Download FAMAT State Convention - Gemini

FAMAT State Convention - Gemini 2011
1. Heracles lies on Monday, Tuesday, and Wednesday. Theseus lies on Thursday, Friday,
and Saturday. At all other times they both tell the truth. “Yesterday was one of my lying
days,” says Heracles. “Yesterday was one of my lying days too,” says Theseus. Which day
of the week is it in the context of the problem?
A. Monday
D. Thursday
B. Tuesday
E. None of the Above
C. Wednesday
2. Which number is the next number in this sequence: 2, 28, 126, 344, 730, _____.
A. 1332
D. 898
B. 860
E. None of the Above
C. 5110
3. The driver of a car glanced at the odometer and saw that it read 15,951 miles. He said to
himself; “That’s interesting. The mileage is a palindrome: it reads the same backward as
forward. It will be a long time before that happens again.” Just two hours later,
however, the mileage shown on the odometer was a new palindrome. How fast was the
car going in those two hours in miles per hour?
A. 25 mph
D. 105 mph
B. 51 mph
E. None of the Above
C. 55 mph
4. Given the function y  2( x  4)3  1 to the parent function y  x3 , which of the following is
NOT true about the graph of y  2( x  4)3  1 ?
A.
B.
C.
5.
It is a function.
D. It is vertically shifted up one unit.
It is a one-to-one function.
E. None of the Above
It is reflected over the y-axis.
Logarithms were given their name by John Napier, the Scottish mathematician credited
with their invention. His book, A Description of the Wonderful Law of Logarithms, was
published in 1614. Solve this problem for b in terms of a, so that John Napier may rest in
peace:
a
log  log b 2  log a  2
b
A. b =
100 a
a
B. b =
12 a
a
C. b =
2 a
a
10
a
E. None of the Above
D. b =
FAMAT State Convention - Gemini 2011
6. A pair of adjacent sides of a triangle form a 60˚ angle. If the adjacent sides have lengths
10 and 12, find the exact area of the triangle.
A. 15 2
B. 30 2
C. 15 3
D. 15 6
E. None of the Above
7. A sector of area 20π is cut from a circle of radius 10 inches; find the exact perimeter of
the sector in inches.
C. 20  2
B. 24
A. 24
D. 20  4
E. None of the Above
8. The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into
segments with lengths 6 and 8. Find the area of the triangle.
A. 7 3
B. 14 3
C. 24 3
D. 28 3
E. None of the Above
9. The lengths of the diagonals of a rhombus are 24 and 32. Find the length of the altitude
of the rhombus.
A.
48
5
B.
96
5
C.
192
5
D. Not Enough Information
E. NOTA
10. A chemist has M ounces of salt water that is M% salt. How many ounces of salt must he
add to make a solution that is 2M% salt? Assume that you cannot have more than 100%
salt in a solution.
A.
M2
50  M
B.
M2
100  2M
C.
M2
100  M
D.
M2
100
E. None of the Above
11. Which of the following equations identify circles:
i. 𝑟 = −3𝑠𝑖𝑛𝜃
ii. 𝑥 2 + 2𝑥𝑦 = 8
iii. 5𝑥 2 + 5𝑦 2 = 4
iv. 2𝑥 2 + 2𝑦 2 + 4 = 4
A. i only
B. iii and iv only
sin
12. Simplify:
A.
3
4
B.
C. i and iii only
D. i, ii, and iii only
E. NOTA



