# 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 ?

## 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?

## Problem 30