# 1995 AHSME 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

Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will

## Problem 2

If , then

## Problem 3

The total in-store price for an appliance is 99.99. A television commercial advertises the same product for three easy payments of 29.98 and a one-time shipping and handling charge of 9.98. How much is saved by buying the appliance from the television advertiser?

## Problem 4

If is of , is of , and is of , then

## Problem 5

A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is

## Problem 6

The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked ?

## Problem 7

The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is:

## Problem 8

In , and . Points and are on and , respectively, and . If , then

## Problem 9

Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is

## Problem 10

The area of the triangle bounded by the lines and is

## Problem 11

How many base 10 four-digit numbers, , satisfy all three of the following conditions?

(i) (ii) is a multiple of 5; (iii) .

## Problem 12

Let be a linear function with the properties that and . Which of the following is true?

## Problem 13

The addition below is incorrect. The display can be made correct by changing one digit , wherever it occurs, to another digit . Find the sum of and .

## Problem 14

If and , then

## Problem 15

Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point

## Problem 16

Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:

i. The actual attendance in Atlanta is within of Anita's estimate. ii. Bob's estimate is within of the actual attendance in Boston.

To the nearest 1,000, the largest possible difference between the numbers attending the two games is

## Problem 17

Given regular pentagon , a circle can be drawn that is tangent to at and to at . The number of degrees in minor arc is

## Problem 18

Two rays with common endpoint forms a angle. Point lies on one ray, point on the other ray, and . The maximum possible length of is

## Problem 19

Equilateral triangle is inscribed in equilateral triangle such that . The reatio of the area of to the area of is

## Problem 20

If and are three (not necessarily different) numbers chosen randomly and with replacement from the set , the probability that is even is

## Problem 21

Two nonadjacent vertices of a rectangle are (4,3) and (-4,-3), and the coordinates of the other two vertices are integers. The number of such rectangles is

## Problem 22

A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13,19,20,25 and 31, although this is not necessarily their order around the pentagon. The area of the pentagon is

## Problem 23

The sides of a triangle have lengths 11,15, and , where is an integer. For how many values of is the triangle obtuse?

## Problem 24

There exist positive integers and , with no common factor greater than 1, such that

What is ?

## Problem 25

A list of five positive integers has mean 12 and range 18. The mode and median are both 8. How many different values are possible for the second largest element of the list?

## Problem 26

In the figure, and are diameters of the circle with center , , and chord intersects at . If and , then the area of the circle is

## Problem 27

Consider the triangular array of numbers with 0,1,2,3,... along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.

\[\begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\ & & & & 1 & & 1 & & & & \\ & & & 2 & & 2 & & 2 & & & \\ & & 3 & & 4 & & 4 & & 3 & & \\ & 4 & & 7 & & 8 & & 7 & & 4 & \\ 5 & & 11 & & 15 & & 15 & & 11 & & 5 & \end{tabular}\] (Error compiling LaTeX. ! Extra alignment tab has been changed to \cr.)

Let denote the sum of the numbers in row . What is the remainder when is divided by 100?

## Problem 28

Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length where is

## Problem 29

For how many three-element sets of positive integers is it true that ?

## Problem 30

A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is