Difference between revisions of "1962 AHSME Problems"
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− | ==Problem 1== | + | == Problem 1 == |
− | |||
The expression <math>\frac{1^{4y-1}}{5^{-1}+3^{-1}}</math> is equal to: | The expression <math>\frac{1^{4y-1}}{5^{-1}+3^{-1}}</math> is equal to: | ||
− | <math> \textbf{(A)}\ \frac{4y-1}{8}\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{15}{2}\qquad\textbf{(D)}\ \frac{15}{8}\qquad\textbf{(E)}\ \frac{1}{8} </math> | + | <math>\textbf{(A)}\ \frac{4y-1}{8} \qquad |
+ | \textbf{(B)}\ 8 \qquad | ||
+ | \textbf{(C)}\ \frac{15}{2} \qquad | ||
+ | \textbf{(D)}\ \frac{15}{8}\qquad | ||
+ | \textbf{(E)}\ \frac{1}{8}</math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 1|Solution]] | ||
− | + | == Problem 2 == | |
− | ==Problem 2== | ||
The expression <math>\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}</math> is equal to: | The expression <math>\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}</math> is equal to: | ||
− | <math> \textbf{(A)}\ \frac{\sqrt{3}}{6}\qquad\textbf{(B)}\ \frac{-\sqrt{3}}{6}\qquad\textbf{(C)}\ \frac{\sqrt{-3}}{6}\qquad\textbf{(D)}\ \frac{5\sqrt{3}}{6}\qquad\textbf{(E)}\ 1 </math> | + | <math>\textbf{(A)}\ \frac{\sqrt{3}}{6} \qquad |
− | + | \textbf{(B)}\ \frac{-\sqrt{3}}{6} \qquad | |
+ | \textbf{(C)}\ \frac{\sqrt{-3}}{6}\qquad | ||
+ | \textbf{(D)}\ \frac{5\sqrt{3}}{6}\qquad | ||
+ | \textbf{(E)}\ 1</math> | ||
+ | |||
[[1962 AHSME Problems/Problem 2|Solution]] | [[1962 AHSME Problems/Problem 2|Solution]] | ||
− | |||
+ | == Problem 3 == | ||
+ | |||
The first three terms of an arithmetic progression are <math>x - 1, x + 1, 2x + 3</math>, in the order shown. The value of <math>x</math> is: | The first three terms of an arithmetic progression are <math>x - 1, x + 1, 2x + 3</math>, in the order shown. The value of <math>x</math> is: | ||
− | <math> \textbf{(A)}\ -2\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{ | + | <math>\textbf{(A)}\ - 2 \qquad |
− | + | \textbf{(B)}\ 0 \qquad | |
+ | \textbf{(C)}\ 2 \qquad | ||
+ | \textbf{(D)}\ 4 \qquad | ||
+ | \textbf{(E)}\ \text{undetermineD}</math> | ||
+ | |||
[[1962 AHSME Problems/Problem 3|Solution]] | [[1962 AHSME Problems/Problem 3|Solution]] | ||
− | |||
− | If <math>8^x = 32</math>, then x equals: | + | == Problem 4 == |
+ | |||
+ | If <math>8^x = 32</math>, then <math>x</math> equals: | ||
− | <math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ \frac{5}{3}\qquad\textbf{(C)}\ \frac{3}{2}\qquad\textbf{(D)}\ \frac{3}{5}\qquad\textbf{(E)}\ \frac{1}{4} </math> | + | <math>\textbf{(A)}\ 4 \qquad |
+ | \textbf{(B)}\ \frac{5}{3} \qquad | ||
+ | \textbf{(C)}\ \frac{3}{2} \qquad | ||
+ | \textbf{(D)}\ \frac{3}{5} \qquad | ||
+ | \textbf{(E)}\ \frac{1}{4} </math> | ||
[[1962 AHSME Problems/Problem 4|Solution]] | [[1962 AHSME Problems/Problem 4|Solution]] | ||
+ | == Problem 5 == | ||
+ | |||
+ | If the radius of a circle is increased by <math>1</math> unit, the ratio of the new circumference to the new diameter is: | ||
+ | |||
+ | <math>\textbf{(A)}\ \pi + 2 \qquad | ||
+ | \textbf{(B)}\ \frac{2 \pi + 1}{2} \qquad | ||
+ | \textbf{(C)}\ \pi \qquad | ||
+ | \textbf{(D)}\ \frac{2\pi-1}{2}\qquad | ||
