2011 Excellence in Mathematics Contest
Individual Competition
Name: _____________________________________________________________
School Name: ______________________________________________________
Filling out the Scantron:
a. In the Name box, neatly print First Name then Last Name.
b. In the Subject box, write in the high school. No initials please.
c. In the Test No. box, write 1 for Level I and 2 for Level II.
 Level I is for students currently enrolled in Precalculus or higher level classes
 Level II is for students currently enrolled in classes below Precalculus
Testing Instructions:
a. Do not open the test until told to do so.
b. No talking.
c. No sharing of calculators.
d. Use a pencil to fill in the scantron.
e. Be careful when filing out the scantron – watch for poorly erased marks, incomplete
marks, skipped lines.
f. Scores are based on the number of correct answers. There is no penalty for wrong
answers.
g. When finished, turn in the scantron to the proctor, but keep the test.
h. Test will end promptly at 11:30.
2
3 is equal to
5 3

6 4
6
1.
a)
204
97
b)
95
12
c)
85
18
d)
80
19
e) none of these
2. The average of two numbers is 7/8. If one is 2/3 then the other is
a) 5/4
b) 3/4
c) 13/12
d) 12/11
e) none of these
3. If 7 x  2 y  5x  7 then x  y is
a) 7/2
4.
c) 5/2
d) 5/3
e) none of these
Two right triangles are shown in the figure. The larger has sides 7
and 3, the smaller has sides 5 and 3.What is the area in square units
of the shaded triangle?
a) 6
5.
b) 3/2
b) 5
c) 2
d) 4
e) none of these
Water is poured into the top of a container at 10 gallons per minute and is drained through
an opening in the bottom at a constant rate. The container is initially empty and after an
hour it contains 280 gallons. The drain rate, in gallons per minute, is
a) 9/2
b) 14/3
c) 16/3
d) 19/3
e) none of these
-1-
6. The sum of the roots of the equation 2 x 2  x  1  0 is
a) 1/2
b) -1/2
c) 3/2
d) -3/2
e) none of these
7. Ice cream fills a cone (right circular) and also fills the hemisphere above the
cone, as shown. If the height of the cone is 3 inches and the radii of the cone
and hemisphere are both 3 inches, then the total volume of ice cream (in
cubic inches) is
a) 27
b) 36
c) 30
d) 54
e) none of these
8. The sum of the positive integers that are less than 21 is
a) 231
b) 210
c) 220
d) 200
e) none of these
9. Sally has a Fahrenheit thermometer and a Centigrade thermometer. She reads both and
records the result as (F,C). The reading at the boiling point of water is (212,100), and at the
freezing point is (32,0). (If you have forgotten the linear relation between F and C, you can
derive it from the given information) When F = C the temperature (in degrees Fahrenheit) is
a) -30
b) -50
c) 30
d) 50
e) none of these
-2-
10.
A set of elements is closed under an operation if when you apply the operation to
elements of the set you always get an element of the set. For example, the sum of two
integers is always an integer, thus the set of integers is closed under the operation of
addition. If S is the set containing the squares of all positive integers, then S is closed under
a) addition
11.
c) multiplication
d) division
e) none of these
The sum of the digits of a two digit number is 10. If the digits are reversed then the
new number exceeds the original number by 36. The product of the digits of the new
number is
a) 16
12.
b) subtraction
b) 24
c) 32
d) 28
e) none of these
Zelma walks at 3 ft/sec along AC on the South side of a 30 ft wide
street. Her husband Herb walks at 5 ft/sec from A to B, where B is
the North side and opposite the midpoint of AC. He then turns and
walks to C at the same rate. The figure is shown but not to scale.
is the length, in feet, of AC if they start together at A and arrive
simultaneously at C?
C
on
B
What
A
a) 30
13.
b) 30 2
c) 60
d) 45
e) none of these
Given a rectangle R1 with area 5/2 and a second rectangle R2 with sides 1 greater than the
corresponding sides of R1. If the area of R2 is numerically equal to the perimeter of R1
then the sum of their perimeters is
a) 16
b) 22
c) 20
d) 18
e) none of these
-3-
14.
The sum of the ages (integers) of Ann and Joe is 100. Joe is now three times as old as Ann
was when Joe was 10 years older than Ann is now. The positive difference in their ages
now is
a) 12
15.
b) 16
d) 18
e) none of these
If a circle is inscribed in a triangle with vertex angles 30 , 60 , and 90 , then the area of
the triangle divided by the area of the circle is
a) (3  3 3 ) / 
e) none of these
16.
c) 26
b) (3  2 3 ) / 
c) (2  3 3 ) / 
d) (1  2 3 ) / 
On a single purchase you are given successive discounts of 5.5%, 10.5%, 14.5% and 12.5%
in any order that you wish. Which discount should you choose first to get the largest total
discount?
a) it does not matter
b) 14.5 %
c) 12.5%
d) 5.5%
e) none of these
17. Alice has a coin that comes up heads 60% of the time, otherwise tails. Bob’s coin is ‘fair’,
50% heads and 50% tails. They each flip 2 times and Alice wins if she has more heads than
Bob. The probability of Alice winning is
a) .32
b) .37
c) .39
d) .41
e) none of these
-4-
18.
You are given 25 coins that look identical; however one of
the coins is heavier than the others. You also have a pan
balance. The minimum number of weighings needed to find
the heavy coin is
a) 4
19.
b) 5
d) 7
e) none of these
A rhombus, ABCD, has sides of length 1. A circle with center A passes through C (the
opposite vertex.) Likewise, a circle with center B passes through D. If the two circles are
tangent to each other, then the length of BD is
a) (2  2 ) / 4
20.
c) 6
b) (1  7 ) / 2
c) 4  5 / 2
d) 5  2 5
e) none of these
A number is palindromic if it the same number when its digits are reversed, for example
14341. The odometer in Sally's car displays only 5 digits and does not display tenths of
miles. When she started her trip, the last three digits formed a palindromic number. One
mile later, the last four digits formed a palindromic number. After another mile, the last
five formed a palindromic number. List all possible initial odometer readings that satisfy
these conditions. The number of times that the digit 9 appears in this list is
a) 5
b) 6
c) 7
d) 8
e) none of these
-5-