Download 2011SeniorInterschool

Rocket City Math League
2011-2012
Inter-School Test
Senior Division
Answers must be written inside the adjacent answer boxes. All answers must be written in exact, reduced, simplified, and rationalized form. All
decimals and mixed numbers must be written as improper fractions. No calculators, books, or other aides may be used.
1. The area of an equilateral triangle is 1.5 times that of a regular hexagon. The length of the side of a square is 3 units greater
than the length of the side of the triangle. If the hexagon has a side length of 2, what is the ratio of the square’s area to the
hexagon’s area?
(1 point)
2. If log 3 x (5x  14)  2 , find x.
(1 point)
3. Four identical spheres with radii of 10 fit snugly in a 40 by 40 by 20 sealed box. What is the volume of the 3-D region that lies
within the box and outside all four spheres?
(1 point)
4. Solve for x: 0   x   x  3 x  7   2 x  3
2
2
10
7 x  70 . Write your answer in interval notation.
5. Convert 45729 to a base 4 number.
(1 point)
(1 point)
6. Johnny only walks up stairs and takes 1 or 2 steps at a time. If the stairway in his space station is 13 steps long, how many
different ways is it possible for Johnny to walk up these steps?
(2 points)
2n
7. Find the sum of the first 4 terms if a1  7 and a n  a n 1   1n   1  .
(2 points)
 2
8. Circle A, with center A, and Circle B, with center B, have radii of lengths 12 and 5 respectively. Circle B lies outside of Circle
A and the circles do not touch. Segment AB is intersected at point C by the common internal tangent of Circles A and B. If this
common internal tangent has a length of 119, what is the positive difference between segments AC and CB?
(2 points)
9. Find all points of intersection between the graph 16x 2  9y 2  96x  72y  2016 and the vertical asymptote(s)
of y 
x2  9 .
x  9 x  18
(2 points)
2
10. In the quadratic equation Ax 2  Bx  C  0 , A is the 15th term of the Fibonacci Sequence, B is the 16th term from the
Fibonacci Sequence, and C is the 17th term from the Fibonacci Sequence. Find the sum of the product of the roots and sum of the
roots of this quadratic equation. Note: Fibonacci sequence starts 1, 1, 2, 3, 5, 8, …
(3 points)
2
th
11. Let the n partial sum of an arithmetic sequence be denoted by S n  n  2n  Q . Find the sum of the first term and Q.
(3 points)
12. A teacher throws a marker along the parabola y  x  C where x represents horizontal distance and y represents height
above the floor. If the marker reaches the vertex of the parabola 7 units horizontally from the teacher, find the value of the
constant, C, such that the marker lands in a waste basket at a point 1 unit high and 35 units horizontally from the teacher
(assume the floor is at y  0 ).
(3 points)
1
4
2
13. Johnny wants to buy nuggets from a lunar cafe. However the cafe only sells nuggets in sets of 8 and 9. What is the greatest
number of nuggets that Johnny can’t buy (assuming he has sufficient money)?
(4 points)
14. Find the volume of the figure generated when the graph of y   1 x  3  8 is rotated about the line y  1 , where x is
2
between -6 and 9.
15. Find R  O  C  K  E  T  1000 if:
R= Number of positive integral divisors of 75937500
O= Reciprocal of the total surface area of a cone with radius 7  and height 15

(4 points)

C= The number of consecutive zeroes at the end of the product 2011!
17 2  82
K=
412  402
E= Sum of the first 8 triangular numbers
T=   2  i  2  i      2    2    1


(5 points)
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