Rocket City Math League
2012-2013
Round 1
Apollo Test Solutions
n
 x2 
1.
1
n(n  1)( 2n  1)
, so
6
10
x
2

1
10  11  21
 385
6
385
2. [ln(1000)][log(𝑒 2 )] = [log(1000)][ln(𝑒 2 )] = 3 * 2 = 6.
6
3. Where a+bi corresponds to a point a in the real direction and b in the imaginary direction, the reflection of this
point over the imaginary axis would correspond to a point –a in the real direction but still b in the imaginary
direction. This reflected point corresponds to the complex number –a+bi.
4. Hyun Su had a 120 mile (3hr*40mph) head start. After that, both rockets went forward—Hyun Su’s at 40 mph
and Raisa’s at 60 mph. Raisa’s rocket gained 20 miles for every hour traveled. That means for it to catch up to
Hyun Su, it would take 120/20, or 6 hours. Alternatively, let x be the number of hours it would take for both
rockets to be at the same location. Then,
-a+bi
6
x  40  ( x  3)  60  40 x  60 x  180  20 x  180  x  6
5. x  6x  y - 10y  22  0  (x  6x  9)  ( y - 10y  25)  -22  9  25  12 , so r2=12 and r=
2
2
2
2
12 = 2√3 using the equation of a circle in the coordinate plane.
√3
6. A = (.5)(a)(b)(sin(C)) = (.5)(6√6)(12)(sin(60)) = (.5)(6√6)(12)( ) = 54√2
2
𝟓𝟒√𝟐
7. The expansion is found using Pascal’s triangle. The fourth line is 1, 4, 6, 4, 1. Thus, the equation is 1(81x4) –
4(27x3y) + 6(9x2y2) – 4(3xy3) + 1(y4), or 81x4 – 108x3y + 54x2y2 – 12xy3 + y4. The sum of the coefficients is then
81 – 108 + 54 – 12 + 1=16. Alternatively, the sum of the coefficients is also the value of (3x-y)4 when x=y=1.
(3(1)-(1))4=(2)4=16.
8. Using the formula for the sum of an infinite geometric series 𝑆
1
𝑥−1
2𝑥
𝑥
=
𝑎1
1−𝑟
,𝑆
2√𝟑
=
1
𝑥
1
1− 2
𝑥
=
𝑥
.
1
𝑥 2 −1 𝑥+1
+
16
110
= 𝑥 2 −1 = 2 (𝑥 2 −1) = 2(55) = 110.
9. The Galaxy goes 330 miles forward for every 8 gallons. That means it goes a total of 1650 miles. The Starchild
just goes 1080 miles. The Moondust goes 210 miles for every 5 gallons. That means it goes 2240 miles total. The
Rumble goes 125 miles for every 3.5 gallons, or 1425 miles. The Moondust is the only rocket ship that can go the
2000 miles.
10. 5𝑥 3 − 11𝑥 2 + 9𝑥 = −6  5𝑥 3 − 11𝑥 2 + 9𝑥 + 6 = 0. 𝑚2 + 𝑛2 + 𝑝2 =
−11
(𝑚 + 𝑛 + 𝑝)(𝑚 + 𝑛 + 𝑝) − 2(𝑚𝑛 + 𝑛𝑝 + 𝑚𝑝). By Vieta’s formulas, 𝑚 + 𝑛 + 𝑝 = −
=
9
11
5
5
𝑚𝑛 + 𝑛𝑝 + 𝑚𝑝 = , so 𝑚2 + 𝑛2 + 𝑝2 =
∗
11
5
9
31
5
25
−2( ) =
5
11
5
and
.
The
Moondust
𝟑𝟏
𝟐𝟓
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