Difference between revisions of "1958 AHSME Problems"
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+ | {{AHSC 50 Problems | ||
+ | |year=1958 | ||
+ | }} | ||
== Problem 1 == | == Problem 1 == | ||
The value of <math> [2 - 3(2 - 3)^{-1}]^{-1}</math> is: | The value of <math> [2 - 3(2 - 3)^{-1}]^{-1}</math> is: | ||
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== Problem 9 == | == Problem 9 == | ||
− | A value of <math> x</math> satisfying the equation <math> x^2 | + | A value of <math> x</math> satisfying the equation <math> x^2 + b^2 = (a - x)^2</math> is: |
− | <math> \textbf{(A)}\ \frac{b^2 | + | <math> \textbf{(A)}\ \frac{b^2 + a^2}{2a}\qquad |
− | \textbf{(B)}\ \frac{b^2 | + | \textbf{(B)}\ \frac{b^2 - a^2}{2a}\qquad |
− | \textbf{(C)}\ \frac{a^2 | + | \textbf{(C)}\ \frac{a^2 - b^2}{2a}\qquad |
− | \textbf{(D)}\ \frac{a | + | \textbf{(D)}\ \frac{a - b}{2}\qquad |
− | \textbf{(E)}\ \frac{a^2 | + | \textbf{(E)}\ \frac{a^2 - b^2}{2}</math> |
[[1958 AHSME Problems/Problem 9|Solution]] | [[1958 AHSME Problems/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
− | For what real values of <math> k</math>, other than <math> k | + | For what real values of <math> k</math>, other than <math> k = 0</math>, does the equation <math> x^2 + kx + k^2 = 0</math> have real roots? |
<math> \textbf{(A)}\ {k < 0}\qquad | <math> \textbf{(A)}\ {k < 0}\qquad | ||
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== Problem 11 == | == Problem 11 == | ||
− | The number of roots satisfying the equation <math> \sqrt{5 | + | The number of roots satisfying the equation <math> \sqrt{5 - x} = x\sqrt{5 - x}</math> is: |
<math> \textbf{(A)}\ \text{unlimited}\qquad | <math> \textbf{(A)}\ \text{unlimited}\qquad | ||
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== Problem 12 == | == Problem 12 == | ||
− | If <math> P | + | If <math> P = \frac{s}{(1 + k)^n}</math> then <math> n</math> equals: |
− | <math> \textbf{(A)}\ \frac{\log{\left(\frac{s}{P}\right)}}{\log{(1 | + | <math> \textbf{(A)}\ \frac{\log{\left(\frac{s}{P}\right)}}{\log{(1 + k)}}\qquad |
− | \textbf{(B)}\ \log{\left(\frac{s}{P(1 | + | \textbf{(B)}\ \log{\left(\frac{s}{P(1 + k)}\right)}\qquad |
− | \textbf{(C)}\ \log{\left(\frac{s | + | \textbf{(C)}\ \log{\left(\frac{s - P}{1 + k}\right)}\qquad \\ |
− | \textbf{(D)}\ \log{\left(\frac{s}{P}\right)} | + | \textbf{(D)}\ \log{\left(\frac{s}{P}\right)} + \log{(1 + k)}\qquad |
− | \textbf{(E)}\ \frac{\log{(s)}}{\log{(P(1 | + | \textbf{(E)}\ \frac{\log{(s)}}{\log{(P(1 + k))}}</math> |
[[1958 AHSME Problems/Problem 12|Solution]] | [[1958 AHSME Problems/Problem 12|Solution]] | ||
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At a dance party a group of boys and girls exchange dances as follows: one boy dances with <math> 5</math> girls, a second boy dances with <math> 6</math> girls, and so on, the last boy dancing with all the girls. If <math> b</math> represents the number of boys and <math> g</math> the number of girls, then: | At a dance party a group of boys and girls exchange dances as follows: one boy dances with <math> 5</math> girls, a second boy dances with <math> 6</math> girls, and so on, the last boy dancing with all the girls. If <math> b</math> represents the number of boys and <math> g</math> the number of girls, then: | ||
