# Difference between revisions of "1975 AHSME Problems"

(Created page with "== Problem 1 == The value of <math>\displaystyle{\frac {1}{2 - \frac {1}{2 - \frac {1}{2 - \frac12}}}</math> is <math>\textbf{(A)}\ 3/4 \qquad \textbf{(B)}\ 4/5 \qquad \text...") |
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== Problem 1 == | == Problem 1 == | ||

− | The value of <math> | + | The value of <math>{\frac {1}{2 - \frac {1}{2 - \frac {1}{2 - \frac12}}}</math> is |

<math>\textbf{(A)}\ 3/4 \qquad | <math>\textbf{(A)}\ 3/4 \qquad | ||

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\textbf{(B)}\ \text{all }m\neq 0\qquad | \textbf{(B)}\ \text{all }m\neq 0\qquad | ||

\textbf{(C)}\ \text{all }m\neq 1/2\qquad | \textbf{(C)}\ \text{all }m\neq 1/2\qquad | ||

− | \textbf{(D)}\ \text{all }m\neq 1\qquad | + | \textbf{(D)}\ \text{all }m\neq 1\qquad \\ |

\textbf{(E)}\ \text{no values of }m </math> | \textbf{(E)}\ \text{no values of }m </math> | ||

Line 81: | Line 81: | ||

== Problem 7 == | == Problem 7 == | ||

− | For which non-zero real numbers x is \frac{|x-|x|\-|}{x} a positive integers? | + | For which non-zero real numbers <math>x</math> is <math>\frac{|x-|x|\-|}{x}</math> a positive integers? |

<math>\textbf{(A)}\ \text{for negative } x \text{ only} \qquad \\ | <math>\textbf{(A)}\ \text{for negative } x \text{ only} \qquad \\ | ||

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I. All shirts in this store are at non-sale prices. | I. All shirts in this store are at non-sale prices. | ||

+ | |||

II. There is some shirt in this store not on sale. | II. There is some shirt in this store not on sale. | ||

+ | |||

III. No shirt in this store is on sale. | III. No shirt in this store is on sale. | ||

+ | |||

IV. Not all shirts in this store are on sale. | IV. Not all shirts in this store are on sale. | ||

+ | |||

<math>\textbf{(A)}\ \text{II only} \qquad | <math>\textbf{(A)}\ \text{II only} \qquad | ||

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− | + | <asy> | |

draw((0,0)--(2.8284,2)--(8,0)--cycle); | draw((0,0)--(2.8284,2)--(8,0)--cycle); | ||

draw((2.8284,2)--(4,0)); | draw((2.8284,2)--(4,0)); | ||

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label("B",(0,0),S); | label("B",(0,0),S); | ||

label("C",(8,0),S); | label("C",(8,0),S); | ||

− | label("M",(4,0),S); | + | label("M",(4,0),S); |

+ | </asy> | ||

<math>\textbf{(A)}\ 2\sqrt{26} \qquad | <math>\textbf{(A)}\ 2\sqrt{26} \qquad | ||

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<math>N</math> is the midpoint of <math>BC</math>; and <math>AN</math> and <math>CM</math> intersect at <math>O</math>. The ratio of the area of <math>AOCD</math> to the area of <math>ABCD</math> is | <math>N</math> is the midpoint of <math>BC</math>; and <math>AN</math> and <math>CM</math> intersect at <math>O</math>. The ratio of the area of <math>AOCD</math> to the area of <math>ABCD</math> is | ||

− | + | <asy> | |

draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--(2,1)--(2,2)--(1,0)); | draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--(2,1)--(2,2)--(1,0)); | ||

label("A", (0,0), S); | label("A", (0,0), S); | ||

Line 324: | Line 329: | ||

label("M", (1,0), S); | label("M", (1,0), S); | ||

label("N", (2,1), E); | label("N", (2,1), E); | ||

− | label("O", (1.2, .8)); | + | label("O", (1.2, .8)); |

+ | </asy> | ||

<math>\textbf{(A)}\ \frac{5}{6}\qquad | <math>\textbf{(A)}\ \frac{5}{6}\qquad | ||

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and <math>AB</math>, respectively, and lines <math>EF</math> and <math>AM</math> intersect at <math>G</math>. If <math>AE=2AF</math> then <math>\frac{EG}{GF}</math> equals | and <math>AB</math>, respectively, and lines <math>EF</math> and <math>AM</math> intersect at <math>G</math>. If <math>AE=2AF</math> then <math>\frac{EG}{GF}</math> equals | ||

− | + | <asy> | |

draw((0,0)--(12,0)--(14,7.75)--(0,0)); | draw((0,0)--(12,0)--(14,7.75)--(0,0)); | ||

draw((0,0)--(13,3.875)); | draw((0,0)--(13,3.875)); | ||

Line 400: | Line 406: | ||

label("F", (5,0), S); | label("F", (5,0), S); | ||

label("M", (13,3.875), E); | label("M", (13,3.875), E); | ||

− | label("G", (7,1)); | + | label("G", (7,1)); |

+ | </asy> | ||

<math>\textbf{(A)}\ \frac{3}{2} \qquad | <math>\textbf{(A)}\ \frac{3}{2} \qquad | ||

