Difference between revisions of "1957 AHSME Problems"

Latest revision as of 10:55, 27 July 2024

1957 AHSC (Answer Key) Printable version: Wiki \| AoPS Resources • PDF
Instructions This is a 50-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. You will receive ? points for each correct answer, ? points for each problem left unanswered, and ? points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers. Figures are not necessarily drawn to scale. You will have ? minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50

Problem 1

The number of distinct lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is:

$\textbf{(A)}\ 9\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 3$

Solution

Problem 2

In the equation $2x^2 - hx + 2k = 0$ , the sum of the roots is $4$ and the product of the roots is $-3$ . Then $h$ and $k$ have the values, respectively:

$\textbf{(A)}\ 8\text{ and }{-6} \qquad \textbf{(B)}\ 4\text{ and }{-3}\qquad \textbf{(C)}\ {-3}\text{ and }4\qquad \textbf{(D)}\ {-3}\text{ and }8\qquad \textbf{(E)}\ 8\text{ and }{-3}$

Solution

Problem 3

The simplest form of $1 - \frac{1}{1 + \frac{a}{1 - a}}$ is:

$\textbf{(A)}\ {a}\text{ if }{a\not= 0} \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ {a}\text{ if }{a\not=-1}\qquad \textbf{(D)}\ {1-a}\text{ with not restriction on }{a}\qquad \textbf{(E)}\ {a}\text{ if }{a\not= 1}$

Solution

Problem 4

The first step in finding the product $(3x + 2)(x - 5)$ by use of the distributive property in the form $a(b + c) = ab + ac$ is:

$\textbf{(A)}\ 3x^2 - 13x - 10 \qquad \textbf{(B)}\ 3x(x - 5) + 2(x - 5)\qquad \\ \textbf{(C)}\ (3x+2)x+(3x+2)(-5)\qquad \textbf{(D)}\ 3x^2-17x-10\qquad \textbf{(E)}\ 3x^2+2x-15x-10$

Solution

Problem 5

Through the use of theorems on logarithms $\log{\frac{a}{b}} + \log{\frac{b}{c}} + \log{\frac{c}{d}} - \log{\frac{ay}{dx}}$ can be reduced to:

$\textbf{(A)}\ \log{\frac{y}{x}}\qquad \textbf{(B)}\ \log{\frac{x}{y}}\qquad \textbf{(C)}\ 1\qquad \\ \textbf{(D)}\ 140x-24x^2+x^3\qquad \textbf{(E)}\ \text{none of these}$

Solution

Problem 6

An open box is constructed by starting with a rectangular sheet of metal $10$ in. by $14$ in. and cutting a square of side $x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is:

$\textbf{(A)}\ 140x - 48x^2 + 4x^3 \qquad \textbf{(B)}\ 140x + 48x^2 + 4x^3\qquad \\ \textbf{(C)}\ 140x+24x^2+x^3\qquad \textbf{(D)}\ 140x-24x^2+x^3\qquad \textbf{(E)}\ \text{none of these}$

Solution

Problem 7

The area of a circle inscribed in an equilateral triangle is $48\pi$ . The perimeter of this triangle is:

$\textbf{(A)}\ 72\sqrt{3} \qquad \textbf{(B)}\ 48\sqrt{3}\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 24\qquad \textbf{(E)}\ 72$

Solution

Problem 8

The numbers $x,\,y,\,z$ are proportional to $2,\,3,\,5$ . The sum of $x, y$ , and $z$ is $100$ . The number y is given by the equation $y = ax - 10$ . Then a is:

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

Solution

Problem 9

The value of $x - y^{x - y}$ when $x = 2$ and $y = -2$ is:

$\textbf{(A)}\ -18 \qquad \textbf{(B)}\ -14\qquad \textbf{(C)}\ 14\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 256$

Solution

Problem 10

The graph of $y = 2x^2 + 4x + 3$ has its:

$\textbf{(A)}\ \text{lowest point at } {(-1,9)}\qquad \textbf{(B)}\ \text{lowest point at } {(1,1)}\qquad \\ \textbf{(C)}\ \text{lowest point at }{(-1,1)}\qquad \textbf{(D)}\ \text{highest point at }{(-1,9)}\qquad\\ \textbf{(E)}\ \text{highest point at }{(-1,1)}$

Solution

Problem 11

The angle formed by the hands of a clock at $2:15$ is:

$\textbf{(A)}\ 30^\circ \qquad \textbf{(B)}\ 27\frac{1}{2}^\circ\qquad \textbf{(C)}\ 157\frac{1}{2}^\circ\qquad \textbf{(D)}\ 172\frac{1}{2}^\circ\qquad \textbf{(E)}\ \text{none of these}$

Solution

Problem 12

Comparing the numbers $10^{-49}$ and $2\cdot 10^{-50}$ we may say:

$\textbf{(A)}\ \text{the first exceeds the second by }{8\cdot 10^{-1}}\qquad\\ \textbf{(B)}\ \text{the first exceeds the second by }{2\cdot 10^{-1}}\qquad\\ \textbf{(C)}\ \text{the first exceeds the second by }{8\cdot 10^{-50}}\qquad\\ \textbf{(D)}\ \text{the second is five times the first}\qquad\\ \textbf{(E)}\ \text{the first exceeds the second by }{5}$

Solution

Problem 13

A rational number between $\sqrt{2}$ and $\sqrt{3}$ is:

$\textbf{(A)}\ \frac{\sqrt{2} + \sqrt{3}}{2} \qquad \textbf{(B)}\ \frac{\sqrt{2} \cdot \sqrt{3}}{2}\qquad \textbf{(C)}\ 1.5\qquad \textbf{(D)}\ 1.8\qquad \textbf{(E)}\ 1.4$

Solution

Problem 14

If $y = \sqrt{x^2 - 2x + 1} + \sqrt{x^2 + 2x + 1}$ , then $y$ is:

$\textbf{(A)}\ 2x\qquad \textbf{(B)}\ 2(x+1)\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ |x-1|+|x+1|\qquad\textbf{(E)}\ \text{none of these}$

Solution

Problem 15

The table below shows the distance $s$ in feet a ball rolls down an inclined plane in $t$ seconds.