− | <math> \textbf{(A)}\ b | + | <math> \textbf{(A)}\ b = g\qquad |
− | \textbf{(B)}\ b | + | \textbf{(B)}\ b = \frac{g}{5}\qquad |
− | \textbf{(C)}\ b | + | \textbf{(C)}\ b = g - 4\qquad |
− | \textbf{(D)}\ b | + | \textbf{(D)}\ b = g - 5\qquad \\ |
− | \textbf{(E)}\ \text{It is impossible to determine a relation between }{b}\text{ and }{g}\text{ without knowing }{b | + | \textbf{(E)}\ \text{It is impossible to determine a relation between }{b}\text{ and }{g}\text{ without knowing }{b + g.}</math> |
[[1958 AHSME Problems/Problem 14|Solution]] | [[1958 AHSME Problems/Problem 14|Solution]] | ||
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== Problem 17 == | == Problem 17 == | ||
− | If <math> x</math> is positive and <math> \log{x} \ge \log{2} | + | If <math> x</math> is positive and <math> \log{x} \ge \log{2} + \frac{1}{2}\log{x}</math>, then: |
<math> \textbf{(A)}\ {x}\text{ has no minimum or maximum value}\qquad \\ | <math> \textbf{(A)}\ {x}\text{ has no minimum or maximum value}\qquad \\ | ||
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The area of a circle is doubled when its radius <math> r</math> is increased by <math> n</math>. Then <math> r</math> equals: | The area of a circle is doubled when its radius <math> r</math> is increased by <math> n</math>. Then <math> r</math> equals: | ||
− | <math> \textbf{(A)}\ n(\sqrt{2} | + | <math> \textbf{(A)}\ n(\sqrt{2} + 1)\qquad |
− | \textbf{(B)}\ n(\sqrt{2} | + | \textbf{(B)}\ n(\sqrt{2} - 1)\qquad |
\textbf{(C)}\ n\qquad | \textbf{(C)}\ n\qquad | ||
− | \textbf{(D)}\ n(2 | + | \textbf{(D)}\ n(2 - \sqrt{2})\qquad |
− | \textbf{(E)}\ \frac{n\pi}{\sqrt{2} | + | \textbf{(E)}\ \frac{n\pi}{\sqrt{2} + 1}</math> |
[[1958 AHSME Problems/Problem 18|Solution]] | [[1958 AHSME Problems/Problem 18|Solution]] | ||
== Problem 19 == | == Problem 19 == | ||
− | The sides of a right triangle are <math> a</math> and <math> b</math> and the hypotenuse is <math> c</math>. A perpendicular from the vertex divides <math> c</math> into segments <math> r</math> and <math> s</math>, adjacent respectively to <math> a</math> and <math> b</math>. If <math> a : b | + | The sides of a right triangle are <math> a</math> and <math> b</math> and the hypotenuse is <math> c</math>. A perpendicular from the vertex divides <math> c</math> into segments <math> r</math> and <math> s</math>, adjacent respectively to <math> a</math> and <math> b</math>. If <math> a : b = 1 : 3</math>, then the ratio of <math> r</math> to <math> s</math> is: |
<math> \textbf{(A)}\ 1 : 3\qquad | <math> \textbf{(A)}\ 1 : 3\qquad | ||
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== Problem 20 == | == Problem 20 == | ||
− | If <math> 4^x | + | If <math> 4^x - 4^{x - 1} = 24</math>, then <math> (2x)^x</math> equals: |
<math> \textbf{(A)}\ 5\sqrt{5}\qquad | <math> \textbf{(A)}\ 5\sqrt{5}\qquad | ||
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== Problem 21 == | == Problem 21 == | ||