+ | \textbf{(E)}\ \pi-2 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 5|Solution]] | ||
+ | |||
+ | == Problem 6 == | ||
+ | |||
+ | A square and an equilateral triangle have equal perimeters. The area of the triangle is <math>9 \sqrt{3}</math> square inches. | ||
+ | Expressed in inches the diagonal of the square is: | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{9}{2} \qquad | ||
+ | \textbf{(B)}\ 2 \sqrt{5} \qquad | ||
+ | \textbf{(C)}\ 4 \sqrt{2} \qquad | ||
+ | \textbf{(D)}\ \frac{9\sqrt{2}}{2}\qquad | ||
+ | \textbf{(E)}\ \text{none of these} </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 6|Solution]] | ||
+ | |||
+ | == Problem 7 == | ||
+ | |||
+ | Let the bisectors of the exterior angles at <math>B</math> and <math>C</math> of <math>\triangle ABC</math> meet at <math>D</math>. Then, if all measurements are in degrees, <math>\angle BDC</math> equals: | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac {1}{2} (90 - A) \qquad | ||
+ | \textbf{(B)}\ 90 - A \qquad | ||
+ | \textbf{(C)}\ \frac {1}{2} (180 - A) \qquad | ||
+ | \textbf{(D)}\ 180-A\qquad | ||
+ | \textbf{(E)}\ 180-2A </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 7|Solution]] | ||
+ | |||
+ | == Problem 8 == | ||
+ | |||
+ | Given the set of <math>n</math> numbers; <math>n > 1</math>, of which one is <math>1 - \frac {1}{n}</math> and all the others are <math>1</math>. The arithmetic mean of the <math>n</math> numbers is: | ||
+ | |||
+ | <math>\textbf{(A)}\ 1 \qquad | ||
+ | \textbf{(B)}\ n - \frac {1}{n} \qquad | ||
+ | \textbf{(C)}\ n - \frac {1}{n^2} \qquad | ||
+ | \textbf{(D)}\ 1-\frac{1}{n^2}\qquad | ||
+ | \textbf{(E)}\ 1-\frac{1}{n}-\frac{1}{n^2} </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 8|Solution]] | ||
+ | |||
+ | == Problem 9 == | ||
+ | |||
+ | When <math>x^9-x</math> is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is: | ||
+ | |||
+ | <math>\textbf{(A)}\ \text{more than 5} \qquad | ||
+ | \textbf{(B)}\ 5 \qquad | ||
+ | \textbf{(C)}\ 4 \qquad | ||
+ | \textbf{(D)}\ 3 \qquad | ||
+ | \textbf{(E)}\ 2 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 9|Solution]] | ||
+ | |||
+ | == Problem 10 == | ||
+ | |||
+ | A man drives <math>150</math> miles to the seashore in <math>3</math> hours and <math>20</math> minutes. He returns from the shore to the starting point in <math>4</math> hours and <math>10</math> minutes. | ||
+ | Let <math>r</math> be the average rate for the entire trip. Then the average rate for the trip going exceeds <math>r</math> in miles per hour, by: | ||
+ | |||
+ | \textbf{(A)}\ 5 \qquad | ||
+ | \textbf{(B)}\ 4 \frac{1}{2} \qquad | ||
+ | \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 2 \qquad | ||
+ | \textbf{(E)}\ 1 | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 10|Solution]] | ||
+ | |||
+ | == Problem 11 == | ||
+ | |||
+ | The difference between the larger root and the smaller root of x^2 - px + (p^2 - 1)/4 = 0 is: | ||
+ | |||
+ | \textbf{(A)}\ 0 \qquad | ||
+ | \textbf{(B)}\ 1 \qquad | ||
+ | \textbf{(C)}\ 2 \qquad | ||
+ | \textbf{(D)}\ p \qquad | ||
+ | \textbf{(E)}\ p+1 | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 11|Solution]] | ||
+ | |||
+ | == Problem 12 == | ||
+ | |||
