Mathematics Contest Worksheet - Bluffton - 2009 Page 3
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1
2
3
4
5
6(n−4)/3
= 3
× 4. If n is two more than a multiple of 3,
one more than a multiple of 3, then P
max
(n−2)/3
= 3
× 2.
then P
max
a + log
c = b. Solve for c in terms of a and b without “log” in the answer.
14. Given: log
b
b
(ac). Exponentiating
Solution: Using the product rule for logarithms, the left side equals log
b
both sides of the equation (with base b) gives ac = b
, so c = b
/a.
b
b
+ 12x + 2y = −7. Express as an ordered pair (x, y).
2
15. Find the vertex of the parabola 4x
+ 3x + 1.5
) − 4 × 1.5
+ 2y = −7. Rearranging terms
2
2
2
Solution 1: Complete the square: 4(x
gives 2y = −4(x + 1.5)
+ 2, or y = −2(x + 1.5)
+ 1. That places the vertex at (−1.5, 1).
2
2
Solution 2: Rearrange terms to get y = −2x
− 6x − 3.5. This is now in “standard form”
2
+ bx + c), so the x-coordinate of the vertex is −
b
= −
6
= −
3
2
(ax
. The y-coordinate is
2a
4
2
therefore −2(−1.5)
− 6(−1.5) − 3.5 = 1.
2
Solution 3: Using calculus, we find the derivative of y = −2x
−6x −3.5, which is y = −4x −6.
2
This equals 0 when x = −1.5; that is the x-coordinate of the vertex. Continue as in Solution 2.
PART II: 30 Minutes; CALCULATORS NEEDED
Section A. Each correct answer is worth 1 point.
1. Latin teacher Mathematicus had 1 penny, 5 nickels, 10 dimes, 25 quarters, 50 50-cent pieces,
and 100 dollar bills. How much money did he have in all?
Solution: He has 1 × 1 + 5 × 5 + 10 × 10 + 25 × 25 + 50 × 50 + 100 × 100 = 13251 cents, or
$132.51.
√
2. Find 2009
2009 to the nearest hundredth.
√
.
= 90047.13615 . . ., which rounds to 90047.14.
Solution: A calculator gives 2009
2009
3. The greatest common factor of x and y is 3, and their lowest common multiple is 66. x is 33.
Find the value of y.
Solution: For any integers x and y, x y = GCF(x, y) × LCM(x, y). Therefore, 33y = (3)(66),
so y = 6.
+ 1.3.
2
4. Express as an improper fraction in simplest form:
5
+ 1.3 =
+
=
=
2
2
4
6+20
26
Solution:
.
5
5
3
15
15
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