$\begin{tabular}{|c|c|c|c|c|c|c|}\hline t & 0 & 1 & 2 & 3 & 4 & 5\\ \hline s & 0 & 10 & 40 & 90 & 160 & 250\\ \hline\end{tabular}$

The distance $s$ for $t = 2.5$ is:

$\textbf{(A)}\ 45\qquad \textbf{(B)}\ 62.5\qquad \textbf{(C)}\ 70\qquad \textbf{(D)}\ 75\qquad \textbf{(E)}\ 82.5$

Solution

Problem 16

Goldfish are sold at $15$ cents each. The rectangular coordinate graph showing the cost of $1$ to $12$ goldfish is:

$\textbf{(A)}\ \text{a straight line segment} \qquad \\ \textbf{(B)}\ \text{a set of horizontal parallel line segments}\qquad\\ \textbf{(C)}\ \text{a set of vertical parallel line segments}\qquad\\ \textbf{(D)}\ \text{a finite set of distinct points} \qquad\textbf{(E)}\ \text{a straight line}$

Solution

Problem 17

A cube is made by soldering twelve $3$ -inch lengths of wire properly at the vertices of the cube. If a fly alights at one of the vertices and then walks along the edges, the greatest distance it could travel before coming to any vertex a second time, without retracing any distance, is:

$\textbf{(A)}\ 24\text{ in.}\qquad \textbf{(B)}\ 12\text{ in.}\qquad \textbf{(C)}\ 30\text{ in.}\qquad \textbf{(D)}\ 18\text{ in.}\qquad\textbf{(E)}\ 36\text{ in.}$

Solution

Problem 18

Circle $O$ has diameters $AB$ and $CD$ perpendicular to each other. $AM$ is any chord intersecting $CD$ at $P$ . Then $AP\cdot AM$ is equal to:

$[asy] defaultpen(linewidth(.8pt)); unitsize(2cm); pair O = origin; pair A = (-1,0); pair B = (1,0); pair C = (0,1); pair D = (0,-1); pair M = dir(45); pair P = intersectionpoint(O--C,A--M); draw(Circle(O,1)); draw(A--B); draw(C--D); draw(A--M); label("$A$",A,W); label("$B$",B,E); label("$C$",C,N); label("$D$",D,S); label("$M$",M,NE); label("$O$",O,NE); label("$P$",P,NW);[/asy]$

$\textbf{(A)}\ AO\cdot OB \qquad \textbf{(B)}\ AO\cdot AB\qquad \\ \textbf{(C)}\ CP\cdot CD \qquad \textbf{(D)}\ CP\cdot PD\qquad \textbf{(E)}\ CO\cdot OP$

Solution

Problem 19

The base of the decimal number system is ten, meaning, for example, that $123 = 1\cdot 10^2 + 2\cdot 10 + 3$ . In the binary system, which has base two, the first five positive integers are $1,\,10,\,11,\,100,\,101$ . The numeral $10011$ in the binary system would then be written in the decimal system as:

$\textbf{(A)}\ 19 \qquad \textbf{(B)}\ 40\qquad \textbf{(C)}\ 10011\qquad \textbf{(D)}\ 11\qquad \textbf{(E)}\ 7$

Solution

Problem 20

A man makes a trip by automobile at an average speed of 50 mph. He returns over the same route at an average speed of $45$ mph. His average speed for the entire trip is:

$\textbf{(A)}\ 47\frac{7}{19}\qquad \textbf{(B)}\ 47\frac{1}{4}\qquad \textbf{(C)}\ 47\frac{1}{2}\qquad \textbf{(D)}\ 47\frac{11}{19}\qquad \textbf{(E)}\ \text{none of these}$

Solution

Problem 21

Start with the theorem "If two angles of a triangle are equal, the triangle is isosceles," and the following four statements:

1. If two angles of a triangle are not equal, the triangle is not isosceles.

2. The base angles of an isosceles triangle are equal.

3. If a triangle is not isosceles, then two of its angles are not equal.

4. A necessary condition that two angles of a triangle be equal is that the triangle be isosceles.

Which combination of statements contains only those which are logically equivalent to the given theorem?

