# 1959 AHSME Problems

## Problem 1

Each edge of a cube is increased by $50$%. The percent of increase of the surface area of the cube is: $\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 750$

## Problem 2

Through a point $P$ inside the $\triangle ABC$ a line is drawn parallel to the base $AB$, dividing the triangle into two equal areas. If the altitude to $AB$ has a length of $1$, then the distance from $P$ to $AB$ is: $\textbf{(A)}\ \frac12 \qquad\textbf{(B)}\ \frac14\qquad\textbf{(C)}\ 2-\sqrt2\qquad\textbf{(D)}\ \frac{2-\sqrt2}{2}\qquad\textbf{(E)}\ \frac{2+\sqrt2}{8}$

## Problem 3

If the diagonals of a quadrilateral are perpendicular to each other, the figure would always be included under the general classification: $\textbf{(A)}\ \text{rhombus} \qquad\textbf{(B)}\ \text{rectangles} \qquad\textbf{(C)}\ \text{square} \qquad\textbf{(D)}\ \text{isosceles trapezoid}\qquad\textbf{(E)}\ \text{none of these}$

## Problem 4

If $78$ is divided into three parts which are proportional to $1, \frac13, \frac16,$ the middle part is: $\textbf{(A)}\ 9\frac13 \qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 17\frac13 \qquad\textbf{(D)}\ 18\frac13\qquad\textbf{(E)}\ 26$

## Problem 5

The value of $\left(256\right)^{.16}\left(256\right)^{.09}$ is:

$\textbf{(A)}\ 4 \qquad \\ \textbf{(B)}\ 16\qquad \\ \textbf{(C)}\ 64\qquad \\ \textbf{(D)}\ 256.25\qquad \\ \textbf{(E)}\ -16$

## Problem 6

Given the true statement: If a quadrilateral is a square, then it is a rectangle. It follows that, of the converse and the inverse of this true statement is:

$\textbf{(A)}\ \text{only the converse is true} \qquad \\ \textbf{(B)}\ \text{only the inverse is true }\qquad \\ \textbf{(C)}\ \text{both are true} \qquad \\ \textbf{(D)}\ \text{neither is true} \qquad \\ \textbf{(E)}\ \text{the inverse is true, but the converse is sometimes true}$

## Problem 7

The sides of a right triangle are $a$, $a+d$, and $a+2d$, with $a$ and $d$ both positive. The ratio of $a$ to $d$ is: $\textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 1:4 \qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 3:1\qquad\textbf{(E)}\ 3:4$

## Problem 8

The value of $x^2-6x+13$ can never be less than:

$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 4.5 \qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 13$

## Problem 9

A farmer divides his herd of $n$cows among his four sons so that one son gets one-half the herd, a second son, one-fourth, a third son, one-fifth, and the fourth son, $7$ cows. Then $n$ is: $\textbf{(A)}\ 80 \qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 140\qquad\textbf{(D)}\ 180\qquad\textbf{(E)}\ 240$

## Problem 10

In $\triangle ABC$ with $\overbar{AB}=\overbar{AC}=3.6$ (Error compiling LaTeX. ! Undefined control sequence.), a point $D$ is taken on $AB$ at a distance $1.2$from $A$. Point $D$ is joined to $E$ in the prolongation of $AC$ so that $\triangle AED$ is equal in area to $ABC$. Then $\overbar{AE}$ (Error compiling LaTeX. ! Undefined control sequence.) is: $\textbf{(A)}\ 4.8 \qquad\textbf{(B)}\ 5.4\qquad\textbf{(C)}\ 7.2\qquad\textbf{(D)}\ 10.8\qquad\textbf{(E)}\ 12.6$

## Problem 11

The logarithm of $.0625$ to the base $2$ is: $\textbf{(A)}\ .025 \qquad\textbf{(B)}\ .25\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ -4\qquad\textbf{(E)}\ -2$

## Problem 12

By adding the same constant to $20,50,100$ a geometric progression results. The common ratio is: $\textbf{(A)}\ \frac53 \qquad\textbf{(B)}\ \frac43\qquad\textbf{(C)}\ \frac32\qquad\textbf{(D)}\ \frac12\qquad\textbf{(E)}\ \frac{1}3$

