DUSO MATHEMATICS LEAGUE

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DUSO MATHEMATICS LEAGUE
INDIVIDUAL QUESTIONS - MEET #1
OCTOBER 15, 2008
1.
EL. ALG (MATH A)
4 MINUTES
Each of the letters A through E are associated with a real number as follows.
A = 1 19
B = 1 2 5
C =  1 20
D = 4
E = 5
Arrange the five letters in decending order of their given values.
(Greatest to least, from left to right)
2.
EL. ALG (MATH A)
4 MINUTES
On the planet Crankios, an astronaut finds that their operation x  y means the
same thing as our y 2  x 2 . So on Crankios, 2 6 = 32. If the Crankians use
grouping symbols and the minus sign just as we do, find the value (simplest form)
of the following expression.
5  4  3  5  4  3
3.
IntALG./TRIG
(MATH B)
4 MINUTES
Find the sum of the digits in the base 10 expansion of the following expression.
700,000,000,0092
DUSO MATHEMATICS LEAGUE
INDIVIDUAL QUESTIONS - MEET #1
OCTOBER 15, 2008
4.
GEOMETRY
(MATH B)
6 MINUTES
P is a point in the interior of equilateral triangle ABC, such that perpendicular
segments from P to each of the sides of ∆ABC measure 2 inches, 9 inches and
13 inches. Find the number of square inches in the area of ∆ABC, and express
your answer in simplest radical form.
B
P
13
A
5.
GEOMETRY
9
2
C
(MATH A)
6 MINUTES
A square is inscribed in isosceles right triangle ABC, so that a side of the square
rests on the hypotenuse of ∆ABC as shown. If leg AB = 12, find the area of
the square and express your answer in simplest form.
A
12
B
6.
IntALG./TRIG
C
(MATH B)
6 MINUTES
When the sum of the first k terms of the series
12  2 2  3 2  ...  n 2  ....
is subtracted from the sum of the first k terms of the series
1  2  2  3  3  4  ...  nn  1  ...
the result is 210.
Find the value of k.
DUSO MATHEMATICS LEAGUE
RELAY TEAM QUESTION- MEET #1
OCTOBER 15, 2008
1.
If 8% of a number is 48, find the number.
2.
If n = 6! - 5!,
find the value of n, and add it to TNYWR.
3.
Find the number of degrees in the reflex angle formed by the minute hand and the
hour hand on a standard twelve hour wall clock at 9 o’clock.
Add that this number to TNYWR.
4.
Find the value of 44  22  7  4 to the nearest whole number.
Add that number to TNYWR.
5.
The squares of three consecutive whole numbers have a sum of 2030. Find the
smallest of these three whole numbers. (Caution, find the smallest whole number,
not the smallest square).
Add this number to TNYWR.
DUSO MATHEMATICS LEAGUE
SOLUTIONS - MEET #1
OCTOBER 15, 2008
Answers for RELAY TEAM QUESTION
1). 600
2). 1200
3). 1470
4). 1477
5). 1502
(Solutions for the Relay question)
8
1).
2). 6! = 720
5! = 120
n = 720 – 120 = 600
48

100
x
8x  4800
600 + TNYRW = 600 + 600 = 1200
x = 600
3).
The reflex angle is 360 – 90 = 270
270 + TNTWR = 270 + 1200 + 1470
4)
44  22 
qq
7
44 
72
22 
=
44  Q22
44 
22  3
44 
25
44  5 
4
=
9
49  7
7 + TNYWR = 7 + 1470 = 1477
5). Let x-1, x , x+1, represent the integers
x  12  x 2  ( x  1) 2  2030
x 2  2 x  1  x 2  x 2  2 x  1  2030
3x 2  2  2030
3x 2  2028 , x 2  676 , x = 26
25,26,27 } so the smallest is 25.
25 2  26 2  27 2  625  676  729  2030
25 + TNYWR =25 + 1477 = 1502
Individual Questions written by Steve Conrad www.mathleague.com ;edited by Dan Flegler & J.S.
Relay team question written for DUSO by J.A.; DUSO Editor J.S.
DUSO MATHEMATICS LEAGUE
SOLUTIONS - MEET #1
OCTOBER 15, 2008
Answers for INDIVIDUAL QUESTIONS
1). BAEDC
2). -96
3). 31
4). 192 3
5). 32
6). 20
Solutions for the Individual questions)
5  4  3  5  4  3
16  25  3  5  9  16
 9  3  5   7
9  81  49  25
 72  24
19  4.something and
1)
2) So
2 5  20  4.somethingbigger
1 19 = 5.something
B = 1 2 5 = 5.somethingbigger
C =  1 20 = 3.something bigger
A =
D =
E =
4
5
So
B then A then E then D then C
3)
=
=
=
=
-96
700,000,000,009 
2
11
4)
B
=
=
7 10  9
7 10   27 10 9  9
49 10   126 10   81
Area triangle APB = (1/2)(13)s
Area triangle BPC = (1/2)(9)s
2
11 2
11
22
11
2
P
13
=
Area triangle APC = (1/2)(2)s
9
2
A
s
C
Area triangle ABC =(24/2)s=12s
s2
Area of an equilateral ∆ ABC =
So the sum of the digits will be
4 + 9 + 1 + 2 + 6 + 8 + 1 = 31
5).
Area ∆ABC=
x
x
x
x
x x
6)
x
B
C
The figure above left consists of
the square and isosceles right
triangles, so the hypotenuse is 3x.
x
12  12  3x 
2
2
2
288  9 x 2
x 2  32
Area of the square is x 2  32
or
The figure at the right is divided into 9  isosceles
right triangles with hypotenuse x, so the area of the
square is 4/9 of the area of ∆ABC = (4/9)(1/2)(12)(12)
= (4/9)(72) = 32
-
4
and
3 =
4
3 = 192 3
1  2  2  3  3  4  ...  k k  1 )
( 12  2 2  3 2  ...  k 2
)
(
-
x
3 =12s
(16 3 ) 2
s2
A
4
s = 16 3
So
12
=
2 + 6 + 12 + … + (k2 + k)
1 + 4 + 9 + … + k2
1 + 2 + 3 + … + k
=
=
So result = 1 + 2 + 3 + … + k = 210
k
Sum = 1  k  = 210
2
k 2  k  420  0
k  21k  20  0
k = -21
k = 20
reject
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