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

Solution

Problem 22

If $\sqrt{x - 1} - \sqrt{x + 1} + 1 = 0$ , then $4x$ equals:

$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 4\sqrt{-1}\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ 1\frac{1}{4}\qquad \textbf{(E)}\ \text{no real value}$

Solution

Problem 23

The graph of $x^2 + y = 10$ and the graph of $x + y = 10$ meet in two points. The distance between these two points is:

$\textbf{(A)}\ \text{less than 1} \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ \sqrt{2}\qquad \textbf{(D)}\ 2\qquad\textbf{(E)}\ \text{more than 2}$

Solution

Problem 24

If the square of a number of two digits is decreased by the square of the number formed by reversing the digits, then the result is not always divisible by:

$\textbf{(A)}\ 9 \qquad \textbf{(B)}\ \text{the product of the digits}\qquad \textbf{(C)}\ \text{the sum of the digits}\qquad \textbf{(D)}\ \text{the difference of the digits}\qquad \textbf{(E)}\ 11$

Solution

Problem 25

The vertices of $\triangle PQR$ have coordinates as follows: $P(0,a),\,Q(b,0),\,R(c,d)$ , where $a,\,b,\,c$ and $d$ are positive. The origin and point $R$ lie on opposite sides of $PQ$ . The area of $\triangle PQR$ may be found from the expression:

$\textbf{(A)}\ \frac{ab + ac + bc + cd}{2} \qquad \textbf{(B)}\ \frac{ac + bd - ab}{2}\qquad \textbf{(C)}\ \frac{ab-ac-bd}{2}\qquad \textbf{(D)}\ \frac{ac+bd+ab}{2}\qquad \textbf{(E)}\ \frac{ac+bd-ab-cd}{2}$

Solution

Problem 26

From a point within a triangle, line segments are drawn to the vertices. A necessary and sufficient condition that the three triangles thus formed have equal areas is that the point be:

$\textbf{(A)}\ \text{the center of the inscribed circle} \qquad \\ \textbf{(B)}\ \text{the center of the circumscribed circle}\qquad\\ \textbf{(C)}\ \text{such that the three angles formed at the point each be }{120^\circ}\qquad\\ \textbf{(D)}\ \text{the intersection of the altitudes of the triangle}\qquad\\ \textbf{(E)}\ \text{the intersection of the medians of the triangle}$

Solution

Problem 27

The sum of the reciprocals of the roots of the equation $x^2 + px + q = 0$ is:

$\textbf{(A)}\ -\frac{p}{q} \qquad \textbf{(B)}\ \frac{q}{p}\qquad \textbf{(C)}\ \frac{p}{q}\qquad \textbf{(D)}\ -\frac{q}{p}\qquad\textbf{(E)}\ pq$

Solution

Problem 28

If $a$ and $b$ are positive and $a\not= 1,\,b\not= 1$ , then the value of $b^{\log_b{a}}$ is:

$\textbf{(A)}\ \text{dependent upon }{b} \qquad \textbf{(B)}\ \text{dependent upon }{a}\qquad \textbf{(C)}\ \text{dependent upon }{a}\text{ and }{b}\qquad\textbf{(D)}\ \text{zero}\qquad\textbf{(E)}\ \text{one}$

Solution

Problem 29

The relation $x^2(x^2 - 1)\ge 0$ is true only for:

$\textbf{(A)}\ x \ge 1\qquad \textbf{(B)}\ - 1 \le x \le 1\qquad \textbf{(C)}\ x = 0,\, x = 1,\, x = - 1\qquad \\\textbf{(D)}\ x = 0,\, x\le-1,\, x\ge 1\qquad\textbf{(E)}\ x\ge 0$

Solution

Problem 30

The sum of the squares of the first n positive integers is given by the expression $\frac{n(n + c)(2n + k)}{6}$ , if $c$ and $k$ are, respectively:

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

Solution

Problem 31

A regular octagon is to be formed by cutting equal isosceles right triangles from the corners of a square. If the square has sides of one unit, the leg of each of the triangles has length:

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

Solution

Problem 32

The largest of the following integers which divides each of the numbers of the sequence $1^5 - 1,\, 2^5 - 2,\, 3^5 - 3,\, \cdots, n^5 - n, \cdots$ is:

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 120\qquad \textbf{(E)}\ 30$

Solution

Problem 33

If $9^{x + 2} = 240 + 9^x$ , then the value of $x$ is:

$\textbf{(A)}\ 0.1 \qquad \textbf{(B)}\ 0.2\qquad \textbf{(C)}\ 0.3\qquad \textbf{(D)}\ 0.4\qquad \textbf{(E)}\ 0.5$

Solution

Problem 34

The points that satisfy the system $x + y = 1,\, x^2 + y^2 < 25$ , constitute the following set:

$\textbf{(A)}\ \text{only two points} \qquad \\ \textbf{(B)}\ \text{an arc of a circle}\qquad \\ \textbf{(C)}\ \text{a straight line segment not including the end-points}\qquad\\ \textbf{(D)}\ \text{a straight line segment including the end-points}\qquad\\ \textbf{(E)}\ \text{a single point}$

Solution

Problem 35

Side $AC$ of right triangle $ABC$ is divided into $8$ equal parts. Seven line segments parallel to $BC$ are drawn to $AB$ from the points of division. If $BC = 10$ , then the sum of the lengths of the seven line segments:

$\textbf{(A)}\ \text{cannot be found from the given information} \qquad \textbf{(B)}\ \text{is }{33}\qquad \textbf{(C)}\ \text{is }{34}\qquad\textbf{(D)}\ \text{is }{35}\qquad\textbf{(E)}\ \text{is }{45}$

Solution

Problem 36

If $x + y = 1$ , then the largest value of $xy$ is:

$\textbf{(A)}\ 1\qquad \textbf{(B)}\ 0.5\qquad \textbf{(C)}\ \text{an irrational number about }{0.4}\qquad \textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0$