Line 433: | Line 440: | ||

[[1975 AHSME Problems/Problem 30|Solution]] | [[1975 AHSME Problems/Problem 30|Solution]] | ||

+ | |||

+ | == See also == | ||

+ | * [[AHSME]] | ||

+ | * [[AHSME Problems and Solutions]] | ||

+ | * [[AMC 12 Problems and Solutions]] | ||

+ | * [[Mathematics competition resources]] | ||

+ | {{MAA Notice}} |

## Revision as of 20:14, 7 October 2014

## 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

The value of ${\frac {1}{2 - \frac {1}{2 - \frac {1}{2 - \frac12}}}$ (Error compiling LaTeX. ! Missing } inserted.) is

## Problem 2

For which real values of m are the simultaneous equations

y = mx + 3 \\ \[ y = (2m - 1)x + 4 (Error compiling LaTeX. ! LaTeX Error: \begin{equation*} on input line 20 ended by \end{document}.)

satisfied by at least one pair of real numbers ?

## Problem 3

Which of the following inequalities are satisfied for all real numbers which satisfy the conditions , and ?

## Problem 4

If the side of one square is the diagonal of a second square, what is the ratio of the area of the first square to the area of the second?

## Problem 5

The polynomial is expanded in decreasing powers of . The second and third terms have equal values when evaluated at and , where and are positive numbers whose sum is one. What is the value of ?

## Problem 6

The sum of the first eighty positive odd integers subtracted from the sum of the first eighty positive even integers is

## Problem 7

For which non-zero real numbers is a positive integers?

## Problem 8

If the statement "All shirts in this store are on sale." is false, then which of the following statements must be true?

I. All shirts in this store are at non-sale prices.

II. There is some shirt in this store not on sale.

III. No shirt in this store is on sale.

IV. Not all shirts in this store are on sale.

## Problem 9

Let and be arithmetic progressions such that , and . Find the sum of the first hundred terms of the progression

## Problem 10

The sum of the digits in base ten of , where is a positive integer, is

## Problem 11

Let be an interior point of circle other than the center of . Form all chords of which pass through , and determine their midpoints. The locus of these midpoints is

## Problem 12

If , and , which of the following conclusions is correct?

## Problem 13

The equation has

## Problem 14

If the is when the is and the and is , what is the when the is , the and is and the is is two ( and are variables taking positive values)?

## Problem 15

In the sequence of numbers each term after the first two is equal to the term preceding it minus the term preceding that. The sum of the first one hundred terms of the sequence is

## Problem 16

If the first term of an infinite geometric series is a positive integer, the common ratio is the reciprocal of a positive integer, and the sum of the series is , then the sum of the first two terms of the series is

## Problem 17

A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of x working days, the man took the bus to work in the morning times, came home by bus in the afternoon times, and commuted by train (either morning or afternoon) times. Find .

## Problem 18

A positive integer with three digits in its base ten representation is chosen at random, with each three digit number having an equal chance of being chosen. The probability that is an integer is

## Problem 19

Which positive numbers satisfy the equation ?

## Problem 20

In the adjoining figure is such that and . If is the midpoint of and , what is the length of ?

## Problem 21

Suppose is defined for all real numbers ; for all , and for all and . Which of the following statements is true?

\textbf{(A)}\ \text{III and IV only} \qquad \textbf{(B)}\ \text{I, III, and IV only} \qquad \textbf{(C)}\ \text{I, II, and IV only} \qquad \textbf{(D)}\ \text{I, II, and III only} \qquad \textbf{(E)}\ \text{All are true.}

## Problem 22

If and are primes and has distinct positive integral roots, then which of the following statements are true?

## Problem 23

In the adjoining figure and are adjacent sides of square ; is the midpoint of ; is the midpoint of ; and and intersect at . The ratio of the area of to the area of is

## Problem 24

In , and , where . The circle with center and radius intersects at and intersects , extended if necessary, at and at ( may coincide with ). Then

## Problem 25

A woman, her brother, her son and her daughter are chess players (all relations by birth). The worst player's twin (who is one of the four players) and the best player are of opposite sex. The worst player and the best player are the same age. Who is the worst player?

## Problem 26

In acute the bisector of meets side at . The circle with center and radius intersects side at ; and the circle with center and radius intersects side at . Then it is always true that

## Problem 27

If and are distinct roots of , then equals

## Problem 28

In shown in the adjoining figure, is the midpoint of side and . Points and are taken on and , respectively, and lines and intersect at . If then equals

## Problem 29

What is the smallest integer larger than ?

## Problem 30

Let . Then equals

## See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.