In the accompanying figure <math> \overline{CE}</math> and <math> \overline{DE}</math> are equal chords of a circle with center <math> O</math>. Arc <math> AB</math> is a quarter-circle. Then the ratio of the area of triangle <math> CED</math> to the area of triangle <math> AOB</math> is: | In the accompanying figure <math> \overline{CE}</math> and <math> \overline{DE}</math> are equal chords of a circle with center <math> O</math>. Arc <math> AB</math> is a quarter-circle. Then the ratio of the area of triangle <math> CED</math> to the area of triangle <math> AOB</math> is: | ||
− | + | ||
+ | <asy> | ||
+ | draw(circle((0,0),10),black+linewidth(.75)); | ||
+ | draw((-10,0)--(0,0)--(10,0)--(0,10)--cycle,dot); | ||
+ | MP("O",(0,0),N);MP("C",(-10,0),W);MP("D",(10,0),E);;MP("E",(0,10),N); | ||
+ | draw((-sqrt(70),-sqrt(30))--(sqrt(30),-sqrt(70))--(0,0)--cycle,dot); | ||
+ | MP("A",(-sqrt(70),-sqrt(30)),SW);MP("B",(sqrt(30),-sqrt(70)),SE); | ||
+ | </asy> | ||
+ | |||
<math> \textbf{(A)}\ \sqrt {2} : 1\qquad \textbf{(B)}\ \sqrt {3} : 1\qquad \textbf{(C)}\ 4 : 1\qquad \textbf{(D)}\ 3 : 1\qquad \textbf{(E)}\ 2 : 1</math> | <math> \textbf{(A)}\ \sqrt {2} : 1\qquad \textbf{(B)}\ \sqrt {3} : 1\qquad \textbf{(C)}\ 4 : 1\qquad \textbf{(D)}\ 3 : 1\qquad \textbf{(E)}\ 2 : 1</math> | ||
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== Problem 22 == | == Problem 22 == | ||
− | A particle is placed on the parabola <math> y | + | A particle is placed on the parabola <math> y = x^2 - x - 6</math> at a point <math> P</math> whose <math> y</math>-coordinate is <math> 6</math>. It is allowed to roll along the parabola until it reaches the nearest point <math> Q</math> whose <math> y</math>-coordinate is <math> -6</math>. The horizontal distance traveled by the particle (the numerical value of the difference in the <math> x</math>-coordinates of <math> P</math> and <math> Q</math>) is: |
<math> \textbf{(A)}\ 5\qquad | <math> \textbf{(A)}\ 5\qquad | ||
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== Problem 23 == | == Problem 23 == | ||
− | If, in the expression <math> x^2 | + | If, in the expression <math> x^2 - 3</math>, <math> x</math> increases or decreases by a positive amount of <math> a</math>, the expression changes by an amount: |
− | <math> \textbf{(A)}\ {\pm 2ax | + | <math> \textbf{(A)}\ {\pm 2ax + a^2}\qquad |
\textbf{(B)}\ {2ax \pm a^2}\qquad | \textbf{(B)}\ {2ax \pm a^2}\qquad | ||
− | \textbf{(C)}\ {\pm a^2 | + | \textbf{(C)}\ {\pm a^2 - 3} \qquad |
− | \textbf{(D)}\ {(x | + | \textbf{(D)}\ {(x + a)^2 - 3}\qquad\\ |
− | \textbf{(E)}\ {(x | + | \textbf{(E)}\ {(x - a)^2 - 3}</math> |
[[1958 AHSME Problems/Problem 23|Solution]] | [[1958 AHSME Problems/Problem 23|Solution]] | ||
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== Problem 25 == | == Problem 25 == | ||
− | If <math> \log_{k}{x}\cdot \log_{5}{k} | + | If <math> \log_{k}{x}\cdot \log_{5}{k} = 3</math>, then <math> x</math> equals: |
<math> \textbf{(A)}\ k^6\qquad | <math> \textbf{(A)}\ k^6\qquad | ||