## Problem 13

The arithmetic mean (average) of a set of $50$ numbers is $38$. If two numbers, namely, $45$ and $55$, are discarded, the mean of the remaining set of numbers is: $\textbf{(A)}\ 36.5 \qquad\textbf{(B)}\ 37\qquad\textbf{(C)}\ 37.2\qquad\textbf{(D)}\ 37.5\qquad\textbf{(E)}\ 37.52$

## Problem 14

Given the set $S$ whose elements are zero and the even integers, positive and negative. Of the five operations applied to any pair of elements: (1) addition (2) subtraction (3) multiplication (4) division (5) finding the arithmetic mean (average), those elements that only yield elements of $S$ are: $\textbf{(A)}\ \text{all} \qquad\textbf{(B)}\ 1,2,3,4\qquad\textbf{(C)}\ 1,2,3,5\qquad\textbf{(D)}\ 1,2,3\qquad\textbf{(E)}\ 1,3,5$

## Problem 15

In a right triangle the square of the hypotenuse is equal to twice the product of the legs. One of the acute angles of the triangle is: $\textbf{(A)}\ 15^{\circ} \qquad\textbf{(B)}\ 30^{\circ} \qquad\textbf{(C)}\ 45^{\circ} \qquad\textbf{(D)}\ 60^{\circ}\qquad\textbf{(E)}\ 75^{\circ}$

## Problem 16

The expression$\frac{x^2-3x+2}{x^2-5x+6}\div \frac{x^2-5x+4}{x^2-7x+12},$ when simplified is: $\textbf{(A)}\ \frac{(x-1)(x-6)}{(x-3)(x-4)} \qquad\textbf{(B)}\ \frac{x+3}{x-3}\qquad\textbf{(C)}\ \frac{x+1}{x-1}\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2$

## Problem 17

If $y=a+\frac{b}{x}$, where $a$ and $b$ are constants, and if $y=1$ when $x=-1$, and $y=5$ when $x=-5$, then $a+b$ equals: $\textbf{(A)}\ -1 \qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11$

## Problem 18

The arithmetic mean (average) of the first $n$ positive integers is: $\textbf{(A)}\ \frac{n}{2} \qquad\textbf{(B)}\ \frac{n^2}{2}\qquad\textbf{(C)}\ n\qquad\textbf{(D)}\ \frac{n-1}{2}\qquad\textbf{(E)}\ \frac{n+1}{2}$

## Problem 19

With the use of three different weights, namely $1$ lb., $3$ lb., and $9$ lb., how many objects of different weights can be weighed, if the objects is to be weighed and the given weights may be placed in either pan of the scale? $\textbf{(A)}\ 15 \qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 11\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 7$

## Problem 20

It is given that $x$ varies directly as $y$ and inversely as the square of $z$, and that $x=10$ when $y=4$ and $z=14$. Then, when $y=16$ and $z=7$, $x$ equals: $\textbf{(A)}\ 180\qquad \textbf{(B)}\ 160\qquad \textbf{(C)}\ 154\qquad \textbf{(D)}\ 140\qquad \textbf{(E)}\ 120$

## Problem 21

If$p$ is the perimeter of an equilateral $\triangle$ inscribed in a circle, the area of the circle is: $\textbf{(A)}\ \frac{\pi p^2}{3} \qquad\textbf{(B)}\ \frac{\pi p^2}{9}\qquad\textbf{(C)}\ \frac{\pi p^2}{27}\qquad\textbf{(D)}\ \frac{\pi p^2}{81}\qquad\textbf{(E)}\ \frac{\pi p^2\sqrt3}{27}$

## Problem 22

The line joining the midpoints of the diagonals of a trapezoid has length $3$. If the longer base is $97,$ then the shorter base is: $\textbf{(A)}\ 94 \qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 89$

## Problem 23

The set of solutions of the equation $\log_{10}\left( a^2-15a\right)=2$ consists of $\textbf{(A)}\ \text{two integers } \qquad\textbf{(B)}\ \text{one integer and one fraction}\qquad \textbf{(C)}\ \text{two irrational numbers }\qquad\textbf{(D)}\ \text{two non-real numbers} \qquad\textbf{(E)}\ \text{no numbers, that is, the empty set}$

## Problem 24

A chemist has m ounces of salt that is $m$% salt. How many ounces of salt must he add to make a solution that is $2m$% salt?