Solution

Problem 37

In right triangle $ABC, BC = 5, AC = 12$ , and $AM = x; \overline{MN} \perp \overline{AC}, \overline{NP} \perp \overline{BC}$ ; $N$ is on $AB$ . If $y = MN + NP$ , one-half the perimeter of rectangle $MCPN$ , then:

$[asy] defaultpen(linewidth(.8pt)); unitsize(2cm); pair A = origin; pair M = (1,0); pair C = (2,0); pair P = (2,0.5); pair B = (2,1); pair Q = (1,0.5); draw(A--B--C--cycle); draw(M--Q--P); label("$A$",A,SW); label("$M$",M,S); label("$C$",C,SE); label("$P$",P,E); label("$B$",B,NE); label("$N$",Q,NW);[/asy]$

$\textbf{(A)}\ y = \frac {1}{2}(5 + 12) \qquad \textbf{(B)}\ y = \frac {5x}{12} + \frac {12}{5}\qquad \textbf{(C)}\ y =\frac{144-7x}{12}\qquad \\ \textbf{(D)}\ y = 12\qquad \qquad\quad\,\, \textbf{(E)}\ y = \frac {5x}{12} + 6$

Solution

Problem 38

From a two-digit number $N$ we subtract the number with the digits reversed and find that the result is a positive perfect cube. Then:

$\textbf{(A)}\ {N}\text{ cannot end in 5}\qquad\\ \textbf{(B)}\ {N}\text{ can end in any digit other than 5}\qquad \\ \textbf{(C)}\ {N}\text{ does not exist}\qquad\\ \textbf{(D)}\ \text{there are exactly 7 values for }{N}\qquad\\ \textbf{(E)}\ \text{there are exactly 10 values for }{N}$

Solution

Problem 39

Two men set out at the same time to walk towards each other from $M$ and $N$ , $72$ miles apart. The first man walks at the rate of $4$ mph. The second man walks $2$ miles the first hour, $2\tfrac {1}{2}$ miles the second hour, $3$ miles the third hour, and so on in arithmetic progression. Then the men will meet:

$\textbf{(A)}\ \text{in 7 hours} \qquad \textbf{(B)}\ \text{in }{8\frac {1}{4}}\text{ hours}\qquad \textbf{(C)}\ \text{nearer }{M}\text{ than }{N}\qquad\\ \textbf{(D)}\ \text{nearer }{N}\text{ than }{M}\qquad \textbf{(E)}\ \text{midway between }{M}\text{ and }{N}$

Solution

Problem 40

If the parabola $y = -x^2 + bx - 8$ has its vertex on the $x$ -axis, then $b$ must be:

$\textbf{(A)}\ \text{a positive integer}\qquad \\ \textbf{(B)}\ \text{a positive or a negative rational number}\qquad \\ \textbf{(C)}\ \text{a positive rational number}\qquad\\ \textbf{(D)}\ \text{a positive or a negative irrational number}\qquad\\ \textbf{(E)}\ \text{a negative irrational number}$

Solution

Problem 41

Given the system of equations $ax + (a - 1)y = 1$ $(a + 1)x - ay = 1$ For which one of the following values of $a$ is there no solution $x$ and $y$ ?

$\textbf{(A)}\ 1\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ - 1\qquad \textbf{(D)}\ \frac {\pm \sqrt {2}}{2}\qquad \textbf{(E)}\ \pm\sqrt{2}$

Solution

Problem 42

If $S = i^n + i^{-n}$ , where $i = \sqrt{-1}$ and $n$ is an integer, then the total number of possible distinct values for $S$ is:

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

Solution

Problem 43

We define a lattice point as a point whose coordinates are integers, zero admitted. Then the number of lattice points on the boundary and inside the region bounded by the $x$ -axis, the line $x = 4$ , and the parabola $y = x^2$ is:

$\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 35\qquad \textbf{(C)}\ 34\qquad \textbf{(D)}\ 30\qquad \textbf{(E)}\ \infty$

Solution

Problem 44

In $\triangle ABC, AC = CD$ and $\angle CAB - \angle ABC = 30^\circ$ . Then $\angle BAD$ is:

$[asy] defaultpen(linewidth(.8pt)); unitsize(2.5cm); pair A = origin; pair B = (2,0); pair C = (0.5,0.75); pair D = midpoint(C--B); draw(A--B--C--cycle); draw(A--D); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,NE);[/asy]$

$\textbf{(A)}\ 30^\circ\qquad\textbf{(B)}\ 20^\circ\qquad\textbf{(C)}\ 22\frac{1}{2}^\circ\qquad\textbf{(D)}\ 10^\circ\qquad\textbf{(E)}\ 15^\circ$

Solution

Problem 45

If two real numbers $x$ and $y$ satisfy the equation $\frac{x}{y} = x - y$ , then:

$\textbf{(A)}\ {x \ge 4}\text{ or }{x \le 0}\qquad \\ \textbf{(B)}\ {y}\text{ can equal }{1}\qquad \\ \textbf{(C)}\ \text{both }{x}\text{ and }{y}\text{ must be irrational}\qquad\\ \textbf{(D)}\ {x}\text{ and }{y}\text{ cannot both be integers}\qquad\\ \textbf{(E)}\ \text{both }{x}\text{ and }{y}\text{ must be rational}$