+ | When \left ( 1 - \frac{1}{a} \right ) ^6 is expanded the sum of the last three coefficients is: | ||
+ | |||
+ | \textbf{(A)}\ 22 \qquad | ||
+ | \textbf{(B)}\ 11 \qquad | ||
+ | \textbf{(C)}\ 10 \qquad | ||
+ | \textbf{(D)}\ -10 \qquad | ||
+ | \textbf{(E)}\ -11 | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 12|Solution]] | ||
+ | |||
+ | == Problem 13 == | ||
+ | |||
+ | R varies directly as S and inverse as T. When R = \frac43 and T = \frac {9}{14}, S = \frac37. Find S when R = \sqrt {48} and T = \sqrt {75}. | ||
+ | |||
+ | \textbf{(A)}\ 28 \qquad | ||
+ | \textbf{(B)}\ 30 \qquad | ||
+ | \textbf{(C)}\ 40 \qquad | ||
+ | \textbf{(D)}\ 42 \qquad | ||
+ | \textbf{(E)}\ 60 | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 13|Solution]] | ||
+ | |||
+ | == Problem 14 == | ||
+ | |||
+ | Let <math>s</math> be the limiting sum of the geometric series <math>4- \frac{8}{3} + \frac{16}{9} - \dots</math>, as the number of terms increases without bound. Then <math>s</math> equals: | ||
+ | |||
+ | <math>\textbf{(A)}\ \text{a number between 0 and 1} \qquad | ||
+ | \textbf{(B)}\ 2.4 \qquad | ||
+ | \textbf{(C)}\ 2.5 \qquad | ||
+ | \textbf{(D)}\ 3.6\qquad | ||
+ | \textbf{(E)}\ 12 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 14|Solution]] | ||
+ | |||
+ | == Problem 15 == | ||
+ | |||
+ | Given <math>\triangle ABC</math> with base <math>AB</math> fixed in length and position. As the vertex <math>C</math> moves on a straight line, | ||
+ | the intersection point of the three medians moves on: | ||
+ | |||
+ | <math>\textbf{(A)}\ \text{a circle} \qquad | ||
+ | \textbf{(B)}\ \text{a parabola} \qquad | ||
+ | \textbf{(C)}\ \text{an ellipse} \qquad | ||
+ | \textbf{(D)}\ \text{a straight line}\qquad | ||
+ | \textbf{(E)}\ \text{a curve here not listed} </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 15|Solution]] | ||
+ | |||
+ | == Problem 16 == | ||
+ | |||
+ | Given rectangle <math>R_1</math> with one side <math>2</math> inches and area <math>12</math> square inches. | ||
+ | Rectangle <math>R_2</math> with diagonal <math>15</math> inches is similar to <math>R_1</math>. Expressed in square inches the area of <math>R_2</math> is: | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{9}{2} \qquad | ||
+ | \textbf{(B)}\ 36 \qquad | ||
+ | \textbf{(C)}\ \frac{135}{2} \qquad | ||
+ | \textbf{(D)}\ 9\sqrt{10}\qquad | ||
+ | \textbf{(E)}\ \frac{27\sqrt{10}}{4} </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 16|Solution]] | ||
+ | |||
+ | == Problem 17 == | ||
+ | |||
+ | If <math>a = \log_8 225</math> and <math>b = \log_2 15</math>, then <math>a</math>, in terms of <math>b</math>, is: | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{b}{2} \qquad | ||
+ | \textbf{(B)}\ \frac{2b}{3}\qquad | ||
+ | \textbf{(C)}\ b \qquad | ||
+ | \textbf{(D)}\ \frac{3b}{2}\qquad | ||
+ | \textbf{(E)}\ 2b</math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 17|Solution]] | ||
+ | |||
+ | == Problem 18 == | ||
+ | |||
+ | A regular dodecagon (<math>12</math> sides) is inscribed in a circle with radius <math>r</math> inches. The area of the dodecagon, in square inches, is: | ||
+ | |||
+ | <math>\textbf{(A)}\ 3r^2 \qquad | ||
+ | \textbf{(B)}\ 2r^2 \qquad | ||
+ | \textbf{(C)}\ \frac{3r^2 \sqrt{3}}{4} \qquad | ||