$\textbf{(A)}\ \frac{m}{100+m} \qquad\textbf{(B)}\ \frac{2m}{100-2m}\qquad\textbf{(C)}\ \frac{m^2}{100-2m}\qquad\textbf{(D)}\ \frac{m^2}{100+2m}\qquad\textbf{(E)}\ \frac{2m}{100+2m}$

## Problem 25

The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if a is less than or equal to zero; the symbol $<$ means "less than"; the symbol $>$ means "greater than." The set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that: $\textbf{(A)}\ x^2<49 \qquad\textbf{(B)}\ x^2>1 \qquad\textbf{(C)}\ 1

## Problem 26

The base of an isosceles triangle is $\sqrt 2$. The medians to the leg intersect each other at right angles. The area of the triangle is: $\textbf{(A)}\ 1.5 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 2.5\qquad\textbf{(D)}\ 3.5\qquad\textbf{(E)}\ 4$

## Problem 27

Which one of the following is not true for the equation$ix^2-x+2i=0$, where $i=\sqrt{-1}$ $\textbf{(A)}\ \text{The sum of the roots is 2} \qquad \textbf{(B)}\ \text{The discriminant is 9}\qquad \textbf{(C)}\ \text{The roots are imaginary}\qquad \textbf{(D)}\ \text{The roots can be found using the quadratic formula}\qquad \textbf{(E)}\ \text{The roots can be found by factoring, using imaginary numbers}$

== Problem 28==$M are on BC and AB, respectively. The sides of$\triangle ABC$are$a,b,$and$c$. Then$\frac{\overbar{AM}}{\overbar{MB}}=k\frac{\overbar{CL}}{\overbar{LB}} $where$k$is:$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac{bc}{a^2}\qquad\textbf{(C)}\ \frac{a^2}{bc}\qquad\textbf{(D)}\ \frac{c}{b}\qquad\textbf{(E)}\ \frac{c}{a} $[[1959 AHSME Problems/Problem 28|Solution]] == Problem 29== On a examination of$ (Error compiling LaTeX. ! Missing $inserted.)n$questions a student answers correctly$15$of the first$20$. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is 50%, how many different values of$(Error compiling LaTeX. ! Missing$ inserted.)n$can there be?$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ \text{the problem cannot be solved} $[[1959 AHSME Problems/Problem 29|Solution]] == Problem 30==$ (Error compiling LaTeX. ! Missing $inserted.)A$can run around a circular track in$40$seconds.$B$, running in the opposite direction, meets$A$every$15$seconds. What is$B$'s time to run around the track, expressed in seconds?$\textbf{(A)}\ 12\frac12 \qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 27\frac12\qquad\textbf{(E)}\ 55$[[1959 AHSME Problems/Problem 30|Solution]]

== Problem 31== A square, with an area of$(Error compiling LaTeX. ! Missing$ inserted.)40$, is inscribed in a semicircle. The area of a square that could be inscribed in the entire circle with the same radius, is:$\textbf{(A)}\ 80 \qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 120\qquad\textbf{(D)}\ 160\qquad\textbf{(E)}\ 200 $[[1959 AHSME Problems/Problem 31|Solution]] == Problem 32== The length$ (Error compiling LaTeX. ! Missing $inserted.)l$of a tangent, drawn from a point$A$to a circle, is$\frac43 $of the radius$r$. The (shortest) distance from A to the circle is:$\textbf{(A)}\ \frac{1}{2}r \qquad\textbf{(B)}\ r\qquad\textbf{(C)}\ \frac{1}{2}l\qquad\textbf{(D)}\ \frac23l \qquad\textbf{(E)}\ \text{a value between r and l.}$[[1959 AHSME Problems/Problem 32|Solution]]

== Problem 33== A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression. Let$(Error compiling LaTeX. ! Missing$ inserted.)S_n$represent the sum of the first$n$terms of the harmonic progression; for example$S_3$represents the sum of the first three terms. If the first three terms of a harmonic progression are$3,4,6$, then:$\textbf{(A)}\ S_4=20 \qquad\textbf{(B)}\ S_4=25\qquad\textbf{(C)}\ S_5=49\qquad\textbf{(D)}\ S_6=49\qquad\textbf{(E)}\ S_2=\frac{1}2 S_4 $[[1959 AHSME Problems/Problem 33|Solution]] == Problem 34== Let the roots of$ (Error compiling LaTeX. ! Missing $inserted.)x^2-3x+1=0$be$r$and$s$. Then the expression$r^2+s^2 $is:$\textbf{(A)}\ \text{a positive integer} \qquad\textbf{(B)}\ \text{a positive fraction greater than 1}\qquad\textbf{(C)}\ \text{a positive fraction less than 1}\qquad\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}$[[1959 AHSME Problems/Problem 34|Solution]]