cos 2
(cot
)
6
3
2
2
1  csc2 (
)
7
1
2
C.
3
2
D.
3
2
E. None of the Above
FAMAT State Convention - Gemini 2011
13. The points (4, 1) and (-2,3) are reflected over the line y = x. Find the number of square
units in the area of the quadrilateral whose vertices are the points and their images.
A. 12
B. 14
C. 16
D. 18
E. None of the Above
𝑥+1
14. If 𝑥 3 +𝑥 2 −6𝑥 is written as three smaller fractions with constant numerators and
1
denominators of the form x  c , then the coefficient of 𝑥+3 is:
A. 0
B. -2/3
C. 3/10
D. -2/15
E. None of the Above
15. A battleship traveled 10 miles on a course of 30 degrees North of East. The ship then
changed its course to one of 30 degrees South of East and traveled 10 more miles. How
far was the battleship from the original starting point in yards?
A. 50, 000 3 yards
B. 17, 600 3 yards
C. 8800 3 yards
D.
5 3
yards E. NOTA
3
16. In the distant future, the FAMAT state convention will be VERY large. At the beginning
of the convention each of the infinite number of students attending the convention will
put their name on a card and turn it in to Mrs. Hillard. At the end of the convention, the
cards will be randomly redistributed at one time (in a “making it rain” fashion) to the
students and whoever receives their own card back, wins a prize. What is the
probability NO ONE wins a prize?
A.
1
6e
B.
e
7
C.
1
e
D.
1
2
E. NOTA
17. For what rational number a do the equations x3  ax 2  7  0 and x 2  ax  2  0 have
a common solution?
A. 
7
2
B.
14
7
C.
41
14
D.
57
14
E. NOTA
18. Let the solutions of equation mx 2  nx  p  0 with integral coefficients be r1 and r2 .
Given that r1  r2  2i , r1 and r2 have integral components (for a  bi , a and b are
integers) and
A. 3
1 1 3
  , find m  n  p.
r1 r2 5
B. 1
C. 3
D. 5
E. NOTA
FAMAT State Convention - Gemini 2011
19. I enjoy a good doodle sometimes and my favorite thing to doodle is fractals. In this
particular drawing, I have an isosceles right triangle with legs of length 8. I then draw an
altitude to the hypotenuse creating two congruent triangles. In one of the smaller
triangles, I draw another altitude creating two smaller triangles. If I continue to draw
altitudes for the rest of time (infinitely), what will the sum of the lengths of all the
altitudes be?
A. 8 2
C. 8  8 2
B. 12 2
D. 16
E. NOTA
20. The integers from 1 to 99 are written in order around a circle in a clockwise direction.
Starting with 1 and counting in a clockwise direction, every 13th number is circled. (13,
26,…91, 5, 18,…) Which will be the 25th integer to be circled?
A. 16
B. 28
C. 72
D. 93

E. NOTA

21. Given that f  3x  5  x2  2 x  1 and g 2 x  1  3x  4 , find f  g  6   .
A. 14
B. 27
C. 34
D. 167
E. NOTA
22. Express the following as a rational number:
4  2 3  28  10 3
15
A. 
4
3
23.If sin 2 15
A. 
31
16
B. 
4
5
 is one root of x
B. 
15
16
C. 
2
4
15
D.
2
5
E. NOTA
 bx  c  0 , find the sum of the rational numbers b and c .
C.
7
8
D.
17
4
E. NOTA
24.In base eight, the four-digit numeral BBCC is the square of the two-digit numeral AA.
Find base ten sum of the digits A, B and C.
A. 11
B. 14
C. 15
D. 20
E. NOTA
FAMAT State Convention - Gemini 2011
25. Given three sets of objects A, B and C. 5
9
of the objects in A are in B; 4
7
of the
objects in B are in A; 3
of the objects in A are in C; 2 of the objects in C are in A. If
4
3
the ratio of the objects in A, B and is expressed as a : b : c , where a , b and c are the
smallest possible positive integers, find a  b  c .
A. 140
B. 152
C. 186
D. 223
E. NOTA
26. A spider is on a wall 6 inches directly above a fly on the same wall. The fly starts to
move horizontally on the wall at a rate of 1 inch per second. After 3 seconds the spider
starts moving in a straight line at a rate of 2 inches per second in order to intercept the
fly. How far in inches will the spider travel in order to intercept the fly assuming the
wall has sufficient width?
A. 5
B. 6
2010
27. Evaluate:
D. 14
E. NOTA
D. 10025
E. NOTA
ln n
 ln  2011n  n 
2
n 1
A. 502
C. 10
B. 1005
C. 2010
28. f  ab   f  a   f b  where a and b are positive real numbers. If f  9  5 , find
 3 .
f 1
A. 
5
2
B.
1
5
C.
4
5
D.
7
2
E. NOTA
29. A triangle of area 5 has vertices 1,6  ,  3, 4  and  x, y  , for which  x, y  is a point
on the line y  3x  4 . If x and y are integral, find the value of x  y .
A. 2
B. 0
C. 10
D. 20
E. NOTA
30. Let a1 , a2 and a3 be consecutive terms in an arithmetic sequence. If a1  a2  a3  60
and a1  a2  a2  a3  a1  a3  1175 find the value of a1  a2  a3
A. 3750
B. 5250
C. 6000
D. 7500
E. NOTA
FAMAT State Convention - Gemini 2011