+ | \textbf{(D)}\ r^2\sqrt{3}\qquad | ||
+ | \textbf{(E)}\ 3r^2\sqrt{3} </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 18|Solution]] | ||
+ | |||
+ | == Problem 19 == | ||
+ | |||
+ | If the parabola <math>y = ax^2 + bx + c</math> passes through the points <math>( - 1, 12), (0, 5)</math>, and <math>(2, - 3)</math>, the value of <math>a + b + c</math> is: | ||
+ | |||
+ | <math>\textbf{(A)}\ - 4 \qquad | ||
+ | \textbf{(B)}\ - 2 \qquad | ||
+ | \textbf{(C)}\ 0 \qquad | ||
+ | \textbf{(D)}\ 1 \qquad | ||
+ | \textbf{(E)}\ 2 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 19|Solution]] | ||
+ | |||
+ | == Problem 20 == | ||
+ | |||
+ | The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be: | ||
+ | |||
+ | <math>\textbf{(A)}\ 108 \qquad | ||
+ | \textbf{(B)}\ 90 \qquad | ||
+ | \textbf{(C)}\ 72 \qquad | ||
+ | \textbf{(D)}\ 54 \qquad | ||
+ | \textbf{(E)}\ 36 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 20|Solution]] | ||
+ | |||
+ | == Problem 21 == | ||
+ | |||
+ | It is given that one root of <math>2x^2 + rx + s = 0</math>, with <math>r</math> and <math>s</math> real numbers, is <math>3+2i (i = \sqrt{-1})</math>. The value of <math>s</math> is: | ||
+ | |||
+ | <math>\textbf{(A)}\ \text{undetermined} \qquad | ||
+ | \textbf{(B)}\ 5 \qquad | ||
+ | \textbf{(C)}\ 6 \qquad | ||
+ | \textbf{(D)}\ -13\qquad | ||
+ | \textbf{(E)}\ 26 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 21|Solution]] | ||
+ | |||
+ | == Problem 22 == | ||
+ | |||
+ | The number 121_b, written in the integral base b, is the square of an integer, for | ||
+ | |||
+ | <math>\textbf{(A)}\ b = 10,\text{ only} \qquad | ||
+ | \textbf{(B)}\ b = 10 \text{ and } b = 5, \text{ only} \qquad | ||
+ | \textbf{(C)}\ 2\leq b\leq 10\qquad | ||
+ | \textbf{(D)}\ b > 2\qquad | ||
+ | \textbf{(E)}\ \text{no value of }b </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 22|Solution]] | ||
+ | |||
+ | == Problem 23 == | ||
+ | |||
+ | In <math>\triangle ABC</math>, <math>CD</math> is the altitude to <math>AB</math> and <math>AE</math> is the altitude to <math>BC</math>. | ||
+ | If the lengths of <math>AB, CD</math>, and <math>AE</math> are known, the length of <math>DB</math> is: | ||
+ | |||
+ | <math>\textbf{(A)}\ \text{not determined by the information given} \qquad \\ | ||
+ | \textbf{(B)}\ \text{determined only if A is an acute angle} \qquad \\ | ||
+ | \textbf{(C)}\ \text{determined only if B is an acute angle} \qquad \\ | ||
+ | \textbf{(D)}\ \text{determined only in ABC is an acute triangle} \qquad \\ | ||
+ | \textbf{(E)}\ \text{none of these is correct} </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 23|Solution]] | ||
+ | |||
+ | == Problem 24 == | ||
+ | |||
+ | Three machines <math>\text{P, Q, and R,}</math> working together, can do a job in <math>x</math> hours. | ||
+ | When working alone, <math>\text{P}</math> needs an additional <math>6</math> hours to do the job; <math>\text{Q}</math>, one additional hour; and <math>R</math>, <math>x</math> additional hours. The value of <math>x</math> is: | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac23 \qquad | ||
+ | \textbf{(B)}\ \frac{11}{12} \qquad | ||
+ | \textbf{(C)}\ \frac32 \qquad | ||