Solution

Problem 46

Two perpendicular chords intersect in a circle. The segments of one chord are $3$ and $4$ ; the segments of the other are $6$ and $2$ . Then the diameter of the circle is:

$\textbf{(A)}\ \sqrt{89}\qquad \textbf{(B)}\ \sqrt{56}\qquad \textbf{(C)}\ \sqrt{61}\qquad \textbf{(D)}\ \sqrt{75}\qquad \textbf{(E)}\ \sqrt{65}$

Solution

Problem 47

In circle $O$ , the midpoint of radius $OX$ is $Q$ ; at $Q$ , $\overline{AB} \perp \overline{XY}$ . The semi-circle with $\overline{AB}$ as diameter intersects $\overline{XY}$ in $M$ . Line $\overline{AM}$ intersects circle $O$ in $C$ , and line $\overline{BM}$ intersects circle $O$ in $D$ . Line $\overline{AD}$ is drawn. Then, if the radius of circle $O$ is $r$ , $AD$ is:

$[asy] defaultpen(linewidth(.8pt)); unitsize(2.5cm); real m = 0; real b = 0; pair O = origin; pair X = (-1,0); pair Y = (1,0); pair Q = midpoint(O--X); pair A = (Q.x, -1*sqrt(3)/2); pair B = (Q.x, -1*A.y); pair M = (Q.x + sqrt(3)/2,0); m = (B.y - M.y)/(B.x - M.x); b = (B.y - m*B.x); pair D = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b)); m = (A.y - M.y)/(A.x - M.x); b = (A.y - m*A.x); pair C = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b)); draw(Circle(O,1)); draw(Arc(Q,sqrt(3)/2,-90,90)); draw(A--B); draw(X--Y); draw(B--D); draw(A--C); draw(A--D); dot(O);dot(M); label("$B$",B,NW); label("$C$",C,NE); label("$Y$",Y,E); label("$D$",D,SE); label("$A$",A,SW); label("$X$",X,W); label("$Q$",Q,SW); label("$O$",O,SW); label("$M$",M,NE+2N);[/asy]$

$\textbf{(A)}\ r\sqrt{2}\qquad\textbf{(B)}\ r\qquad\textbf{(C)}\ \text{not a side of an inscribed regular polygon}\qquad\textbf{(D)}\ \frac{r\sqrt{3}}{2}\qquad\textbf{(E)}\ r\sqrt{3}$

Solution

Problem 48

Let $ABC$ be an equilateral triangle inscribed in circle $O$ . $M$ is a point on arc $BC$ . Lines $\overline{AM}$ , $\overline{BM}$ , and $\overline{CM}$ are drawn. Then $AM$ is:

$[asy] defaultpen(linewidth(.8pt)); unitsize(2cm); pair O = origin; pair B = (1,0); pair C = dir(120); pair A = dir(240); pair M = dir(90 - 18); draw(Circle(O,1)); draw(A--C--M--B--cycle); draw(B--C); draw(A--M); dot(O); label("$A$",A,SW); label("$B$",B,E); label("$M$",M,NE); label("$C$",C,NW); label("$O$",O,SE);[/asy]$

$\textbf{(A)}\ \text{equal to }{BM + CM}\qquad \textbf{(B)}\ \text{less than }{BM + CM}\qquad \\ \textbf{(C)}\ \text{greater than }{BM+CM}\qquad \\ \textbf{(D)}\ \text{equal, less than, or greater than }{BM + CM}\text{, depending upon the position of } {M}\qquad \\ \textbf{(E)}\ \text{none of these}$

Solution

Problem 49

The parallel sides of a trapezoid are $3$ and $9$ . The non-parallel sides are $4$ and $6$ . A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is:

$[asy] defaultpen(linewidth(.8pt)); unitsize(2cm); pair A = origin; pair B = (2.25,0); pair C = (2,1); pair D = (1,1); pair E = waypoint(A--D,0.25); pair F = waypoint(B--C,0.25); draw(A--B--C--D--cycle); draw(E--F); label("6",midpoint(A--D),NW); label("3",midpoint(C--D),N); label("4",midpoint(C--B),NE); label("9",midpoint(A--B),S);[/asy]$

$\textbf{(A)}\ 4: 3\qquad\textbf{(B)}\ 3: 2\qquad\textbf{(C)}\ 4: 1\qquad\textbf{(D)}\ 3: 1\qquad\textbf{(E)}\ 6: 1$

Solution

Problem 50

In circle $O$ , $G$ is a moving point on diameter $\overline{AB}$ . $\overline{AA'}$ is drawn perpendicular to $\overline{AB}$ and equal to $\overline{AG}$ . $\overline{BB'}$ is drawn perpendicular to $\overline{AB}$ , on the same side of diameter $\overline{AB}$ as $\overline{AA'}$ , and equal to $BG$ . Let $O'$ be the midpoint of $\overline{A'B'}$ . Then, as $G$ moves from $A$ to $B$ , point $O'$ :

$\textbf{(A)}\ \text{moves on a straight line parallel to }{AB}\qquad \\ \textbf{(B)}\ \text{remains stationary}\qquad\\ \textbf{(C)}\ \text{moves on a straight line perpendicular to }{AB}\qquad\\ \textbf{(D)}\ \text{moves in a small circle intersecting the given circle}\qquad\\ \textbf{(E)}\ \text{follows a path which is neither a circle nor a straight line}$