== Problem 35== The symbol$(Error compiling LaTeX. ! Missing$ inserted.)\ge$means "greater than or equal to"; the symbol$\le$means "less than or equal to". In the eqeuation$(x-m)^2-(x-n)^2=(m-n)^2; m$is a fixed positive number, and$n$is a fixed negative number. The set of values x satisfying the equation is:$\textbf{(A)}\ x\ge 0 \qquad\textbf{(B)}\ x\le n\qquad\textbf{(C)}\ x=0\qquad\textbf{(D)}\ \text{the set of all real numbers}\qquad\textbf{(E)}\ \text{none of these} $[[1959 AHSME Problems/Problem 35|Solution]] == Problem 36== The base of a triangle is$ (Error compiling LaTeX. ! Missing $inserted.)80$, and one side of the base angle is$60^\circ$. The sum of the lengths of the other two sides is$90$. The shortest side is:$\textbf{(A)}\ 45 \qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 12$[[1959 AHSME Problems/Problem 36|Solution]]

== Problem 37== When simplified the product$(Error compiling LaTeX. ! Missing$ inserted.)\left(1-\frac13\right)\left(1-\frac14\right)\left(1-\frac15\right)\cdots\left(1-\frac1n\right)$becomes:$\textbf{(A)}\ \frac1n \qquad\textbf{(B)}\ \frac2n\qquad\textbf{(C)}\ \frac{2(n-1)}{n}\qquad\textbf{(D)}\ \frac{2}{n(n+1)}\qquad\textbf{(E)}\ \frac{3}{n(n+1)} $[[1959 AHSME Problems/Problem 37|Solution]] == Problem 38== If$ (Error compiling LaTeX. ! Missing $inserted.)4x+\sqrt{2x}=1$, then$x$:$\textbf{(A)}\ \text{is an integer} \qquad\textbf{(B)}\ \text{is fractional}\qquad\textbf{(C)}\ \text{is irrational}\qquad\textbf{(D)}\ \text{is imaginary}\qquad\textbf{(E)}\ \text{may have two different values}$[[1959 AHSME Problems/Problem 38|Solution]]

== Problem 39== Let S be the sum of the first nine terms of the sequence$(Error compiling LaTeX. ! Missing$ inserted.)x+a, x^2+2a, x^3+3a, \cdots.$Then S equals:$\textbf{(A)}\ \frac{50a+x+x^8}{x+1} \qquad\textbf{(B)}\ 50a-\frac{x+x^{10}}{x-1}\qquad\textbf{(C)}\ \frac{x^9-1}{x+1}+45a\qquaud\textbf{(D)}\ \frac{x^{10}-x}{x-1}+45a\qquad\textbf{(E)}\ \frac{x^{11}-x}{x-1}+45a $[[1959 AHSME Problems/Problem 39|Solution]] == Problem 40== In$ (Error compiling LaTeX. ! Missing $inserted.)\triangle ABC$,$BD$is a median.$CF$intersects$BD$at$E$so that$\overbar{BE}=\overbar{ED}$. Point$F$is on$AB$. Then, if$\overbar{BF}=5$,$\overbar{BA}$equals:$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ \text{none of these}$[[1959 AHSME Problems/Problem 40|Solution]]

== Problem 41== On the same side of a straight line three circles are drawn as follows: a circle with a radius of$(Error compiling LaTeX. ! Missing$ inserted.)4$inches is tangent to the line, the other two circles are equal, and each is tangent to the line and to the other two circles. The radius of the equal circles is:$\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 12 $[[1959 AHSME Problems/Problem 41|Solution]] == Problem 42== Given three positive integers$ (Error compiling LaTeX. ! Missing $inserted.)a,b,$and$c$. Their greatest common divisor is$D$; their least common multiple is$m$. Then, which two of the following statements are true?$\text{(1)}\ \text{the product MD cannot be less than abc} \qquad \\ \text{(2)}\ \text{the product MD cannot be greater than abc}\qquad \\ \text{(3)}\ \text{MD equals abc if and only if a,b,c are each prime}\qquad \\ \text{(4)}\ \text{MD equals abc if and only if a,b,c are each relatively prime in pairs} \text{ (This means: no two have a common factor greater than 1.)}$$(Error compiling LaTeX. ! Missing$ inserted.)\textbf{(A)}\ 1,2 \qquad\textbf{(B)}\ 1,3\qquad\textbf{(C)}\ 1,4\qquad\textbf{(D)}\ 2,3\qquad\textbf{(E)}\ 2,4 $[[1959 AHSME Problems/Problem 42|Solution]] == Problem 43== The sides of a triangle are$ (Error compiling LaTeX. ! Missing $inserted.)25,39$, and$40$. The diameter of the circumscribed circle is:$\textbf{(A)}\ \frac{133}{3}\qquad\textbf{(B)}\ \frac{125}{3}\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 41\qquad\textbf{(E)}\ 40$[[1959 AHSME Problems/Problem 43|Solution]]