+ | \textbf{(D)}\ 2\qquad | ||
+ | \textbf{(E)}\ 3 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 24|Solution]] | ||
+ | |||
+ | == Problem 25 == | ||
+ | |||
+ | Given square <math>ABCD</math> with side <math>8</math> feet. A circle is drawn through vertices <math>A</math> and <math>D</math> and tangent to side <math>BC</math>. The radius of the circle, in feet, is: | ||
+ | |||
+ | <math>\textbf{(A)}\ 4 \qquad | ||
+ | \textbf{(B)}\ 4 \sqrt{2} \qquad | ||
+ | \textbf{(C)}\ 5 \qquad | ||
+ | \textbf{(D)}\ 5 \sqrt{2} \qquad | ||
+ | \textbf{(E)}\ 6 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 25|Solution]] | ||
+ | |||
+ | == Problem 26 == | ||
+ | |||
+ | For any real value of <math>x</math> the maximum value of <math>8x - 3x^2</math> is: | ||
+ | |||
+ | <math>\textbf{(A)}\ 0 \qquad | ||
+ | \textbf{(B)}\ \frac83 \qquad | ||
+ | \textbf{(C)}\ 4 \qquad | ||
+ | \textbf{(D)}\ 5 \qquad | ||
+ | \textbf{(E)}\ \frac{16}{3} </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 26|Solution]] | ||
+ | |||
+ | == Problem 27 == | ||
+ | |||
+ | Let <math>a @ b</math> represent the operation on two numbers, <math>a</math> and <math>b</math>, which selects the larger of the two numbers, | ||
+ | with <math>a@a = a</math>. Let <math>a ! b</math> represent the operator which selects the smaller of the two numbers, with <math>a ! a = a</math>. | ||
+ | Which of the following three rules is (are) correct? | ||
+ | |||
+ | <math>\textbf{(1)}\ a@b = b@a \qquad \\ | ||
+ | \textbf{(2)}\ a@(b@c) = (a@b)@c \qquad \\ | ||
+ | \textbf{(3)}\ a ! (b@c) = (a ! b) @ (a ! c) </math> | ||
+ | |||
+ | |||
+ | <math>\textbf{(A)}\ (1)\text{ only} \qquad | ||
+ | \textbf{(B)}\ (2) \text{ only} \qquad | ||
+ | \textbf{(C)}\ \text{(1) and (2) only}\qquad | ||
+ | \textbf{(D)}\ \text{(1) and (3) only}\qquad | ||
+ | \textbf{(E)}\ \text{all three} </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 27|Solution]] | ||
+ | |||
+ | == Problem 28 == | ||
+ | |||
+ | The set of <math>x</math>-values satisfying the equation <math>x^{\log_{10} x} = \frac{x^3}{100}</math> consists of: | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{1}{10} \qquad | ||
+ | \textbf{(B)}\ \text{10, only} \qquad | ||
+ | \textbf{(C)}\ \text{100, only} \qquad | ||
+ | \textbf{(D)}\ \text{10 or 100, only}\qquad | ||
+ | \textbf{(E)}\ \text{more than two real numbers.} </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 28|Solution]] | ||
+ | |||
+ | == Problem 29 == | ||
+ | |||
+ | Which of the following sets of <math>x</math>-values satisfy the inequality <math>2x^2 + x < 6</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ - 2 < x < \frac{3}{2} \qquad | ||
+ | \textbf{(B)}\ x > \frac32 \text{ or }x < - 2 \qquad | ||
+ | \textbf{(C)}\ x <\frac{3}2\qquad\\ | ||
+ | \textbf{(D)}\ \frac{3}2 < x < 2\qquad | ||
+ | \textbf{(E)}\ x <-2 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 29|Solution]] | ||
+ | |||
+ | == Problem 30 == | ||
+ | |||
+ | Consider the statements: | ||
+ | |||
+ | <math> \textbf{(1)}\ \text{p and q are both true}\qquad\\ | ||
+ | \textbf{(2)}\ \text{p is true and q is false}\qquad\\ | ||
+ | \textbf{(3)}\ \text{p is false and q is true}\qquad\\ | ||
+ | \textbf{(4)}\ \text{p is false and q is false.} </math> | ||