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A set of <math> n</math> numbers has the sum <math> s</math>. Each number of the set is increased by <math> 20</math>, then multiplied by <math> 5</math>, and then decreased by <math> 20</math>. The sum of the numbers in the new set thus obtained is: | A set of <math> n</math> numbers has the sum <math> s</math>. Each number of the set is increased by <math> 20</math>, then multiplied by <math> 5</math>, and then decreased by <math> 20</math>. The sum of the numbers in the new set thus obtained is: | ||
− | <math> \textbf{(A)}\ s | + | <math> \textbf{(A)}\ s + 20n\qquad |
− | \textbf{(B)}\ 5s | + | \textbf{(B)}\ 5s + 80n\qquad |
\textbf{(C)}\ s\qquad | \textbf{(C)}\ s\qquad | ||
\textbf{(D)}\ 5s\qquad | \textbf{(D)}\ 5s\qquad | ||
− | \textbf{(E)}\ 5s | + | \textbf{(E)}\ 5s + 4n</math> |
[[1958 AHSME Problems/Problem 26|Solution]] | [[1958 AHSME Problems/Problem 26|Solution]] | ||
== Problem 27 == | == Problem 27 == | ||
− | The points <math> (2, | + | The points <math> (2,-3)</math>, <math> (4,3)</math>, and <math> (5, k/2)</math> are on the same straight line. The value(s) of <math> k</math> is (are): |
<math> \textbf{(A)}\ 12\qquad | <math> \textbf{(A)}\ 12\qquad | ||
− | \textbf{(B)}\ | + | \textbf{(B)}\ -12\qquad |
\textbf{(C)}\ \pm 12\qquad | \textbf{(C)}\ \pm 12\qquad | ||
\textbf{(D)}\ {12}\text{ or }{6}\qquad | \textbf{(D)}\ {12}\text{ or }{6}\qquad | ||
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== Problem 29 == | == Problem 29 == | ||
In a general triangle <math> ADE</math> (as shown) lines <math> \overline{EB}</math> and <math> \overline{EC}</math> are drawn. Which of the following angle relations is true? | In a general triangle <math> ADE</math> (as shown) lines <math> \overline{EB}</math> and <math> \overline{EC}</math> are drawn. Which of the following angle relations is true? | ||
− | + | ||
− | <math> \textbf{(A)}\ x | + | <asy> |
− | \textbf{(D)}\ x | + | draw((-8,0)--(-2,0)--(4,0)--(10,0)--(0,10)--cycle,dot); |
+ | draw((-2,0)--(0,10),dot);draw((4,0)--(0,10),dot); | ||
+ | MP("A",(-8,0),S);MP("B",(-2,0),S);MP("C",(4,0),S);MP("D",(10,0),S);MP("E",(0,10),N); | ||
+ | MP("x",(-7.9,.4),E);MP("z",(-2,.4),W);MP("m",(-2,.4),E);MP("n",(4,.4),W);MP("c",(4,.4),E);MP("a",(9.9,.4),W); | ||
+ | MP("y",(-.2,8.8),SW);MP("w",(.1,8.8),S);MP("b",(.7,9),SE); | ||
+ | </asy> | ||
+ | |||
+ | <math> \textbf{(A)}\ x + z = a + b\qquad \textbf{(B)}\ y + z = a + b\qquad \textbf{(C)}\ m + x = w + n\qquad \\ | ||
+ | \textbf{(D)}\ x + z + n = w + c + m\qquad \textbf{(E)}\ x + y + n = a + b + m</math> | ||
[[1958 AHSME Problems/Problem 29|Solution]] | [[1958 AHSME Problems/Problem 29|Solution]] | ||
== Problem 30 == | == Problem 30 == | ||
− | If <math> xy | + | If <math> xy = b</math> and <math> \frac{1}{x^2} + \frac{1}{y^2} = a</math>, then <math> (x + y)^2</math> equals: |
− | <math> \textbf{(A)}\ (a | + | <math> \textbf{(A)}\ (a + 2b)^2\qquad |
− | \textbf{(B)}\ a^2 | + | \textbf{(B)}\ a^2 + b^2\qquad |
− | \textbf{(C)}\ b(ab | + | \textbf{(C)}\ b(ab + 2)\qquad |
− | \textbf{(D)}\ ab(b | + | \textbf{(D)}\ ab(b + 2)\qquad |
− | \textbf{(E)}\ \frac{1}{a} | + | \textbf{(E)}\ \frac{1}{a} + 2b</math> |
[[1958 AHSME Problems/Problem 30|Solution]] | [[1958 AHSME Problems/Problem 30|Solution]] | ||
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== Problem 32 == | == Problem 32 == | ||
