1958 AHSME Problems

From AoPSWiki

Problem 1

The value of $[2 - 3(2 - 3)^{-1}]^{-1}$ is:

$\textbf{(A)}\ 5\qquad \textbf{(B)}\ -5\qquad \textbf{(C)}\ \frac{1}{5}\qquad \textbf{(D)}\ -\frac{1}{5}\qquad \textbf{(E)}\ \...$

Solution

Problem 2

If $\frac {1}{x} - \frac {1}{y} = \frac {1}{z}$ , then $z$ equals:

$\textbf{(A)}\ y - x\qquad \textbf{(B)}\ x - y\qquad \textbf{(C)}\ \frac {y - x}{xy}\qquad \textbf{(D)}\ \frac {xy}{y - x}\qqu...$

Solution

Problem 3

Of the following expressions the one equal to $\frac{a^{-1}b^{-1}}{a^{-3} - b^{-3}}$ is:

$\textbf{(A)}\ \frac{a^2b^2}{b^2 - a^2}\qquad \textbf{(B)}\ \frac{a^2b^2}{b^3 - a^3}\qquad \textbf{(C)}\ \frac{ab}{b^3 - a^3}\...$

Solution

Problem 4

In the expression $\frac{x + 1}{x - 1}$ each $x$ is replaced by $\frac{x + 1}{x - 1}$ . The resulting expression, evaluated for $x = \frac{1}{2}$ , equals:

$\textbf{(A)}\ 3\qquad \textbf{(B)}\ -3\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ -1\qquad \textbf{(E)}\ \text{none of these}$

Solution

Problem 5

The expression $2 + \sqrt{2} + \frac{1}{2 + \sqrt{2}} + \frac{1}{\sqrt{2} - 2}$ equals:

$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 2 - \sqrt{2}\qquad \textbf{(C)}\ 2 + \sqrt{2}\qquad \textbf{(D)}\ 2\sqrt{2}\qquad \textbf...$

Solution

Problem 6

The arithmetic mean between $\frac {x + a}{x}$ and $\frac {x - a}{x}$ , when $x \neq 0$ , is:

$\textbf{(A)}\ {2}\text{, if }{a \neq 0}\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ {1}\text{, only if }{a = 0}\qquad \textbf{(...$

Solution

Problem 7

A straight line joins the points $(-1,1)$ and $(3,9)$ . Its $x$ -intercept is:

$\textbf{(A)}\ -\frac{3}{2}\qquad \textbf{(B)}\ -\frac{2}{3}\qquad \textbf{(C)}\ \frac{2}{5}\qquad \textbf{(D)}\ 2\qquad \text...$

Solution

Problem 8

Which of these four numbers $\sqrt{\pi^2},\,\sqrt[3]{.8},\,\sqrt[4]{.00016},\,\sqrt[3]{-1}\cdot \sqrt{(.09)^{-1}}$ , is (are) rational:

$\textbf{(A)}\ \text{none}\qquad \textbf{(B)}\ \text{all}\qquad \textbf{(C)}\ \text{the first and fourth}\qquad \textbf{(D)}\ ...$

Solution

Problem 9

A value of $x$ satisfying the equation $x^2 + b^2 = (a - x)^2$ is:

$\textbf{(A)}\ \frac{b^2 + a^2}{2a}\qquad \textbf{(B)}\ \frac{b^2 - a^2}{2a}\qquad \textbf{(C)}\ \frac{a^2 - b^2}{2a}\qquad \t...$

Solution

Problem 10

For what real values of $k$ , other than $k = 0$ , does the equation $x^2 + kx + k^2 = 0$ have real roots?

