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Alpha Equations and Inequalities
FAMAT State Convention 2011
For all questions, the answer choice “E) NOTA” denotes “None of These Answers is correct.”
Let
i  1 where applicable.
1. Find the values of x such that x  5  2 x  3 holds true.
A) x  2
C) 2  x  8
B) x  2
3
D) x  8
3
E) NOTA
2. Find the sum of the values of x that satisfy the equation  x  2   9  x  2   20  0 .
2
A) 5
B) 4
C) 5
A) 0
B) 2
C) 3
D) 9
E) NOTA
x
3. Consider the inequality x 
, for some nonnegative x . If the solution to this inequality is the set
x 1
of all x  such that x   or 1  x   , find the value of    .
D) 5
E) NOTA
4. Consider the equation 22 x 3  2 x 1  15 . If x   is the only solution to this equality, find the value
of  . If more than one solution exists, find their sum.
B) log 2 1.25
A) 0
C) log 2 1.5
D) log 2 6
E) NOTA
 2 5   x   4
      , find x  y .
 3 4  y  17 
5. Given 
A) 5
B) 1
C) 1
D) 5
E) NOTA
1 x a  a x
, where a is an arbitrarily large natural number.
a
x
0 x  a
For problems 6-7, let g  x   
6. Find the value of g 1  g  2  g  3  ...  g  a  .
A) 0
B) 1
C) 2
D) a
E) NOTA
7. Suppose g  a   g  1 . Which of the following is true about a ?
A)
B)
C)
D)
E)
a cannot be even.
a cannot be odd.
There are no restrictions on the values of a that satisfy this equality.
There is not enough information to determine anything about the possible values of a .
NOTA
8. Let a  b  be a fixed value. For what value(s) of c does a 2  b 2  c 2  2ab  8c  4  20 NOT
yield solutions, regardless of the value of a  b ?
A) c  4 only
E) NOTA
B) c  4 only
C) c  4 and c  4 only
Page 1 of 4
D) c  16 only
Alpha Equations and Inequalities
FAMAT State Convention 2011
9. Given two points A  0, 2  and B  0, 2  , which of the following represents the equation of the locus
of points P  x, y  such that for lengths PA and PB, PA  PB  8 ?
A)
x2 y 2
x2 y 2
x2 y 2
x2 y 2

 1 B)

 1 C)

 1 D)

1
12 16
12 16
16 12
16 12
10. Find the values of x that satisfy the inequality
A)
 , 2   2, 
B)
 2, 2
11. Find the area bounded by the system
A) 4
C)
x3  6 x 2  12 x  8
0.
x3  2 x 2  4 x  8
 2, 2   2,  
x2  y 2  8x  7  0
x2  y 2  8x  9  0
C) 16
B) 9
E) NOTA
D)
 2,  
E) NOTA
.
D) 25
E) NOTA
12. For what values of x does 9 x 2  12 x  9  5 hold true?
A)
 , 3 2    3 2 , 
D) All real numbers
B)
 , 2 3    2 3 , 
C)
 , 2 3    2 3 , 
E) NOTA
For problems 13-14, arithmetic mean  geometric mean , where the geometric mean is defined as the
square root of the product of two numbers.
13. Given two numbers 2k 1 and k  3 , what value of k will make their arithmetic and geometric
means equivalent?
11  1
5 1
C)
D) 4
E) NOTA
2
2
14. Given two numbers cos  k  and sin  k  in quadrant I, which of the following conclusions can be
A) 4
B)
made based on the inequality formed by their arithmetic and geometric means?
A) sin k  cos k  sin  2k   0
B) sin  k  1  sin  k   sin  2k   1
C) sin k  cos k  2sin  2k 
D) 1  sin  2k 
2
E) NOTA
 log 9 log 4 

  log 3 6 hold true?
 log 2 log 3 
15. For what value of z does the equality z 
A)
1
log 1.5
B)
1
2 log 1.5 
C) log1.5 2
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D) log1.5 4
E) NOTA
Alpha Equations and Inequalities
FAMAT State Convention 2011
16. Let an denote the nth term of the Fibonacci sequence (1, 1, 2, 3, 5, 8, …), and let f  n  be defined