− | With <math> | + | With \$ <math> 1000</math> a rancher is to buy steers at \$ <math>25</math> each and cows at \$ <math>26</math> each. If the number of steers <math> s</math> and the number of cows <math> c</math> are both positive integers, then: |
<math> \textbf{(A)}\ \text{this problem has no solution}\qquad\\ | <math> \textbf{(A)}\ \text{this problem has no solution}\qquad\\ | ||
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== Problem 33 == | == Problem 33 == | ||
− | For one root of <math> ax^2 | + | For one root of <math> ax^2 + bx + c = 0</math> to be double the other, the coefficients <math> a,\,b,\,c</math> must be related as follows: |
− | <math> \textbf{(A)}\ 4b^2 | + | <math> \textbf{(A)}\ 4b^2 = 9c\qquad |
− | \textbf{(B)}\ 2b^2 | + | \textbf{(B)}\ 2b^2 = 9ac\qquad |
− | \textbf{(C)}\ 2b^2 | + | \textbf{(C)}\ 2b^2 = 9a\qquad \\ |
− | \textbf{(D)}\ b^2 | + | \textbf{(D)}\ b^2 - 8ac = 0\qquad |
− | \textbf{(E)}\ 9b^2 | + | \textbf{(E)}\ 9b^2 = 2ac</math> |
[[1958 AHSME Problems/Problem 33|Solution]] | [[1958 AHSME Problems/Problem 33|Solution]] | ||
== Problem 34 == | == Problem 34 == | ||
− | The numerator of a fraction is <math> 6x | + | The numerator of a fraction is <math> 6x + 1</math>, then denominator is <math> 7 - 4x</math>, and <math> x</math> can have any value between <math> -2</math> and <math> 2</math>, both included. The values of <math> x</math> for which the numerator is greater than the denominator are: |
<math> \textbf{(A)}\ \frac{3}{5} < x \le 2\qquad | <math> \textbf{(A)}\ \frac{3}{5} < x \le 2\qquad | ||
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\textbf{(C)}\ 0 < x \le 2\qquad \\ | \textbf{(C)}\ 0 < x \le 2\qquad \\ | ||
\textbf{(D)}\ 0 \le x \le 2\qquad | \textbf{(D)}\ 0 \le x \le 2\qquad | ||
− | \textbf{(E)}\ | + | \textbf{(E)}\ -2 \le x \le 2</math> |
[[1958 AHSME Problems/Problem 34|Solution]] | [[1958 AHSME Problems/Problem 34|Solution]] | ||
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== Problem 37 == | == Problem 37 == | ||
− | The first term of an arithmetic series of consecutive integers is <math> k^2 | + | The first term of an arithmetic series of consecutive integers is <math> k^2 + 1</math>. The sum of <math> 2k + 1</math> terms of this series may be expressed as: |
− | <math> \textbf{(A)}\ k^3 | + | <math> \textbf{(A)}\ k^3 + (k + 1)^3\qquad |
− | \textbf{(B)}\ (k | + | \textbf{(B)}\ (k - 1)^3 + k^3\qquad |
− | \textbf{(C)}\ (k | + | \textbf{(C)}\ (k + 1)^3\qquad \\ |
− | \textbf{(D)}\ (k | + | \textbf{(D)}\ (k + 1)^2\qquad |
− | \textbf{(E)}\ (2k | + | \textbf{(E)}\ (2k + 1)(k + 1)^2</math> |
[[1958 AHSME Problems/Problem 37|Solution]] | [[1958 AHSME Problems/Problem 37|Solution]] | ||
== Problem 38 == | == Problem 38 == | ||
− | Let <math> r</math> be the distance from the origin to a point <math> P</math> with coordinates <math> x</math> and <math> y</math>. Designate the ratio <math> \frac{y}{r}</math> by <math> s</math> and the ratio <math> \frac{x}{r}</math> by <math> c</math>. Then the values of <math> s^2 | + | Let <math> r</math> be the distance from the origin to a point <math> P</math> with coordinates <math> x</math> and <math> y</math>. Designate the ratio <math> \frac{y}{r}</math> by <math> s</math> and the ratio <math> \frac{x}{r}</math> by <math> c</math>. Then the values of <math> s^2 - c^2</math> are limited to the numbers: |
