Download Lassiter Varsity Test 2005

2005 Lassiter Invitational Varsity Math Tournament
Page 1 of 7
1. In Triangle ABC, AB=2x+5 , AC= 3x-2, and BC= 4x-8, which of the following is the smallest
possible integral value for x?
a)
b)
c)
d)
e)
1
2
3
4
5
2. What two-digit number has the following property: the number is equal to twice the sum of
its digits.
a)
b)
c)
d)
e)
0
18
23
27
73
3. In a circle of radius 2, certain line segments of length 2 can be drawn tangent to the circle
at their respective midpoints. What is the area of the region that encompasses all of these
line segments?
a)
b)
c)
d)
e)

2
2

4
5
4. If xy - 3x + 5y = 159 , then how many solutions (x,y) exist in integers?
a)
b)
c)
d)
e)
3
9
16
18
30
2005 Lassiter Invitational Varsity Math Tournament
Page 2 of 7
5. Let (n) be a real number so that : n  = n - 2 + n - n , where n is the greatest integer
2
less that or equal to n. This means (n) must have the following properties:
a)
b)
c)
d)
e)
1  n < 0
2n<3
1n<2
1  n < 0 or 2  n < 3
1n<3
6. Determine the number of zeros at the end of 2005! .
a)
b)
c)
d)
e)
401
481
500
515
516
7. Define a trifect number to be a number that, when expressed in base 3, the sum of the
digits is equal to 8. How many trifect numbers exist less than or equal to 243 (base 10) ?
a)
b)
c)
d)
e)
10
13
15
18
27
8. In an equilateral triangle, with side length x, the ratio of the area to the perimeter is 17.
Find the side length.
a) 204 3
b) 136 3
c) 68 3
d) 17 3
e) 4.25
2005 Lassiter Invitational Varsity Math Tournament
Page 3 of 7
9. If you averaged 60 mph to get to this tournament, and x mph to return, and your average
rate for the entire trip is 50 mph, what is x?
a) 40
b) 42
6
7
c) 44
d) 48
1
7
e) 55
10. Find the infinite sum:
a)
b)
c)
d)
e)
1
2
3
4
+
+ 3 + 4 ...
10 100 10 10
10
81
1
8
1
5
5
27
14
65
11. Phillip dips a 4 X 4 X 4 cube into paint. After the paint dries, he cuts the cube into
1 X 1 X 1 pieces. He then picks a unit cube from this set at random and rolls the unit
cube. What is the probability that the face that lands up is painted?
a)
b)
c)
d)
e)
1
16
5
24
1
4
15
64
19
64
2005 Lassiter Invitational Varsity Math Tournament
Page 4 of 7
12. Let rhombus TINA have diagonals of length 14 and 48. Circle P is inscribed inside of this
rhombus. Find the radius of P.
a)
b)
c)
d)
e)
28
5
168
25
50
7
10
25
2
13. If (log x)2 + log x 2 + 1= 0 , find x.
a) -1
1
b)
10
c) 1
d) 2
e) 10
sin 2x 
 sin 2x
+
14. Simplify 

 2 cos x 2 sin x 
a) 1
b) tan 2x
c) sin x + cos x
d) sin 2x + cos 2x
e) 1 + sin 2x
2
15. In ∆ABC, AB=13, BC=15, AC=14. Let BD be the median to AC and BE be the altitude to
AC. What is the area of ∆BDE ?
a)
b)
c)
d)
e)
9
10
11
12
13
2005 Lassiter Invitational Varsity Math Tournament
Page 5 of 7
16. If x=465, some number, p, exists so that the sum of all possible positive integer factors of
p is equal to x. Find p.
a)
b)
c)
d)
e)
144
200
225
250
256
17. In some triangle, the csc θ = 2 cos x . Find the cot θ.
a)
b)
c)
d)
e)
cos 2x
csc 2x
sec 3x
tan x
1
18. How many clock times from 12:00 AM to 12:00 PM exist, so that the hour evenly divides
the number of minutes? (2:30, 8:00, and 6:48 work, but 3:16, 7:15, and 10:09 don’t).
a)
b)
c)
d)
e)
12
156
167
176
188
19. A target has values of 5 and 7 on it. What is the largest positive integer value that you can
not obtain by throwing n darts and hitting the target? (Note: n is a whole number.)
a)
b)
c)
d)
e)
20
21
22
23
34
20. If ( ab +ac +ad+ae +bc +bd.....de = 15 ) and ( a2 + b2 + c2 + d2 + e2 = 370 ) find
( a +b +c +d+e ).
a)
b)
c)
d)
e)
15
20
25
30
35
2005 Lassiter Invitational Varsity Math Tournament
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21. If there exists some number, n , so that n = (ab )(ba ) where a and b are distinct positive
integers greater than 1, find the minimum number of factors that n can have.
a)
b)
c)
d)
e)
9
4
12
14
16
22. Three numbers are chosen from the set 1,2,....10 with replacement. What is the
probability that the product of the three numbers is even, but not divisible by 4?
a)
b)
c)
d)
e)
3
40
1
8
9
40
13
20
1
38
6-n
where 1£ n£5 and
15
n is an integer, what is the probability he passes all 3 of them? (Passing is 3 or better)
23. If Matt takes 3 AP tests, and the probability that he gets an n is
a)
b)
c)
d)
e)
1
8
27
125
2
5
8
27
8
125
2005 Lassiter Invitational Varsity Math Tournament
Page 7 of 7
Free Response:
Write the answers to each of the following on the back of your GradeMaster sheet.
26. If Mrs. Poss and Mr. Slater each roll his/her own dice, what is the probability that Slater’s
number evenly divides Poss’s number?
27. For the following cubic equation, find the sum of the squares of the reciprocals of the
roots: 2x 3 - x 2 - 2x + 1= 0
28. In the Quadrant I of the xy-plane, a line with slope –1 defines a triangle bound by x, y-axis
and the line with area of 405000. This line undergoes a translation of all points
(x, y)  ( x, y) . What is the area of the new region bound by the translated line, the
x-axis, and the y-axis? (Express answer as A where A is an integer)
29. A new kids’ puzzle has 4 distinct pieces. The machine that makes the puzzles randomly
puts either 0, 1, 2, 3, or 4 different pieces in Bag A and either 0, 1, 2, 3, or 4 different
pieces in Bag B. If you are given both Bag A and Bag B, and the probability that you can
a
assemble the puzzle is
(when written in reduced form) , find a .
b
30. Allow ¡(n)! to represent n! (n -1)! (n -2)! .... 3!2!1!.
How many zeroes are at the end of ¡(26)! ?