1959 AHSME Problems

Problem 1

Each edge of a cube is increased by %. The percent of increase of the surface area of the cube is:

Solution

Problem 2

Through a point inside the a line is drawn parallel to the base , dividing the triangle into two equal areas. If the altitude to has a length of , then the distance from to is:

Solution

Problem 3

If the diagonals of a quadrilateral are perpendicular to each other, the figure would always be included under the general classification:

Solution

Problem 4

If is divided into three parts which are proportional to the middle part is:

Solution

Problem 5

The value of is:

Solution

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:

Solution

Problem 7

The sides of a right triangle are , , and , with and both positive. The ratio of to is:

Solution

Problem 8

The value of can never be less than:

Solution

Problem 9

A farmer divides his herd of 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, cows. Then is:

Solution

Problem 10

In with $\overbar{AB}=\overbar{AC}=3.6$ (Error compiling LaTeX. ! Undefined control sequence.), a point is taken on at a distance from . Point is joined to in the prolongation of so that is equal in area to . Then $\overbar{AE}$ (Error compiling LaTeX. ! Undefined control sequence.) is:

Solution

Problem 11

The logarithm of to the base is:

Solution

Problem 12

By adding the same constant to a geometric progression results. The common ratio is:

Solution

Problem 13

The arithmetic mean (average) of a set of numbers is . If two numbers, namely, and , are discarded, the mean of the remaining set of numbers is:

Solution

Problem 14

Given the set 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 are:

Solution

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:

Solution

Problem 16

The expression when simplified is:

Solution

Problem 17

If , where and are constants, and if when , and x=-5a+b\textbf{(A)}\ -1 \qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 $[[1959 AHSME Problems/Problem 17|Solution]]

== Problem 18== The arithmetic mean (average) of the first$ (Error compiling LaTeX. ! Missing $ inserted.)n\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} $[[1959 AHSME Problems/Problem 18|Solution]]

== Problem 19== With the use of three different weights, namely$ (Error compiling LaTeX. ! Missing $ inserted.)139\textbf{(A)}\ 15 \qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 11\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 7 $[[1959 AHSME Problems/Problem 19|Solution]]

== Problem 20== It is given that$ (Error compiling LaTeX. ! Missing $ inserted.)xyzx=10y=4z=14y=16z=7x\textbf{(A)}\ 180\qquad\textbf{(B)}\ 160\qquad\textbf{(C)}\ 154\qquad\textbf{(D)}\ 140\qquad\textbf{(E)}\ 120 $[[1959 AHSME Problems/Problem 20|Solution]]

== Problem 21== If$ (Error compiling LaTeX. ! Missing $ inserted.) p\triangle\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} $[[1959 AHSME Problems/Problem 21|Solution]]

== Problem 22== The line joining the midpoints of the diagonals of a trapezoid has length$ (Error compiling LaTeX. ! Missing $ inserted.)397,\textbf{(A)}\ 94 \qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 89 $[[1959 AHSME Problems/Problem 22|Solution]]

== Problem 23== The set of solutions of the equation$ (Error compiling LaTeX. ! Missing $ inserted.)\log_{10}\left( a^2-15a\right)=2\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} $[[1959 AHSME Problems/Problem 23|Solution]]

== Problem 24== A chemist has m ounces of salt that is$ (Error compiling LaTeX. ! Missing $ inserted.)m$% salt. How many ounces of salt must he add to make a solution that is$ (Error compiling LaTeX. ! Missing $ inserted.)2m$% salt?$ (Error compiling LaTeX. ! Missing $ inserted.)\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}$[[1959 AHSME Problems/Problem 24|Solution]]

== Problem 25== The symbol$ (Error compiling LaTeX. ! Missing $ inserted.)|a|+aa-a<>x|3-x|<4x\textbf{(A)}\ x^2<49 \qquad\textbf{(B)}\ x^2>1 \qquad\textbf{(C)}\ 1<x^2<49\qquad\textbf{(D)}\ -1<x<7\qquad\textbf{(E)}\ -7<x<1$[[1959 AHSME Problems/Problem 25|Solution]]

== Problem 26== The base of an isosceles triangle is$ (Error compiling LaTeX. ! Missing $ inserted.)\sqrt 2\textbf{(A)}\ 1.5 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 2.5\qquad\textbf{(D)}\ 3.5\qquad\textbf{(E)}\ 4 $[[1959 AHSME Problems/Problem 26|Solution]]

== Problem 27== Which one of the following is not true for the equation$ (Error compiling LaTeX. ! Missing $ inserted.) ix^2-x+2i=0i=\sqrt{-1}

Solution

== Problem 28==\triangle ABCa,b,c\frac{\overbar{AM}}{\overbar{MB}}=k\frac{\overbar{CL}}{\overbar{LB}} k\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.)n1520$. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is 50%, how many different values of$ (Error compiling LaTeX. ! Missing $ inserted.)n\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.)A40BA15B\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\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.)lA\frac43 r\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_nnS_33,4,6\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=0rsr^2+s^2 \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\le(x-m)^2-(x-n)^2=(m-n)^2; mn\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.)8060^\circ90\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)\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}=1x\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.\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 ABCBDCFBDE\overbar{BE}=\overbar{ED}FAB\overbar{BF}=5\overbar{BA}\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\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,cDm\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.)25,3940\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=01s=b+c+1s\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 y\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.)d7d\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.}\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_nna_0ah=n+a_0+|a_1|+|a_2|+\cdots+|a_n||x|h=3\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}-\cdotsSS\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} \qquad \\

Solution

See also

• AHSME
• AHSME Problems and Solutions
• 1959 AHSME
• Mathematics competition resources

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