− | <math> \textbf{(A)}\ \text{less than }{ | + | <math> \textbf{(A)}\ \text{less than }{-1}\text{ are greater than }{+1}\text{, both excluded}\qquad\\ |
− | \textbf{(B)}\ \text{less than }{ | + | \textbf{(B)}\ \text{less than }{-1}\text{ are greater than }{+1}\text{, both included}\qquad \\ |
− | \textbf{(C)}\ \text{between }{ | + | \textbf{(C)}\ \text{between }{-1}\text{ and }{+1}\text{, both excluded}\qquad \\ |
− | \textbf{(D)}\ \text{between }{ | + | \textbf{(D)}\ \text{between }{-1}\text{ and }{+1}\text{, both included}\qquad \\ |
− | \textbf{(E)}\ { | + | \textbf{(E)}\ {-1}\text{ and }{+1}\text{ only}</math> |
[[1958 AHSME Problems/Problem 38|Solution]] | [[1958 AHSME Problems/Problem 38|Solution]] | ||
== Problem 39 == | == Problem 39 == | ||
− | We may say concerning the solution of | + | We may say concerning the solution of <math>|x|^2 + |x| - 6 =0</math> that: |
− | |||
− | |||
− | that: | ||
<math> \textbf{(A)}\ \text{there is only one root}\qquad | <math> \textbf{(A)}\ \text{there is only one root}\qquad | ||
− | \textbf{(B)}\ \text{the sum of the roots is }{ | + | \textbf{(B)}\ \text{the sum of the roots is }{+1}\qquad |
\textbf{(C)}\ \text{the sum of the roots is }{0}\qquad \\ | \textbf{(C)}\ \text{the sum of the roots is }{0}\qquad \\ | ||
− | \textbf{(D)}\ \text{the product of the roots is }{ | + | \textbf{(D)}\ \text{the product of the roots is }{+4}\qquad |
− | \textbf{(E)}\ \text{the product of the roots is }{ | + | \textbf{(E)}\ \text{the product of the roots is }{-6}</math> |
[[1958 AHSME Problems/Problem 39|Solution]] | [[1958 AHSME Problems/Problem 39|Solution]] | ||
== Problem 40 == | == Problem 40 == | ||
− | Given <math> a_0 | + | Given <math> a_0 = 1</math>, <math> a_1 = 3</math>, and the general relation <math> a_n^2 - a_{n - 1}a_{n + 1} = (-1)^n</math> for <math> n \ge 1</math>. Then <math> a_3</math> equals: |
<math> \textbf{(A)}\ \frac{13}{27}\qquad | <math> \textbf{(A)}\ \frac{13}{27}\qquad | ||
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\textbf{(C)}\ 21\qquad | \textbf{(C)}\ 21\qquad | ||
\textbf{(D)}\ 10\qquad | \textbf{(D)}\ 10\qquad | ||
− | \textbf{(E)}\ | + | \textbf{(E)}\ -17</math> |
[[1958 AHSME Problems/Problem 40|Solution]] | [[1958 AHSME Problems/Problem 40|Solution]] | ||
== Problem 41 == | == Problem 41 == | ||
− | The roots of <math> Ax^2 | + | The roots of <math> Ax^2 + Bx + C = 0</math> are <math> r</math> and <math> s</math>. For the roots of |
− | + | <math> x^2+px +q =0</math> | |
− | + | ||
to be <math> r^2</math> and <math> s^2</math>, <math> p</math> must equal: | to be <math> r^2</math> and <math> s^2</math>, <math> p</math> must equal: | ||
− | <math> \textbf{(A)}\ \frac{B^2 | + | <math> \textbf{(A)}\ \frac{B^2 - 4AC}{A^2}\qquad |
− | \textbf{(B)}\ \frac{B^2 | + | \textbf{(B)}\ \frac{B^2 - 2AC}{A^2}\qquad |
− | \textbf{(C)}\ \frac{2AC | + | \textbf{(C)}\ \frac{2AC - B^2}{A^2}\qquad \\ |
− | \textbf{(D)}\ B^2 | + | \textbf{(D)}\ B^2 - 2C\qquad |
− | \textbf{(E)}\ 2C | + | \textbf{(E)}\ 2C - B^2</math> |