== Problem 44== The roots of$(Error compiling LaTeX. ! Missing$ inserted.)x^2+bx+c=0$are both real and greater than$1$. Let$s=b+c+1$. Then$s$:$\textbf{(A)}\ \text{may be less than zero}\qquad\textbf{(B)}\ \text{may be equal to zero}\qquad \textbf{(C)}\ \text{must be greater than zero}\qquad\textbf{(D)}\ \text{must be less than zero}\qquad \textbf{(E)}\text{ must be between -1 and +1}$[[1959 AHSME Problems/Problem 44|Solution]] == Problem 45== If$ (Error compiling LaTeX. ! Missing $inserted.)\left(\log_3 x\right)\left(\log_x 2x\right)\left( \log_{2x} y\right)=\log_{x}x^2$, then$ y$equals:$\textbf{(A)}\ \frac92\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 81$[[1959 AHSME Problems/Problem 45|Solution]]

== Problem 46== A student on vacation for$(Error compiling LaTeX. ! Missing$ inserted.)d$days observed that (1) it rained$7$times, morning or afternoon (2) when it rained in the afternoon, it was clear in the morning (3) there were five clear afternoons (4) there were six clear mornings. Then$d$equals:$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12 $[[1959 AHSME Problems/Problem 46|Solution]] == Problem 47== Assume that the following three statements are true: (I). All freshmen are human. (II). All students are human. (III). Some students think. Given the following four statements:$ (Error compiling LaTeX. ! Missing $inserted.)\textbf{(1)}\ \text{All freshmen are students.}\qquad \\ \textbf{(2)}\ \text{Some humans think.}\qquad \\ \textbf{(3)}\ \text{No freshmen think.}\qquad \\ \textbf{(4)}\ \text{Some humans who think are not students.}$Those which are logical consequences of I,II, and III are:$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 2,3\qquad\textbf{(D)}\ 2,4\qquad\textbf{(E)}\ 1,2$[[1959 AHSME Problems/Problem 47|Solution]]

== Problem 48== Given the polynomial$(Error compiling LaTeX. ! Missing$ inserted.)a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$, where$n$is a positive integer or zero, and$a_0$is a positive integer. The remaining$a$'s are integers or zero. Set$h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|$. [See example 25 for the meaning of$|x|$.] The number of polynomials with$h=3$is:$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 9 $[[1959 AHSME Problems/Problem 48|Solution]] == Problem 49== For the infinite series$ (Error compiling LaTeX. ! Missing $inserted.)1-\frac12-\frac14+\frac18-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots$let$S$be the (limiting) sum. Then$S$equals:$\textbf{(A)}\ 0\qquad\textbf{(B)}\ \frac27\qquad\textbf{(C)}\ \frac67\qquad\textbf{(D)}\ \frac{9}{32}\qquad\textbf{(E)}\ \frac{27}{32}$[[1959 AHSME Problems/Problem 49|Solution]]

== Problem 50== A club with$(Error compiling LaTeX. ! Missing$ inserted.)x$members is organized into four committees in accordance with these two rules: \text{(1)}\ \text{Each member belongs to two and only two committees}\qquad \\ \text{(2)}\ \text{Each pair of committees has one and only one member in common} Then x: \textbf{(A)} \ \text{cannont be determined} \qquad \\ \textbf{(B)} \ \text{has a single value between 8 and 16} \qquad \\ \textbf{(C)} \ \text{has two values between 8 and 16} \qquad \\ \textbf{(D)} \ \text{has a single value between 4 and 8} \qquad \\ \textbf{(E)} \ \text{has two values between 4 and 8}$