+ | |||
+ | How many of these imply the negative of the statement "<math>p</math> and <math>q</math> are both true?" | ||
+ | |||
+ | <math>\textbf{(A)}\ 0 \qquad | ||
+ | \textbf{(B)}\ 1 \qquad | ||
+ | \textbf{(C)}\ 2 \qquad | ||
+ | \textbf{(D)}\ 3 \qquad | ||
+ | \textbf{(E)}\ 4 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 30|Solution]] | ||
+ | |||
+ | == Problem 31 == | ||
+ | |||
+ | The ratio of the interior angles of two regular polygons with sides of unit length is <math>3: 2</math>. How many such pairs are there? | ||
+ | |||
+ | <math>\textbf{(A)}\ 1 \qquad | ||
+ | \textbf{(B)}\ 2 \qquad | ||
+ | \textbf{(C)}\ 3 \qquad | ||
+ | \textbf{(D)}\ 4 \qquad | ||
+ | \textbf{(E)}\ \infty</math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 31|Solution]] | ||
+ | |||
+ | == Problem 32 == | ||
+ | |||
+ | If <math>x_{k+1} = x_k + \frac12</math> for <math>k=1, 2, \dots, n-1</math> and <math>x_1=1</math>, find <math>x_1 + x_2 + \dots + x_n</math>. | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{n+1}{2} \qquad | ||
+ | \textbf{(B)}\ \frac{n+3}{2} \qquad | ||
+ | \textbf{(C)}\ \frac{n^2-1}{2} \qquad | ||
+ | \textbf{(D)}\ \frac{n^2+n}{4}\qquad | ||
+ | \textbf{(E)}\ \frac{n^2+3n}{4} </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 32|Solution]] | ||
+ | |||
+ | == Problem 33 == | ||
+ | |||
+ | The set of <math>x</math>-values satisfying the inequality <math>2 \leq |x-1| \leq 5</math> is: | ||
+ | |||
+ | <math> \textbf{(A)}\ -4\leq x\leq-1\text{ or }3\leq x\leq 6\qquad | ||
+ | \textbf{(B)}\ 3\leq x\leq 6\text{ or }-6\leq x\leq-3\qquad\\ | ||
+ | \textbf{(C)}\ x\leq-1\text{ or }x\geq 3\qquad | ||
+ | \textbf{(D)}\ -1\leq x\leq 3\qquad | ||
+ | \textbf{(E)}\ -4\leq x\leq 6 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 33|Solution]] | ||
+ | |||
+ | == Problem 34 == | ||
+ | |||
+ | For what real values of <math>K</math> does <math>x = K^2 (x-1)(x-2)</math> have real roots? | ||
+ | |||
+ | <math> \textbf{(A)}\ \text{none}\qquad | ||
+ | \textbf{(B)}\ -2<K<1\qquad | ||
+ | \textbf{(C)}\ -2\sqrt{2}< K < 2\sqrt{2}\qquad\\ | ||
+ | \textbf{(D)}\ K>1\text{ or }K<-2\qquad | ||
+ | \textbf{(E)}\ \text{all} </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 34|Solution]] | ||
+ | |||
+ | == Problem 35 == | ||
+ | |||
+ | A man on his way to dinner short after 6: 00 p.m. observes that the hands of his watch form an angle of <math>110^{\circ}</math>. | ||
+ | Returning before 7: 00 p.m. he notices that again the hands of his watch form an angle of <math>110^{\circ}</math>. | ||
+ | The number of minutes that he has been away is: | ||
+ | |||
+ | <math>\textbf{(A)}\ 36 \frac23 \qquad | ||
+ | \textbf{(B)}\ 40 \qquad | ||
+ | \textbf{(C)}\ 42 \qquad | ||
+ | \textbf{(D)}\ 42.4 \qquad | ||
+ | \textbf{(E)}\ 45 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 35|Solution]] | ||
+ | |||
+ | == Problem 36 == | ||
+ | |||
+ | If both <math>x</math> and <math>y</math> are both integers, how many pairs of solutions are there of the equation <math>(x-8)(x-10) = 2^y</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ 0 \qquad | ||
+ | \textbf{(B)}\ 1 \qquad | ||
+ | \textbf{(C)}\ 2 \qquad | ||
+ | \textbf{(D)}\ 3 \qquad | ||