Solution

@@ Line 1: / Line 1: @@
+{{AHSC 50 Problems
+|year=1957
+}}
 == Problem 1==
@@ Line 5: / Line 8: @@
 <math>\textbf{(A)}\ 9\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 3 </math>
-[[1957 AHSME Problems/Problem 2|Solution]]
+[[1957 AHSME Problems/Problem 1|Solution]]
 == Problem 2==
@@ Line 213: / Line 216: @@
 label("$P$",P,NW);</asy>
-\textbf{(A)}\ AO\cdot OB \qquad \textbf{(B)}\ AO\cdot AB\qquad \textbf{(C)}\ CP\cdot CD \qquad
+<math>\textbf{(A)}\ AO\cdot OB \qquad \textbf{(B)}\ AO\cdot AB\qquad \\
+\textbf{(C)}\ CP\cdot CD \qquad
 \textbf{(D)}\ CP\cdot PD\qquad
-\textbf{(E)}\ CO\cdot OP
+\textbf{(E)}\ CO\cdot OP    </math>
 [[1957 AHSME Problems/Problem 18|Solution]]
@@ Line 247: / Line 251: @@
 . If two angles of a triangle are not equal, the triangle is not isosceles.
 . The base angles of an isosceles triangle are equal.
 . If a triangle is not isosceles, then two of its angles are not equal.
 . A necessary condition that two angles of a triangle be equal is that the triangle be isosceles.
@@ Line 306: / Line 313: @@
 <math>\textbf{(A)}\ \text{the center of the inscribed circle} \qquad \\
 \textbf{(B)}\ \text{the center of the circumscribed circle}\qquad\\
-\textbf{(C)}\ \text{such that the three angles fromed at the point each be }{120^\circ}\qquad\\
+\textbf{(C)}\ \text{such that the three angles formed at the point each be }{120^\circ}\qquad\\
 \textbf{(D)}\ \text{the intersection of the altitudes of the triangle}\qquad\\
 \textbf{(E)}\ \text{the intersection of the medians of the triangle} </math>
@@ Line 324: / Line 331: @@
 If <math>a</math> and <math>b</math> are positive and <math>a\not= 1,\,b\not= 1</math>, then the value of <math>b^{\log_b{a}}</math> is:
-\textbf{(A)}\ \text{dependent upon }{b} \qquad \textbf{(B)}\ \text{dependent upon }{a}\qquad \textbf{(C)}\ \text{dependent upon }{a}\text{ and }{b}\qquad\textbf{(D)}\ \text{zero}\qquad\textbf{(E)}\ \text{one}   <math>
+<math>\textbf{(A)}\ \text{dependent upon }{b} \qquad \textbf{(B)}\ \text{dependent upon }{a}\qquad \textbf{(C)}\ \text{dependent upon }{a}\text{ and }{b}\qquad\textbf{(D)}\ \text{zero}\qquad\textbf{(E)}\ \text{one}   </math>
 [[1957 AHSME Problems/Problem 28|Solution]]
@@ Line 330: / Line 337: @@
 == Problem 29==
-The relation </math>x^2(x^2 - 1)\ge 0<math> is true only for:
+The relation <math>x^2(x^2 - 1)\ge 0</math> is true only for:
-</math>\textbf{(A)}\ x \ge 1\qquad \textbf{(B)}\ - 1 \le x \le 1\qquad \textbf{(C)}\ x = 0,\, x = 1,\, x = - 1\qquad \\\textbf{(D)}\ x = 0,\, x\le-1,\, x\ge 1\qquad\textbf{(E)}\ x\ge 0  <math>
+<math>\textbf{(A)}\ x \ge 1\qquad \textbf{(B)}\ - 1 \le x \le 1\qquad \textbf{(C)}\ x = 0,\, x = 1,\, x = - 1\qquad \\\textbf{(D)}\ x = 0,\, x\le-1,\, x\ge 1\qquad\textbf{(E)}\ x\ge 0  </math>
 [[1957 AHSME Problems/Problem 29|Solution]]
@@ Line 338: / Line 345: @@
 == Problem 30==
-The sum of the squares of the first n positive integers is given by the expression </math>\frac{n(n + c)(2n + k)}{6}<math>,
+The sum of the squares of the first n positive integers is given by the expression <math>\frac{n(n + c)(2n + k)}{6}</math>,
-if </math>c<math> and </math>k<math> are, respectively:
+if <math>c</math> and <math>k</math> are, respectively:
-</math>\textbf{(A)}\ {1}\text{ and }{2} \qquad \textbf{(B)}\ {3}\text{ and }{5}\qquad \textbf{(C)}\ {2}\text{ and }{2}\qquad\textbf{(D)}\ {1}\text{ and }{1}\qquad\textbf{(E)}\ {2}\text{ and }{1}    <math>
+<math>\textbf{(A)}\ {1}\text{ and }{2} \qquad \textbf{(B)}\ {3}\text{ and }{5}\qquad \textbf{(C)}\ {2}\text{ and }{2}\qquad\textbf{(D)}\ {1}\text{ and }{1}\qquad\textbf{(E)}\ {2}\text{ and }{1}    </math>
 [[1957 AHSME Problems/Problem 30|Solution]]
@@ Line 350: / Line 357: @@
 If the square has sides of one unit, the leg of each of the triangles has length:
-</math>\textbf{(A)}\ \frac{2 + \sqrt{2}}{3} \qquad \textbf{(B)}\ \frac{2 - \sqrt{2}}{2}\qquad \textbf{(C)}\ \frac{1+\sqrt{2}}{2}\qquad\textbf{(D)}\ \frac{1+\sqrt{2}}{3}\qquad\textbf{(E)}\ \frac{2-\sqrt{2}}{3} <math>
