# Difference between revisions of "1976 AHSME Problems"

## Problem 1

If one minus the reciprocal of $(1-x)$ equals the reciprocal of $(1-x)$, then $x$ equals

$\textbf{(A) }-2\qquad \textbf{(B) }-1\qquad \textbf{(C) }1/2\qquad \textbf{(D) }2\qquad \textbf{(E) }3$

## Problem 2

For how many real numbers $x$ is $\sqrt{-(x+1)^2}$ a real number?

$\textbf{(A) }\text{none}\qquad \textbf{(B) }\text{one}\qquad \textbf{(C) }\text{two}\qquad\\ \textbf{(D) }\text{a finite number greater than two}\qquad \textbf{(E) }\infty$

## Problem 3

The sum of the distances from one vertex of a square with sides of length $2$ to the midpoints of each of the sides of the square is

$\textbf{(A) }2\sqrt{5}\qquad \textbf{(B) }2+\sqrt{3}\qquad \textbf{(C) }2+2\sqrt{3}\qquad \textbf{(D) }2+\sqrt{5}\qquad \textbf{(E) }2+2\sqrt{5}$

## Problem 4

Let a geometric progression with n terms have first term one, common ratio $r$ and sum $s$, where $r$ and $s$ are not zero. The sum of the geometric progression formed by replacing each term of the original progression by its reciprocal is

$\textbf{(A) }\frac{1}{s}\qquad \textbf{(B) }\frac{1}{r^ns}\qquad \textbf{(C) }\frac{s}{r^{n-1}}\qquad \textbf{(D) }\frac{r^n}{s}\qquad \textbf{(E) } \frac{r^{n-1}}{s}$

## Problem 5

How many integers greater than $10$ and less than $100$, written in base-$10$ notation, are increased by $9$ when their digits are reversed?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

## Problem 6

If $c$ is a real number and the negative of one of the solutions of $x^2-3x+c=0$ is a solution of $x^2+3x-c=0$, then the solutions of $x^2-3x+c=0$ are

$\textbf{(A) }1,~2\qquad \textbf{(B) }-1,~-2\qquad \textbf{(C) }0,~3\qquad \textbf{(D) }0,~-3\qquad \textbf{(E) }\frac{3}{2},~\frac{3}{2}$

## Problem 7

If $x$ is a real number, then the quantity $(1-|x|)(1+x)$ is positive if and only if

$\textbf{(A) }|x|<1\qquad \textbf{(B) }|x|>1\qquad \textbf{(C) }x<-1\text{ or }-1

## Problem 8

A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. What is the probability that the distance from the point to the origin is at most two units?

$\textbf{(A) }\frac{13}{81}\qquad \textbf{(B) }\frac{15}{81}\qquad \textbf{(C) }\frac{13}{64}\qquad \textbf{(D) }\frac{\pi}{16}\qquad \textbf{(E) }\text{the square of a rational number}$

## Problem 9

In triangle $ABC$, $D$ is the midpoint of $AB$; $E$ is the midpoint of $DB$; and $F$ is the midpoint of $BC$. If the area of $\triangle ABC$ is $96$, then the area of $\triangle AEF$ is

$\textbf{(A) }16\qquad \textbf{(B) }24\qquad \textbf{(C) }32\qquad \textbf{(D) }36\qquad \textbf{(E) }48$

## Problem 10

If $m,~n,~p$, and $q$ are real numbers and $f(x)=mx+n$ and $g(x)=px+q$, then the equation $f(g(x))=g(f(x))$ has a solution

$\textbf{(A) }\text{for all choices of }m,~n,~p, \text{ and } q\qquad\\ \textbf{(B) }\text{if and only if }m=p\text{ and }n=q\qquad\\ \textbf{(C) }\text{if and only if }mq-np=0\qquad\\ \textbf{(D) }\text{if and only if }n(1-p)-q(1-m)=0\qquad\\ \textbf{(E) }\text{if and only if }(1-n)(1-p)-(1-q)(1-m)=0$