1

such that f  n   
 1

n
n
i 1
i 1
n
n
 ai   ai
a  a
i 1
i
i 1
, for n
(for example, f 1  1, since the sum of the first 1
i
terms of the sequence is equal to the product of the first 1 terms). Find the smallest value of n such that
f  n  0 .
A) n  4
B) n  5
C) n  7
D) n  9
E) NOTA
17. Determine the number of solutions to the equality cos  tan on the interval 0,2  .
A) 0
B) 1
C) 2
D) 3
E) NOTA
18. Find the sum of the integral solutions to the equation x 4  x3  32 x 2  16 x  256  0 .
A) 1
C) 1
B) 0
D) 4
E) NOTA
19. How many integral values of x satisfy the equality x 4  10 x 2  9  0 ?
A) 1
B) 2
C) 3
D) 4
E) NOTA
 y  2x 1
, and let  x, y    a, b  be a solution to
2
2
 x  y  29
this system (where a and b are integers). Find the value of a  b .
20. Consider the system of equations defined by 
A) 7
B) 3
C) 3
D) 7
E) NOTA
21. Currently, the largest boy band in the world is known to have a total of twenty-one members.
Suppose at a given concert, one member is required to wear seven different outfits, some of the other
members are required to wear six, while the rest are required to wear eight. If a total of 145 outfits are
prepared for that concert, determine the number of members that need to wear six outfits. You may
assume members do not share outfits.
A) 9
B) 10
C) 11
D) 12
E) NOTA
22. Find the sum of the solutions to the equality 4sin x sin 2 x  3  4sin 2 x  6 cos x on the interval
0,   .
A) 
3
C) 4
B) 
3
D) 2
E) NOTA
23. Find the value of x that satisfies the equation cis 2x   3cis  x   4 on 0,2  ,
where cis  x   cos  x   i sin  x  .
A) 0
B) 
2
C) 
D) 3
2
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E) NOTA
Alpha Equations and Inequalities
FAMAT State Convention 2011
24. Given y  2x 2  2  3x  1 k and y  x 2  2x   x  4 k , find the positive value of k such that
the two equations have exactly one common solution  x , y  .
A)
1 5
2
B)
5  29
5  29
C)
2
2
D)
4  4 10
9
E) NOTA
25. Alice, Gabe and Justin are playing a number game such that w  x and y  z . After Alice fixes the
values of w and z, Gabe gives Alice a positive integer y, while independently Justin gives her another
positive integer x. She then tells her friends that all variables chosen have values less than ten and that
Justin has chosen a larger-valued integer than Gabe has. Which of the following statements is NOT
possible?
A)
B)
C)
D)
E)
Justin’s number is the largest of the four numbers.
The sum of Gabe and Justin’s numbers is smaller than the sum of Alice’s two numbers.
The sum of Gabe and Justin’s numbers is larger than the sum of Alice’s two numbers.
Justin and Alice have a number in common.
NOTA
26. Suppose f  x 
g  x  only if f  x   g  x  for all x where both functions are defined. Which of
the followings statements is true?
A) x 2
3x
B) x 2
2x 2
C) x 3
2x 2  7
D) 2x  9 2x  3
E) NOTA
27. For 0  x  2 , suppose c  cos x  sin x  d . If C is the maximum value of c that satisfies the
inequality, and if D is the minimum value of d that satisfies the inequality, find the value of C  D .
A) 2
B) 1
D) 2
C) 0
E) NOTA
28. Find the values of x that do NOT satisfy the inequality x 2  5x  75  9 .
A)
 7,6
B)
 12, 11   7, 6
C)
 7, 6  11,12
D) 
E) NOTA
 2 7 5 

4  nonnegative.
29. Determine the value of k that makes the determinant of the matrix  1 k
 1 2 8 
A) k  30
7
B) k  26
7
C) k  34
11
D) k  54
30. Which of the following statements is FALSE?
A) If a and b are nonnegative numbers, then a  b implies a2  b2 .
B) If a and b are nonnegative numbers, then a2  b2 implies a  b .
C) If a and b are negative numbers, then a  b implies a2  b2 .
D) If a and b are nonnegative numbers, then a  b implies
E) NOTA
Page 4 of 4
1 1
 .
a b
11
E) NOTA