+<math>\textbf{(A)}\ \frac{2 + \sqrt{2}}{3} \qquad \textbf{(B)}\ \frac{2 - \sqrt{2}}{2}\qquad \textbf{(C)}\ \frac{1+\sqrt{2}}{2}\qquad\textbf{(D)}\ \frac{1+\sqrt{2}}{3}\qquad\textbf{(E)}\ \frac{2-\sqrt{2}}{3} </math>
 [[1957 AHSME Problems/Problem 31|Solution]]
@@ Line 356: / Line 363: @@
 == Problem 32==
-The largest of the following integers which divides each of the numbers of the sequence </math>1^5 - 1,\, 2^5 - 2,\, 3^5 - 3,\, \cdots, n^5 - n, \cdots<math> is:
+The largest of the following integers which divides each of the numbers of the sequence <math>1^5 - 1,\, 2^5 - 2,\, 3^5 - 3,\, \cdots, n^5 - n, \cdots</math> is:
-</math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 120\qquad \textbf{(E)}\ 30  <math>
+<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 120\qquad \textbf{(E)}\ 30  </math>
 [[1957 AHSME Problems/Problem 32|Solution]]
@@ Line 364: / Line 371: @@
 == Problem 33==
-If </math>9^{x + 2} = 240 + 9^x<math>, then the value of </math>x<math> is:
+If <math>9^{x + 2} = 240 + 9^x</math>, then the value of <math>x</math> is:
-</math>\textbf{(A)}\ 0.1 \qquad \textbf{(B)}\ 0.2\qquad \textbf{(C)}\ 0.3\qquad \textbf{(D)}\ 0.4\qquad \textbf{(E)}\ 0.5  <math>
+<math>\textbf{(A)}\ 0.1 \qquad \textbf{(B)}\ 0.2\qquad \textbf{(C)}\ 0.3\qquad \textbf{(D)}\ 0.4\qquad \textbf{(E)}\ 0.5  </math>
 [[1957 AHSME Problems/Problem 33|Solution]]
@@ Line 372: / Line 379: @@
 == Problem 34==
-The points that satisfy the system </math>x + y = 1,\, x^2 + y^2 < 25<math>, constitute the following set:
+The points that satisfy the system <math>x + y = 1,\, x^2 + y^2 < 25</math>, constitute the following set:
-</math>\textbf{(A)}\ \text{only two points} \qquad \\
+<math>\textbf{(A)}\ \text{only two points} \qquad \\
 \textbf{(B)}\ \text{an arc of a circle}\qquad \\
 \textbf{(C)}\ \text{a straight line segment not including the end-points}\qquad\\
 \textbf{(D)}\ \text{a straight line segment including the end-points}\qquad\\
-\textbf{(E)}\ \text{a single point} <math>
+\textbf{(E)}\ \text{a single point} </math>
 [[1957 AHSME Problems/Problem 34|Solution]]
@@ Line 384: / Line 391: @@
 == Problem 35==
-Side </math>AC<math> of right triangle </math>ABC<math> is divide into </math>8<math> equal parts. Seven line segments parallel to </math>BC<math> are drawn to </math>AB<math> from the points of division.
+Side <math>AC</math> of right triangle <math>ABC</math> is divided into <math>8</math> equal parts. Seven line segments parallel to <math>BC</math> are drawn to <math>AB</math> from the points of division.
-If </math>BC = 10<math>, then the sum of the lengths of the seven line segments:
+If <math>BC = 10</math>, then the sum of the lengths of the seven line segments:
-</math>\textbf{(A)}\ \text{cannot be found from the given information} \qquad \textbf{(B)}\ \text{is }{33}\qquad \textbf{(C)}\ \text{is }{34}\qquad\textbf{(D)}\ \text{is }{35}\qquad\textbf{(E)}\ \text{is }{45}  <math>
+<math>\textbf{(A)}\ \text{cannot be found from the given information} \qquad \textbf{(B)}\ \text{is }{33}\qquad \textbf{(C)}\ \text{is }{34}\qquad\textbf{(D)}\ \text{is }{35}\qquad\textbf{(E)}\ \text{is }{45}  </math>
 [[1957 AHSME Problems/Problem 35|Solution]]
@@ Line 393: / Line 400: @@
 == Problem 36==
-If </math>x + y = 1<math>, then the largest value of </math>xy<math> is:
+If <math>x + y = 1</math>, then the largest value of <math>xy</math> is:
-</math>\textbf{(A)}\ 1\qquad \textbf{(B)}\ 0.5\qquad \textbf{(C)}\ \text{an irrational number about }{0.4}\qquad \textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0<math>
+<math>\textbf{(A)}\ 1\qquad \textbf{(B)}\ 0.5\qquad \textbf{(C)}\ \text{an irrational number about }{0.4}\qquad \textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0</math>
 [[1957 AHSME Problems/Problem 36|Solution]]
@@ Line 401: / Line 408: @@
 == Problem 37==