[[1958 AHSME Problems/Problem 41|Solution]] | [[1958 AHSME Problems/Problem 41|Solution]] | ||
== Problem 42 == | == Problem 42 == | ||
− | In a circle with center <math> O</math>, chord <math> \overline{AB}</math> equals chord <math> \overline{AC}</math>. Chord <math> \overline{AD}</math> cuts <math> \overline{BC}</math> in <math> E</math>. If <math> AC | + | In a circle with center <math> O</math>, chord <math> \overline{AB}</math> equals chord <math> \overline{AC}</math>. Chord <math> \overline{AD}</math> cuts <math> \overline{BC}</math> in <math> E</math>. If <math> AC = 12</math> and <math> AE = 8</math>, then <math> AD</math> equals: |
<math> \textbf{(A)}\ 27\qquad | <math> \textbf{(A)}\ 27\qquad | ||
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== Problem 45 == | == Problem 45 == | ||
− | A check is written for <math> x</math> dollars and <math> y</math> cents, <math> x</math> and <math> y</math> both two-digit numbers. In error it is cashed for <math> y</math> dollars and <math> x</math> cents, the incorrect amount exceeding the correct amount by <math> | + | A check is written for <math> x</math> dollars and <math> y</math> cents, <math> x</math> and <math> y</math> both two-digit numbers. In error it is cashed for <math> y</math> dollars and <math> x</math> cents, the incorrect amount exceeding the correct amount by <math> \$17.82</math>. Then: |
<math> \textbf{(A)}\ {x}\text{ cannot exceed }{70}\qquad \\ | <math> \textbf{(A)}\ {x}\text{ cannot exceed }{70}\qquad \\ | ||
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== Problem 46 == | == Problem 46 == | ||
− | For values of <math> x</math> less than <math> 1</math> but greater than <math> | + | For values of <math> x</math> less than <math> 1</math> but greater than <math> -4</math>, the expression |
− | <math>\frac{x^2 | + | <math>\frac{x^2 - 2x + 2}{2x - 2}</math> |
has: | has: | ||
<math> \textbf{(A)}\ \text{no maximum or minimum value}\qquad \\ | <math> \textbf{(A)}\ \text{no maximum or minimum value}\qquad \\ | ||
− | \textbf{(B)}\ \text{a minimum value of }{ | + | \textbf{(B)}\ \text{a minimum value of }{+1}\qquad \\ |
− | \textbf{(C)}\ \text{a maximum value of }{ | + | \textbf{(C)}\ \text{a maximum value of }{+1}\qquad \\ |
− | \textbf{(D)}\ \text{a minimum value of }{ | + | \textbf{(D)}\ \text{a minimum value of }{-1}\qquad \\ |
− | \textbf{(E)}\ \text{a maximum value of }{ | + | \textbf{(E)}\ \text{a maximum value of }{-1}</math> |
[[1958 AHSME Problems/Problem 46|Solution]] | [[1958 AHSME Problems/Problem 46|Solution]] | ||
== Problem 47 == | == Problem 47 == | ||
− | <math> ABCD</math> is a rectangle (see the accompanying diagram) with <math> P</math> any point on <math> \overline{AB}</math>. <math> \overline{PS} \perp \overline{BD}</math> and <math> \overline{PR} \perp \overline{AC}</math>. <math> \overline{AF} \perp \overline{BD}</math> and <math> \overline{PQ} \perp \overline{AF}</math>. Then <math> PR | + | <math> ABCD</math> is a rectangle (see the accompanying diagram) with <math> P</math> any point on <math> \overline{AB}</math>. <math> \overline{PS} \perp \overline{BD}</math> and <math> \overline{PR} \perp \overline{AC}</math>. <math> \overline{AF} \perp \overline{BD}</math> and <math> \overline{PQ} \perp \overline{AF}</math>. Then <math> PR + PS</math> is equal to: |