+ | \textbf{(E)}\ \text{more than 3} </math> | ||
+ | [[1962 AHSME Problems/Problem 36|Solution]] | ||
+ | == Problem 37 == | ||
+ | |||
+ | <math>ABCD</math> is a square with side of unit length. Points <math>E</math> and <math>F</math> are taken respectively on sides <math>AB</math> and <math>AD</math> | ||
+ | so that <math>AE = AF</math> and the quadrilateral <math>CDFE</math> has maximum area. In square units this maximum area is: | ||
+ | <math>\textbf{(A)}\ \frac12 \qquad | ||
+ | \textbf{(B)}\ \frac {9}{16} \qquad | ||
+ | \textbf{(C)}\ \frac{19}{32} \qquad | ||
+ | \textbf{(D)}\ \frac{5}{8}\qquad | ||
+ | \textbf{(E)}\ \frac{2}3 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 37|Solution]] | ||
+ | == Problem 38 == | ||
+ | |||
+ | The population of Nosuch Junction at one time was a perfect square. | ||
+ | Later, with an increase of <math>100</math>, the population was one more than a perfect square. | ||
+ | Now, with an additional increase of <math>100</math>, the population is again a perfect square. | ||
+ | The original population is a multiple of: | ||
+ | <math>\textbf{(A)}\ 3 \qquad | ||
+ | \textbf{(B)}\ 7 \qquad | ||
+ | \textbf{(C)}\ 9 \qquad | ||
+ | \textbf{(D)}\ 11 \qquad | ||
+ | \textbf{(E)}\ 17 </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 38|Solution]] | ||
+ | == Problem 39 == | ||
+ | |||
+ | Two medians of a triangle with unequal sides are <math>3</math> inches and <math>6</math> inches. | ||
+ | Its area is <math>3 \sqrt{15}</math> square inches. The length of the third median in inches, is: | ||
+ | <math>\textbf{(A)}\ 4 \qquad | ||
+ | \textbf{(B)}\ 3 \sqrt{3} \qquad | ||
+ | \textbf{(C)}\ 3 \sqrt{6} \qquad | ||
+ | \textbf{(D)}\ 6\sqrt{3}\qquad | ||
+ | \textbf{(E)}\ 6\sqrt{6} </math> | ||
+ | |||
+ | [[1962 AHSME Problems/Problem 39|Solution]] | ||
+ | == Problem 40 == | ||
+ | |||
+ | The limiting sum of the infinite series, <math>\frac{1}{10} + \frac{2}{10^2} + \frac{3}{10^3} + \dots</math> whose <math>n</math>th term is <math>\frac{n}{10^n}</math> is: | ||
+ | <math> \textbf{(A)}\ \frac{1}9\qquad | ||
+ | \textbf{(B)}\ \frac{10}{81}\qquad | ||
+ | \textbf{(C)}\ \frac{1}8\qquad | ||
+ | \textbf{(D)}\ \frac{17}{72}\qquad | ||
+ | \textbf{(E)}\ \text{larger than any finite quantity} </math> | ||
− | ==See | + | == See also == |
− | [[ | + | * [[AHSME]] |
+ | * [[AHSME Problems and Solutions]] | ||
+ | * [[AMC 12 Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 22:04, 3 October 2014
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 Problem 31
- 32 Problem 32
- 33 Problem 33
- 34 Problem 34
- 35 Problem 35
- 36 Problem 36
- 37 Problem 37
- 38 Problem 38
- 39 Problem 39
- 40 Problem 40
- 41 See also
Problem 1
The expression is equal to:
Problem 2
The expression $\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}$ (Error compiling LaTeX. Unknown error_msg) is equal to:
Problem 3
The first three terms of an arithmetic progression are , in the order shown. The value of
is:
Problem 4
If , then
equals:
Problem 5
If the radius of a circle is increased by unit, the ratio of the new circumference to the new diameter is:
Problem 6
A square and an equilateral triangle have equal perimeters. The area of the triangle is square inches.