$\textbf{(A)}\ {k < 0}\qquad \textbf{(B)}\ {k > 0} \qquad \textbf{(C)}\ {k \ge 1} \qquad \textbf{(D)}\ \text{all values ...$

Solution

Problem 11

The number of roots satisfying the equation $\sqrt{5 - x} = x\sqrt{5 - x}$ is:

$\textbf{(A)}\ \text{unlimited}\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 0$

Solution

Problem 12

If $P = \frac{s}{(1 + k)^n}$ then $n$ equals:

$\textbf{(A)}\ \frac{\log{\left(\frac{s}{P}\right)}}{\log{(1 + k)}}\qquad \textbf{(B)}\ \log{\left(\frac{s}{P(1 + k)}\right)}\...$

Solution

Problem 13

The sum of two numbers is $10$ ; their product is $20$ . The sum of their reciprocals is:

$\textbf{(A)}\ \frac{1}{10}\qquad \textbf{(B)}\ \frac{1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ 2\qquad \textbf{(E)}\ 4$

Solution

Problem 14

At a dance party a group of boys and girls exchange dances as follows: one boy dances with $5$ girls, a second boy dances with $6$ girls, and so on, the last boy dancing with all the girls. If $b$ represents the number of boys and $g$ the number of girls, then:

$\textbf{(A)}\ b = g\qquad \textbf{(B)}\ b = \frac{g}{5}\qquad \textbf{(C)}\ b = g - 4\qquad \textbf{(D)}\ b = g - 5\qquad \\\...$

Solution

Problem 15

A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:

$\textbf{(A)}\ 1080\qquad \textbf{(B)}\ 900\qquad \textbf{(C)}\ 720\qquad \textbf{(D)}\ 540\qquad \textbf{(E)}\ 360$

Solution

Problem 16

The area of a circle inscribed in a regular hexagon is $100\pi$ . The area of hexagon is:

$\textbf{(A)}\ 600\qquad \textbf{(B)}\ 300\qquad \textbf{(C)}\ 200\sqrt{2}\qquad \textbf{(D)}\ 200\sqrt{3}\qquad \textbf{(E)}\...$

Solution

Problem 17

If $x$ is positive and $\log{x} \ge \log{2} + \frac{1}{2}\log{x}$ , then:

$\textbf{(A)}\ {x}\text{ has no minimum or maximum value}\qquad \\\textbf{(B)}\ \text{the maximum value of }{x}\text{ is }{1}\...$

Solution

Problem 18

The area of a circle is doubled when its radius $r$ is increased by $n$ . Then $r$ equals:

$\textbf{(A)}\ n(\sqrt{2} + 1)\qquad \textbf{(B)}\ n(\sqrt{2} - 1)\qquad \textbf{(C)}\ n\qquad \textbf{(D)}\ n(2 - \sqrt{2})\q...$

Solution

Problem 19

The sides of a right triangle are $a$ and $b$ and the hypotenuse is $c$ . A perpendicular from the vertex divides $c$ into segments $r$ and $s$ , adjacent respectively to $a$ and $b$ . If $a : b = 1 : 3$ , then the ratio of $r$ to $s$ is:

$\textbf{(A)}\ 1 : 3\qquad \textbf{(B)}\ 1 : 9\qquad \textbf{(C)}\ 1 : 10\qquad \textbf{(D)}\ 3 : 10\qquad \textbf{(E)}\ 1 : \...$

Solution

Problem 20

If $4^x - 4^{x - 1} = 24$ , then $(2x)^x$ equals:

$\textbf{(A)}\ 5\sqrt{5}\qquad \textbf{(B)}\ \sqrt{5}\qquad \textbf{(C)}\ 25\sqrt{5}\qquad \textbf{(D)}\ 125\qquad \textbf{(E)...$

Solution

Problem 21

In the accompanying figure $\overline{CE}$ and $\overline{DE}$ are equal chords of a circle with center $O$ . Arc $AB$ is a quarter-circle. Then the ratio of the area of triangle $CED$ to the area of triangle $AOB$ is:

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$\textbf{(A)}\ \sqrt {2} : 1\qquad \textbf{(B)}\ \sqrt {3} : 1\qquad \textbf{(C)}\ 4 : 1\qquad \textbf{(D)}\ 3 : 1\qquad \text...$

Solution

Problem 22

A particle is placed on the parabola $y = x^2 - x - 6$ at a point $P$ whose $y$ -coordinate is $6$ . It is allowed to roll along the parabola until it reaches the nearest point $Q$ whose $y$ -coordinate is $-6$ . The horizontal distance traveled by the particle (the numerical value of the difference in the $x$ -coordinates of $P$ and $Q$ ) is:

$\textbf{(A)}\ 5\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 2\qquad \textbf{(E)}\ 1$

Solution

Problem 23

If, in the expression $x^2 - 3$ , $x$ increases or decreases by a positive amount of $a$ , the expression changes by an amount:

$\textbf{(A)}\ {\pm 2ax + a^2}\qquad \textbf{(B)}\ {2ax \pm a^2}\qquad \textbf{(C)}\ {\pm a^2 - 3} \qquad \textbf{(D)}\ {(x + ...$

Solution

Problem 24

A man travels $m$ feet due north at $2$ minutes per mile. He returns due south to his starting point at $2$ miles per minute. The average rate in miles per hour for the entire trip is:

$\textbf{(A)}\ 75\qquad \textbf{(B)}\ 48\qquad \textbf{(C)}\ 45\qquad \textbf{(D)}\ 24\qquad\\ \textbf{(E)}\ \text{impossible ...$

Solution

Problem 25

If $\log_{k}{x}\cdot \log_{5}{k} = 3$ , then $x$ equals:

$\textbf{(A)}\ k^6\qquad \textbf{(B)}\ 5k^3\qquad \textbf{(C)}\ k^3\qquad \textbf{(D)}\ 243\qquad \textbf{(E)}\ 125$

Solution

Problem 26

A set of $n$ numbers has the sum $s$ . Each number of the set is increased by $20$ , then multiplied by $5$ , and then decreased by $20$ . The sum of the numbers in the new set thus obtained is:

$\textbf{(A)}\ s + 20n\qquad \textbf{(B)}\ 5s + 80n\qquad \textbf{(C)}\ s\qquad \textbf{(D)}\ 5s\qquad \textbf{(E)}\ 5s + 4n$

Solution

Problem 27

The points $(2,-3)$ , $(4,3)$ , and $(5, k/2)$ are on the same straight line. The value(s) of $k$ is (are):

$\textbf{(A)}\ 12\qquad \textbf{(B)}\ -12\qquad \textbf{(C)}\ \pm 12\qquad \textbf{(D)}\ {12}\text{ or }{6}\qquad \textbf{(E)}...$

Solution

Problem 28

A $16$ -quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This is done a third and a fourth time. The fractional part of the final mixture that is water is:

$\textbf{(A)}\ \frac{1}{4}\qquad \textbf{(B)}\ \frac{81}{256}\qquad \textbf{(C)}\ \frac{27}{64}\qquad \textbf{(D)}\ \frac{37}{...$

Solution

Problem 29

In a general triangle $ADE$ (as shown) lines $\overline{EB}$ and $\overline{EC}$ are drawn. Which of the following angle relations is true?

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$\textbf{(A)}\ x + z = a + b\qquad \textbf{(B)}\ y + z = a + b\qquad \textbf{(C)}\ m + x = w + n\qquad \\\textbf{(D)}\ x + z +...$

Solution

Problem 30

If $xy = b$ and $\frac{1}{x^2} + \frac{1}{y^2} = a$ , then $(x + y)^2$ equals:

$\textbf{(A)}\ (a + 2b)^2\qquad \textbf{(B)}\ a^2 + b^2\qquad \textbf{(C)}\ b(ab + 2)\qquad \textbf{(D)}\ ab(b + 2)\qquad \tex...$

Solution

Problem 31

The altitude drawn to the base of an isosceles triangle is $8$ , and the perimeter $32$ . The area of the triangle is:

$\textbf{(A)}\ 56\qquad \textbf{(B)}\ 48\qquad \textbf{(C)}\ 40\qquad \textbf{(D)}\ 32\qquad \textbf{(E)}\ 24$

Solution

Problem 32

With $$1000$ a rancher is to buy steers at $$25$ each and cows at $$26$ each. If the number of steers $s$ and the number of cows $c$ are both positive integers, then:

$\textbf{(A)}\ \text{this problem has no solution}\qquad\\ \textbf{(B)}\ \text{there are two solutions with }{s}\text{ exceedi...$