## Problem 11

Which of the following statements is (are) equivalent to the statement "If the pink elephant on planet alpha has purple eyes, then the wild pig on planet beta does not have a long nose"?

$$\textbf{I. "If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha has purple eyes."}\\ \textbf{II. "If the pink elephant on planet alpha does not have purple eyes, then the wild pig on planet beta does not have a long nose."}\\ \textbf{III. "If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha does not have purple eyes."}\\ \textbf{IV. "The pink elephant on planet alpha does not have purple eyes, or the wild pig on planet beta does not have a long nose."}$$

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

## Problem 12

A supermarket has $128$ crates of apples. Each crate contains at least $120$ apples and at most $144$ apples. What is the largest integer $n$ such that there must be at least $n$ crates containing the same number of apples?

$\textbf{(A) }4\qquad \textbf{(B) }5\qquad \textbf{(C) }6\qquad \textbf{(D) }24\qquad \textbf{(E) }25$

## Problem 13

If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk?

$\textbf{(A) }\frac{x(x+2)(x+5)}{(x+1)(x+3)}\qquad \textbf{(B) }\frac{x(x+1)(x+5)}{(x+2)(x+3)}\qquad\\ \textbf{(C) }\frac{(x+1)(x+3)(x+5)}{x(x+2)}\qquad \textbf{(D) }\frac{(x+1)(x+3)}{x(x+2)(x+5)}\qquad \\ \textbf{(E) }\text{none of these}$

## Problem 14

The measures of the interior angles of a convex polygon are in arithmetic progression. If the smallest angle is $100^\circ$, and the largest is $140^\circ$, then the number of sides the polygon has is

$\textbf{(A) }6\qquad \textbf{(B) }8\qquad \textbf{(C) }10\qquad \textbf{(D) }11\qquad \textbf{(E) }12$

## Problem 15

If $r$ is the remainder when each of the numbers $1059,~1417$, and $2312$ is divided by $d$, where $d$ is an integer greater than $1$, then $d-r$ equals

$\textbf{(A) }1\qquad \textbf{(B) }15\qquad \textbf{(C) }179\qquad \textbf{(D) }d-15\qquad \textbf{(E) }d-1$

## Problem 16

In triangles $ABC$ and $DEF$, lengths $AC,~BC,~DF$, and $EF$ are all equal. Length $AB$ is twice the length of the altitude of $\triangle DEF$ from $F$ to $DE$. Which of the following statements is (are) true?

## Problem 27

If $N=\frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}-\sqrt{3-2\sqrt{2}}$, then $N$ equals

$\textbf{(A) }1\qquad \textbf{(B) }2\sqrt{2}-1\qquad \textbf{(C) }\frac{\sqrt{5}}{2}\qquad \textbf{(D) }\sqrt{\frac{5}{2}}\qquad \textbf{(E) }\text{none of these}$

## Problem 28

Lines $\mathit{L}_1,\mathit{L}_2,\dots,\mathit{L}_{100}$ are distinct. All lines $\mathit{L}_{4n}, n$ a positive integer, are parallel to each other. All lines $\mathit{L}_{4n-3}$, $n$ a positive integer, pass through a given point $\mathit{A}$. The maximum number of points of intersection of pairs of lines from the complete set $\{\mathit{L}_1,\mathit{L}_2,\dots,\mathit{L}_{100}\}$ is

$\textbf{(A) }4350\qquad \textbf{(B) }4351\qquad \textbf{(C) }4900\qquad \textbf{(D) }4901\qquad \textbf{(E) }9851$

## Problem 29

Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is $44$ years, then Ann's age is

$\textbf{(A) }22\qquad \textbf{(B) }24\qquad \textbf{(C) }25\qquad \textbf{(D) }26\qquad \textbf{(E) }28$

## Problem 30

How many distinct ordered triples $(x,y,z)$ satisfy the equations $$x+2y+4z=12 \\ xy+4yz+2xz=22 \\ xyz=6\quad?$$

$\textbf{(A) }\text{none}\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }4\qquad \textbf{(E) }6$