Expressed in inches the diagonal of the square is:
Problem 7
Let the bisectors of the exterior angles at and
of
meet at
. Then, if all measurements are in degrees,
equals:
Problem 8
Given the set of numbers;
, of which one is
and all the others are
. The arithmetic mean of the
numbers is:
Problem 9
When is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is:
Problem 10
A man drives miles to the seashore in
hours and
minutes. He returns from the shore to the starting point in
hours and
minutes.
Let
be the average rate for the entire trip. Then the average rate for the trip going exceeds
in miles per hour, by:
\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 4 \frac{1}{2} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 1
Problem 11
The difference between the larger root and the smaller root of x^2 - px + (p^2 - 1)/4 = 0 is:
\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ p \qquad \textbf{(E)}\ p+1
Problem 12
When \left ( 1 - \frac{1}{a} \right ) ^6 is expanded the sum of the last three coefficients is:
\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ -10 \qquad \textbf{(E)}\ -11
Problem 13
R varies directly as S and inverse as T. When R = \frac43 and T = \frac {9}{14}, S = \frac37. Find S when R = \sqrt {48} and T = \sqrt {75}.
\textbf{(A)}\ 28 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 60
Problem 14
Let be the limiting sum of the geometric series
, as the number of terms increases without bound. Then
equals:
Problem 15
Given with base
fixed in length and position. As the vertex
moves on a straight line,
the intersection point of the three medians moves on:
Problem 16
Given rectangle with one side
inches and area
square inches.
Rectangle
with diagonal
inches is similar to
. Expressed in square inches the area of
is:
Problem 17
If and
, then
, in terms of
, is:
Problem 18
A regular dodecagon ( sides) is inscribed in a circle with radius
inches. The area of the dodecagon, in square inches, is:
Problem 19
If the parabola passes through the points
, and
, the value of
is:
Problem 20
The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be:
Problem 21
It is given that one root of , with
and
real numbers, is
. The value of
is:
Problem 22
The number 121_b, written in the integral base b, is the square of an integer, for
Problem 23
In ,
is the altitude to
and
is the altitude to
.
If the lengths of
, and
are known, the length of
is:
Problem 24
Three machines working together, can do a job in
hours.
When working alone,
needs an additional
hours to do the job;
, one additional hour; and
,
additional hours. The value of
is:
Problem 25
Given square with side
feet. A circle is drawn through vertices
and
and tangent to side
. The radius of the circle, in feet, is:
Problem 26
For any real value of the maximum value of
is:
Problem 27
Let represent the operation on two numbers,
and
, which selects the larger of the two numbers,
with
. Let
represent the operator which selects the smaller of the two numbers, with
.
Which of the following three rules is (are) correct?
Problem 28
The set of -values satisfying the equation
consists of:
Problem 29
Which of the following sets of -values satisfy the inequality
?
Problem 30
Consider the statements:
How many of these imply the negative of the statement " and
are both true?"
Problem 31
The ratio of the interior angles of two regular polygons with sides of unit length is . How many such pairs are there?
Problem 32
If for
and
, find
.
Problem 33
The set of -values satisfying the inequality
is:
Problem 34
For what real values of does
have real roots?
Problem 35
A man on his way to dinner short after 6: 00 p.m. observes that the hands of his watch form an angle of .
Returning before 7: 00 p.m. he notices that again the hands of his watch form an angle of
.
The number of minutes that he has been away is:
Problem 36
If both and
are both integers, how many pairs of solutions are there of the equation
?
Problem 37
is a square with side of unit length. Points
and
are taken respectively on sides
and
so that
and the quadrilateral
has maximum area. In square units this maximum area is:
Problem 38
The population of Nosuch Junction at one time was a perfect square.
Later, with an increase of , the population was one more than a perfect square.
Now, with an additional increase of
, the population is again a perfect square.
The original population is a multiple of:
Problem 39
Two medians of a triangle with unequal sides are inches and
inches.
Its area is
square inches. The length of the third median in inches, is:
Problem 40
The limiting sum of the infinite series, whose
th term is
is:
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.