Solution

Problem 33

For one root of $ax^2 + bx + c = 0$ to be double the other, the coefficients $a,\,b,\,c$ must be related as follows:

$\textbf{(A)}\ 4b^2 = 9c\qquad \textbf{(B)}\ 2b^2 = 9ac\qquad \textbf{(C)}\ 2b^2 = 9a\qquad \\\textbf{(D)}\ b^2 - 8ac = 0\qqua...$

Solution

Problem 34

The numerator of a fraction is $6x + 1$ , then denominator is $7 - 4x$ , and $x$ can have any value between $-2$ and $2$ , both included. The values of $x$ for which the numerator is greater than the denominator are:

$\textbf{(A)}\ \frac{3}{5} < x \le 2\qquad \textbf{(B)}\ \frac{3}{5} \le x \le 2\qquad \textbf{(C)}\ 0 < x \le 2\qquad \...$

Solution

Problem 35

A triangle is formed by joining three points whose coordinates are integers. If the $x$ -coordinate and the $y$ -coordinate each have a value of $1$ , then the area of the triangle, in square units:

$\textbf{(A)}\ \text{must be an integer}\qquad \textbf{(B)}\ \text{may be irrational}\qquad \textbf{(C)}\ \text{must be irrati...$

Solution

Problem 36

The sides of a triangle are $30$ , $70$ , and $80$ units. If an altitude is dropped upon the side of length $80$ , the larger segment cut off on this side is:

$\textbf{(A)}\ 62\qquad \textbf{(B)}\ 63\qquad \textbf{(C)}\ 64\qquad \textbf{(D)}\ 65\qquad \textbf{(E)}\ 66$

Solution

Problem 37

The first term of an arithmetic series of consecutive integers is $k^2 + 1$ . The sum of $2k + 1$ terms of this series may be expressed as:

$\textbf{(A)}\ k^3 + (k + 1)^3\qquad \textbf{(B)}\ (k - 1)^3 + k^3\qquad \textbf{(C)}\ (k + 1)^3\qquad \\\textbf{(D)}\ (k + 1)...$

Solution

Problem 38

Let $r$ be the distance from the origin to a point $P$ with coordinates $x$ and $y$ . Designate the ratio $\frac{y}{r}$ by $s$ and the ratio $\frac{x}{r}$ by $c$ . Then the values of $s^2 - c^2$ are limited to the numbers:

$\textbf{(A)}\ \text{less than }{-1}\text{ are greater than }{+1}\text{, both excluded}\qquad\\\textbf{(B)}\ \text{less than }...$

Solution

Problem 39

We may say concerning the solution of \[ |x|^2 \plus{} |x| \minus{} 6 \equal{} 0 \] that:

$\textbf{(A)}\ \text{there is only one root}\qquad \textbf{(B)}\ \text{the sum of the roots is }{+1}\qquad \textbf{(C)}\ \text...$

Solution

Problem 40

Given $a_0 = 1$ , $a_1 = 3$ , and the general relation $a_n^2 - a_{n - 1}a_{n + 1} = (-1)^n$ for $n \ge 1$ . Then $a_3$ equals:

$\textbf{(A)}\ \frac{13}{27}\qquad \textbf{(B)}\ 33\qquad \textbf{(C)}\ 21\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ -17$

Solution

Problem 41

The roots of $Ax^2 + Bx + C = 0$ are $r$ and $s$ . For the roots of \[ x^2 \plus{} px \plus{} q \equal{} 0 \] to be $r^2$ and $s^2$ , $p$ must equal:

$\textbf{(A)}\ \frac{B^2 - 4AC}{A^2}\qquad \textbf{(B)}\ \frac{B^2 - 2AC}{A^2}\qquad \textbf{(C)}\ \frac{2AC - B^2}{A^2}\qquad...$

Solution

Problem 42

In a circle with center $O$ , chord $\overline{AB}$ equals chord $\overline{AC}$ . Chord $\overline{AD}$ cuts $\overline{BC}$ in $E$ . If $AC = 12$ and $AE = 8$ , then $AD$ equals:

$\textbf{(A)}\ 27\qquad \textbf{(B)}\ 24\qquad \textbf{(C)}\ 21\qquad \textbf{(D)}\ 20\qquad \textbf{(E)}\ 18$

