University of South Carolina High School Math Contest/1993 Exam/Problems
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 See also
Problem 1
If the width of a particular rectangle is doubled and the length is increased by 3, then the area is tripled. What is the length of the rectangle?
Problem 2
Suppose the operation is defined by
If
then
Problem 3
If 3 circles of radius 1 are mutually tangent as shown, what is the area of the gap they enclose?
Problem 4
If is expanded and written in the form
where
and
are real numbers, then
Problem 5
Suppose that is a function with the property that for all
and
and
What is the value of
?
Problem 6
After a price reduction, what increase does it take to restore the original price?
Problem 7
Each card below covers up a number. The number written below each card is the sum of all the numbers covered by all of the other cards. What is the sum of all of the hidden numbers?
Problem 8
What is the coefficient of in the expansion of
Problem 9
Suppose that and
are integers such that
and
. Then one possible value of
is
Problem 10
Problem 11
Suppose that 4 cards labeled 1 to 4 are placed randomly into 4 boxes also labeled 1 to 4, one card per box. What is the probability that no card gets placed into a box having the same label as the card?
Problem 12
If the equations and
have exactly one root in common, and
then the other root of equation
is
Problem 13
Suppose that and
are numbers such that
and
. Then
Problem 14
How many permutations of 1, 2, 3, 4, 5, 6, 7, 8, 9 have:
- 1 appearing somewhere to the left of 2,
- 3 somewhere to the left of 4, and
- 5 somewhere to the left of 6?
For example, 8 1 5 7 2 3 9 4 6 would be such a permutation.
Problem 15
If we express the sum
as a rational number in reduced form, then the denominator will be
Problem 16
In the triangle below, and
are the midpoints of
and
respectively.
and
intersect at
. If the length of
is 4, then what is the length of
?
Problem 17
Let represent the greatest integer that is less than or equal to
. For example,
and
. Then what is the value of
Problem 18
The minimum value of the function
as varies over all numbers in the largest possible domain of
, is
Problem 19
In the figure below, there are 4 distinct dots and
, joined by edges. Each dot is to be colored either red, blue, green, or yellow. No two dots joined by an edge are to be colored with the same color. How many completed colorings are possible?
Problem 20
Let be the 63 nonempty subsets of
. For each of these sets
, let
denote the product of all the elements in
. Then what is the value of
?
Problem 21
Suppose that each pair of eight tennis players either played exactly one game last week or did not play at all. Each player participated in all but 12 games. How many games were played among the eight players?
Problem 22
Let
and
Then which of the following inequalities is true?
Problem 23
The relation between the sets
and
is
Problem 24
If and in general
then
Problem 25
What is the center of the circle passing through the point and tangent to the circle
at
? (Two circles are tangent at a point
if they intersect at
and at no other point.)
Problem 26
Let . Then the first nonzero digit in the decimal expansion of
is
Problem 27
Suppose is a triangle with area 24 and that there is a point
inside
which is distance 2 from each of the sides of
. What is the perimeter of
?
\[ \mathrm{(A) \ } 12 \qquad \mathrm{(B) \ }24 \qquad \mathrm{(C) \ }36 \qquad \mathrm{(D) \ }12\sqrt{2} \qquad \mathrm{(E) \ }12\sqrt{3} \]
Problem 28
Suppose is a triangle with 3 acute angles
and
. Then the point
(A) can be in the 1st quadrant and can be in the 2nd quadrant only
(B) can be in the 3rd quadrant and can be in the 4th quadrant only
(C) can be in the 2nd quadrant and can be in the 3rd quadrant only
(D) can be in the 2nd quadrant only
(E) can be in any of the 4 quadrants
Problem 29
If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the circle circumscribing the triangle?
\[ \mathrm{(A) \ } 2 \qquad \mathrm{(B) \ } 8/\sqrt{15} \qquad \mathrm{(C) \ } 5/2 \qquad \mathrm{(D) \ } \sqrt{6} \qquad \mathrm{(E) \ } (\sqrt{6} + 1)/2 \]
Problem 30