-In right triangle </math>ABC, BC = 5, AC = 12<math>, and </math>AM = x; \overline{MN} \perp \overline{AC}, \overline{NP} \perp \overline{BC}<math>;
+In right triangle <math>ABC, BC = 5, AC = 12</math>, and <math>AM = x; \overline{MN} \perp \overline{AC}, \overline{NP} \perp \overline{BC}</math>;
-</math>N<math> is on </math>AB<math>. If </math>y = MN + NP<math>, one-half the perimeter of rectangle </math>MCPN$, then:
+<math>N</math> is on <math>AB</math>. If <math>y = MN + NP</math>, one-half the perimeter of rectangle <math>MCPN</math>, then:
 <asy>
@@ Line 471: / Line 478: @@
 Given the system of equations
-<math>ax + (a - 1)y = 1 \\ (a + 1)x - ay = 1</math>.
+<cmath>ax + (a - 1)y = 1</cmath>
+<cmath>(a + 1)x - ay = 1</cmath>
 For which one of the following values of <math>a</math> is there no solution <math>x</math> and <math>y</math>?
@@ Line 497: / Line 505: @@
 <math>\textbf{(A)}\ 24 \qquad
-\textbf{(B)}\ 35\qquad \
+\textbf{(B)}\ 35\qquad
-textbf{(C)}\ 34\qquad
+\textbf{(C)}\ 34\qquad
 \textbf{(D)}\ 30\qquad
 \textbf{(E)}\ \infty</math>
@@ Line 522: / Line 530: @@
 label("$D$",D,NE);</asy>
+<math> \textbf{(A)}\ 30^\circ\qquad\textbf{(B)}\ 20^\circ\qquad\textbf{(C)}\ 22\frac{1}{2}^\circ\qquad\textbf{(D)}\ 10^\circ\qquad\textbf{(E)}\ 15^\circ </math>
 [[1957 AHSME Problems/Problem 44|Solution]]
@@ Line 528: / Line 538: @@
 If two real numbers <math>x</math> and <math>y</math> satisfy the equation <math>\frac{x}{y} = x - y</math>, then:
-<math>\textbf{(A)}\ {x \ge 4}\text{ and }{x \le 0}\qquad \\
+<math>\textbf{(A)}\ {x \ge 4}\text{ or }{x \le 0}\qquad \\
 \textbf{(B)}\ {y}\text{ can equal }{1}\qquad \\
 \textbf{(C)}\ \text{both }{x}\text{ and }{y}\text{ must be irrational}\qquad\\
@@ Line 590: / Line 600: @@
 label("$O$",O,SW);
 label("$M$",M,NE+2N);</asy>
+<math> \textbf{(A)}\ r\sqrt{2}\qquad\textbf{(B)}\ r\qquad\textbf{(C)}\ \text{not a side of an inscribed regular polygon}\qquad\textbf{(D)}\ \frac{r\sqrt{3}}{2}\qquad\textbf{(E)}\ r\sqrt{3} </math>
 [[1957 AHSME Problems/Problem 47|Solution]]
@@ Line 617: / Line 629: @@
 label("$O$",O,SE);</asy>
-<math>\textbf{(A)}\ \text{equal to }{BM + CM}\qquad \textbf{(B)}\ \text{less than }{BM + CM}\qquad \textbf{(C)}\ \text{greater than }{BM+CM}\qquad
+<math>\textbf{(A)}\ \text{equal to }{BM + CM}\qquad
-\textbf{(D)}\ \text{equal, less than, or greater than }{BM + CM}\text{, depending upon the position of }{ {M}\qquad
+\textbf{(B)}\ \text{less than }{BM + CM}\qquad \\
-\textbf{(E)}\ \text{none of these}
+\textbf{(C)}\ \text{greater than }{BM+CM}\qquad \\
+\textbf{(D)}\ \text{equal, less than, or greater than }{BM + CM}\text{, depending upon the position of } {M}\qquad  \\
+\textbf{(E)}\ \text{none of these}     </math>
 [[1957 AHSME Problems/Problem 48|Solution]]
@@ Line 625: / Line 639: @@
 == Problem 49==
-The parallel sides of a trapezoid are </math>3<math> and </math>9<math>. The non-parallel sides are </math>4<math> and </math>6$.
+The parallel sides of a trapezoid are <math>3</math> and <math>9</math>. The non-parallel sides are <math>4</math> and <math>6</math>.
 A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters.
 The ratio in which each of the non-parallel sides is divided is:
@@ Line 644: / Line 658: @@
 label("4",midpoint(C--B),NE);
 label("9",midpoint(A--B),S);</asy>
+<math> \textbf{(A)}\ 4: 3\qquad\textbf{(B)}\ 3: 2\qquad\textbf{(C)}\ 4: 1\qquad\textbf{(D)}\ 3: 1\qquad\textbf{(E)}\ 6: 1 </math>
 [[1957 AHSME Problems/Problem 49|Solution]]
@@ Line 662: / Line 678: @@
 == See also ==
-* [[AHSME]]
-* [[AHSME Problems and Solutions]]
+* [[AMC 12 Problems and Solutions]]
-* [[1957 AHSME]]
 * [[Mathematics competition resources]]
+{{AHSME 50p box|year=1957|before=[[1956 AHSME|1956 AHSME]]|after=[[1958 AHSME|1958 AHSME]]}}
 {{MAA Notice}}

1957 AHSC (Problems • Answer Key • Resources)
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Difference between revisions of "1957 AHSME Problems"

Latest revision as of 10:55, 27 July 2024

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