Solution

Problem 43

$\overline{AB}$ is the hypotenuse of a right triangle $ABC$ . Median $\overline{AD}$ has length $7$ and median $\overline{BE}$ has length $4$ . The length of $\overline{AB}$ is:

$\textbf{(A)}\ 10\qquad \textbf{(B)}\ 5\sqrt{3}\qquad \textbf{(C)}\ 5\sqrt{2}\qquad \textbf{(D)}\ 2\sqrt{13}\qquad \textbf{(E)...$

Solution

Problem 44

Given the true statements: (1) If $a$ is greater than $b$ , then $c$ is greater than $d$ (2) If $c$ is less than $d$ , then $e$ is greater than $f$ . A valid conclusion is:

$\textbf{(A)}\ \text{If }{a}\text{ is less than }{b}\text{, then }{e}\text{ is greater than }{f}\qquad \\\textbf{(B)}\ \text{I...$

Solution

Problem 45

A check is written for $x$ dollars and $y$ cents, $x$ and $y$ both two-digit numbers. In error it is cashed for $y$ dollars and $x$ cents, the incorrect amount exceeding the correct amount by $$17.82$ . Then:

$\textbf{(A)}\ {x}\text{ cannot exceed }{70}\qquad \\\textbf{(B)}\ {y}\text{ can equal }{2x}\qquad\\\textbf{(C)}\ \text{the am...$

Solution

Problem 46

For values of $x$ less than $1$ but greater than $-4$ , the expression $\frac{x^2 - 2x + 2}{2x - 2}$ has:

$\textbf{(A)}\ \text{no maximum or minimum value}\qquad \\\textbf{(B)}\ \text{a minimum value of }{+1}\qquad \\\textbf{(C)}\ \...$

Solution

Problem 47

$ABCD$ is a rectangle (see the accompanying diagram) with $P$ any point on $\overline{AB}$ . $\overline{PS} \perp \overline{BD}$ and $\overline{PR} \perp \overline{AC}$ . $\overline{AF} \perp \overline{BD}$ and $\overline{PQ} \perp \overline{AF}$ . Then $PR + PS$ is equal to:

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$\textbf{(A)}\ PQ\qquad \textbf{(B)}\ AE\qquad \textbf{(C)}\ PT + AT\qquad \textbf{(D)}\ AF\qquad \textbf{(E)}\ EF$

Solution

Problem 48

Diameter $\overline{AB}$ of a circle with center $O$ is $10$ units. $C$ is a point $4$ units from $A$ , and on $\overline{AB}$ . $D$ is a point $4$ units from $B$ , and on $\overline{AB}$ . $P$ is any point on the circle. Then the broken-line path from $C$ to $P$ to $D$ :

$\textbf{(A)}\ \text{has the same length for all positions of }{P}\qquad\\ \textbf{(B)}\ \text{exceeds }{10}\text{ units for a...$

Solution

Problem 49

In the expansion of $(a + b)^n$ there are $n + 1$ dissimilar terms. The number of dissimilar terms in the expansion of $(a + b + c)^{10}$ is:

$\textbf{(A)}\ 11\qquad \textbf{(B)}\ 33\qquad \textbf{(C)}\ 55\qquad \textbf{(D)}\ 66\qquad \textbf{(E)}\ 132$

Solution

Problem 50

In this diagram a scheme is indicated for associating all the points of segment $\overline{AB}$ with those of segment $\overline{A'B'}$ , and reciprocally. To described this association scheme analytically, let $x$ be the distance from a point $P$ on $\overline{AB}$ to $D$ and let $y$ be the distance from the associated point $P'$ of $\overline{A'B'}$ to $D'$ . Then for any pair of associated points, if $x = a,\, x + y$ equals:

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$\textbf{(A)}\ 13a\qquad \textbf{(B)}\ 17a - 51\qquad \textbf{(C)}\ 17 - 3a\qquad \textbf{(D)}\ \frac {17 - 3a}{4}\qquad \text...$

Solution

1958 AHSME Problems

From AoPSWiki

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

See also