Download A square is divided into two rectangles whose areas are in the ration

Senior High Mathletes
Solutions 10/9/12
1.
200
The circle has a radius of 10 units and a diameter of 20. The diameter of the
circle is a diagonal of the square. Divide by 2 to get a side of the square.


2
s = 10 2 . A = 10 2 = 200 square units.
2.
574
x + 6y = 2009
6x + y = 2009
7x + 7y = 4018 Divide by 7 to get x + y = 574
3.
120
One diagonal of the rectangle creates a 30-60-90 right triangle. When two
diagonals intersect, each obtuse triangle has two 30 degree base angles.
4.
60
Since the diameter of the smaller circle is half, so is the circumference: 20 inches
Convert 100 ft to 1200 inches and divide by 20 inches.
5.
30
90 – x = .40(180 – x)
90 – x = 72 – .40x
18 = .60x
6.
3
Replace 4 with 22 and 2(22x) = 4x + 64 becomes 2(22x) = 22x + 64
So 22x = 64
7.
8
1(1)  2(2)  3(3)  ...  10(10) 385

7
55
55
The median is the 28th score. It is easier to start with the highest value. 27 scores
account for all the 10’s, 9’s, and 8’s, so the median is 7.
The mode is clearly 10.
7  7  10
8
3
mean:
8.
40
x
3x
The rectangles will have the same height (or base), so the other dimension for the
two rectangles must be in the ratio 3:1 as shown.
Smaller perimeter = 4x+4x+x+x = 10x
Larger perimeter = 4x + 4x + 3x + 3x = 14x
9.
4
A number is divisible by 6 if it is divisible by 3 and 2. So n must contain 3 1’s so
the sum of digits is 3 and n must end in 0: 1110, 11100, 11010, 10110 are the
only possibilities.
10.
18
The fourth vertex may be (6, 8), (-4, 8), or (6, -6)
11.
50
Using multiplication there are 38 possible answers:
9 multiples of 9. 8 multiples of 8 (excluding 72). 7 multiples of 7 (excluding 63
and 56). 4 multiples of 6 (excluding 18,24,42,48,54). 5 multiples of 5 (excluding
30,35,40,45). Include 4,3,2,1,0. [38 total]
Using addition, there are 3 answers that have not been addressed yet: (11,13,17)
Using subtraction, we may get any of the 9 negative integers -1 through -9.
Using division, all possible integer quotients have already been accounted for.
12.
9
P(HHH or TTT) =
2
8
P(111 or 222 or 333 or 444 or 555 or 666) =
2
6

9
8 216
13.
7
Multiply both sides by the LCD (x + 1)(x + 2)(x + 3) to obtain
(x + 2)(x + 3) + (x + 1)(x + 3) = (x + 1)(x + 2)
2x2 + 9x + 9 = x2 + 3x + 2
c
x2 + 6x + 7 = 0. The product of roots is  7
a
6
216
14.
89
Find the last two digits of the number 92009 ?
There is a 10-part cycle when 9 is raised to the 1st, 2nd, 3rd, …, 10th power. These
numbers end in 09, 81, 29, 61, 49, 41, 69, 21, 89, 01. Raising 9 to the 2009th
power will end in the same digits as raising to the 9th power.
15.
600
Form 2 isosceles triangles as shown. These triangles have a hypotenuse of 10 and
legs of 5 2 . Now the measurement needed is a hypotenuse of a right triangle
whose legs are 10 and 10  10 2 . Pythagorean theorem gives this hypotenuse as
400  200 2
5 2
10
5 2
10