 2 + 3 + 5 + 7 + 11 + 13 = 41 (the number 1 is not a prime).

2. Give the official ( and more common ) name for a regular quadrilateral.
Solution: “Regular” means equal sides and equal interior angles, so it’s a square.

3. Express the repeating decimal as a ratio of two positive integers in simplest form.
Solution 1: Let x = ; then 10x = , and 10x − x = 1.777777 . . . − 0.177777 . . . = 1.6.
If 9x = 1.6, then x = 1.6/9 = 16/90 = 8/45 .

Solution 2: Recall that 7/9 = , so we want 7/9 − 3/5. (Or 1/10 + 7/90, or 1/6 + 1/90 .)

4. If 2x + 1 = 2003, find the value of 3x − 1000.
Solution: Solve for x: If 2x + 1 = 2003, then 2x = 2002, so x = 1001. Then 3x − 1000 =
3(1001) − 1000 = 2003.

5. A non-isosceles triangle has integral sides of 4, 5, and x. Find all possible values of x.
Solution: The triangle inequality theorem says that the sum of any two sides must be greater than
the third side, so 5−4 < x < 4+5, or 1 < x < 9. But the sides are integral (they are integers),
so x must belong to the set {2, 3, 4, 5, 6, 7, 8}. Additionally, the triangle is non-isosceles (no
two sides can be equal). Therefore, x is in {2, 3, 6, 7, 8}.

6. Using some or all of the digits 0–9 (no digit more than once ), construct the largest possible
six-digit odd number with a 9 in the tens place.
Solution:
9 must be in the tens place.
Biggest possible number.
Biggest possible number.
Biggest possible number, number must be odd.

7. 7 + (−7) = 0. This is an example of what basic property of addition ?
Solution: Additive inverse property or Opposites property.

Section B. Each correct answer is worth 2 points.

8. If (x5 + x4 + x − 5) is divided by (x + 1), find the remainder.
Solution 1 (Long division):

Solution 2 (Synthetic division): Works when dividing by x −c. Rewrite x +1 as x −(−1), then
write out the coefficients , multiply and add ; the remainder is the last number on the bottom.

Solution 3 ( Math knowledge ): The remainder theorem says you can just plug in −1 to get the
remainder: (−1)5 + (−1)4 + (−1) − 5 = −6

9. Find the 2003rd digit after the decimal point in the decimal representation of 4/7.
Solution: Rational numbers (such as fractions) either stop or repeat. If you do long division far
enough, the pattern is clear:
4 ÷ 7 = 0.57142857142857... or
So it repeats every six numbers. Now divide 6 into 2003, and observe that the remainder is 5,
so we choose the 5th repeated digit: The answer is 2.

10. In the figure on the right, the length of tangent is 12,
PD = 8, and chord bisects chord . If EB = 3, find
the length of
Solution: (AP)(AP) = (PD)(PC), so 122 = (8)(PC),
or PC = 18. Then CD = 10, and because bisects
, CE = ED = 5. Now (AE)(EB) = (CE)(ED), so
(AE)(3) = (5)(5), and we conclude that AE = 25/3 .

11. Express in simplest radical form (no radicals in the denominator ):

Solution: This is sometimes called “rationalizing the denominator”:

12. Write the exact numerical value of x if log8 128 = x. Express in simplest form.
Solution 1: Transform to exponential form: 8x = 128. Now rewrite both 8 and 128 as powers of
2: 8 = 23 and 128 = 27, so 8x = (23)x = 23x = 27. Therefore, 3x = 7, or x = 7/3.
Solution 2: Use the change-of-base formula and properties of logarithms:

Section C. Each correct answer is worth 3 points.

13. The ellipse x2 + 2y2 + 12y − 10x − 57 = 0 has a major axis with two endpoints. Find the
coordinates of the endpoint that lies in quadrant IV. Express in ordered pair form, (x, y).
Solution: Put in standard form by completing the squares:

The ellipse is centered at (5,−3), and the major axis is therefore horizontal with length 2a = 20;
the endpoint in quadrant IV is (5 + 10,−3) or (15,−3).

14. Softball player Berni Williams has 120 hits in 300 at-bats for a current batting average of .400.
In today’s game, she will have 5 at-bats. What is the probability that she will get exactly 2 hits?
Solution: The number of hits among her next five at-bats has a binomial distribution (more on this
below), so P(exactly 2 hits) = (0.4)2(0.6)3, where is “5 choose 2” (sometimes written
as ), which is equal to 10. This evaluates to

If you have never heard of the binomial distribution , here are two ways to think about it:
1. Write out every possible sequence of two hits and three outs:

HHooo, HoHoo, HooHo, HoooH, oHHoo, oHoHo, oHooH, ooHHo, ooHoH, oooHH

There are 10 such sequences, and each has probability (0.4)2(0.6)3.
2. The binomial distribution gets its name from the terms in the expansion of the binomial
(q + p)n, where p is the probability of success (a hit), q = 1 − p is the probability of failure
(an out), and n is the number of attempts. Thus:

(0.6 + 0.4)5 = 0.65 + 5(0.640.41) + 10(0.630.42) + 10(0.620.43) + 5(0.610.44) + 0.45.

Each term in this sum corresponds to the probability of a certain number of failures and successes;
we want the third term.

15. Solve the inequality and graph its solution on the given number line.
Solution: Either , so that x > 1 or x < −5. This can be drawn either
of these ways:

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