− | + | ||
− | <math> \textbf{(A)}\ PQ\qquad \textbf{(B)}\ AE\qquad \textbf{(C)}\ PT | + | <asy> |
+ | draw((-2,-1)--(-2,1)--(2,1)--(2,-1)--cycle,dot); | ||
+ | draw((-2,-1)--(2,1)--(2,-1)--(-2,1),dot); | ||
+ | draw((-2,1)--(-6/5,-3/5),black+linewidth(.75)); | ||
+ | draw((6/5,3/5)--(1,1)--(-3/2+1/10,-2/10),black+linewidth(.75)); | ||
+ | draw((1,1)--(1-3/5,1-6/5),black+linewidth(.75)); | ||
+ | MP("A",(-2,1),NW);MP("B",(2,1),NE);MP("C",(2,-1),SE);MP("D",(-2,-1),SW); | ||
+ | MP("Q",(-3/2+1/10,-2/10),W);MP("T",(-2/5,1/5),N);MP("P",(1,1),N); | ||
+ | MP("F",(-6/5,-3/5),SE);MP("E",(0,0),S);MP("S",(6/5,3/5),S);MP("R",(1-3/5,1-6/5),S); | ||
+ | </asy> | ||
+ | |||
+ | <math> \textbf{(A)}\ PQ\qquad | ||
+ | \textbf{(B)}\ AE\qquad | ||
+ | \textbf{(C)}\ PT + AT\qquad | ||
+ | \textbf{(D)}\ AF\qquad | ||
+ | \textbf{(E)}\ EF</math> | ||
[[1958 AHSME Problems/Problem 47|Solution]] | [[1958 AHSME Problems/Problem 47|Solution]] | ||
Line 517: | Line 548: | ||
== Problem 49 == | == Problem 49 == | ||
− | In the expansion of <math> (a | + | In the expansion of <math> (a + b)^n</math> there are <math> n + 1</math> dissimilar terms. The number of dissimilar terms in the expansion of <math> (a + b + c)^{10}</math> is: |
<math> \textbf{(A)}\ 11\qquad | <math> \textbf{(A)}\ 11\qquad | ||
Line 528: | Line 559: | ||
== Problem 50 == | == Problem 50 == | ||
− | In this diagram a scheme is indicated for associating all the points of segment <math> \overline{AB}</math> with those of segment <math> \overline{A'B'}</math>, and reciprocally. To described this association scheme analytically, let <math> x</math> be the distance from a point <math> P</math> on <math> \overline{AB}</math> to <math> D</math> and let <math> y</math> be the distance from the associated point <math> P'</math> of <math> \overline{A'B'}</math> to <math> D'</math>. Then for any pair of associated points, if <math> x | + | In this diagram a scheme is indicated for associating all the points of segment <math> \overline{AB}</math> with those of segment <math> \overline{A'B'}</math>, and reciprocally. To described this association scheme analytically, let <math> x</math> be the distance from a point <math> P</math> on <math> \overline{AB}</math> to <math> D</math> and let <math> y</math> be the distance from the associated point <math> P'</math> of <math> \overline{A'B'}</math> to <math> D'</math>. Then for any pair of associated points, if <math> x = a,\, x + y</math> equals: |
− | + | ||
− | <math> \textbf{(A)}\ 13a\qquad \textbf{(B)}\ 17a | + | <asy> |
+ | draw((0,-3)--(0,3),black+linewidth(.75)); | ||
+ | draw((1,-2.5)--(5,-2.5),black+linewidth(.75)); | ||
+ | draw((3,2.5)--(4,2.5),black+linewidth(.75)); | ||
+ | draw((1,-2.5)--(4,2.5),black+linewidth(.75)); | ||
+ | draw((5,-2.5)--(3,2.5),black+linewidth(.75)); | ||
+ | draw((2.6,-2.5)--(3.6,2.5),black+linewidth(.75)); | ||
+ | dot((0,2.5));dot((1,2.5));dot((2,2.5));dot((3,2.5));dot((4,2.5));dot((5,2.5)); | ||
+ | dot((0,-2.5));dot((1,-2.5));dot((2,-2.5));dot((3,-2.5));dot((4,-2.5));dot((5,-2.5)); | ||
+ | MP("D",(0,2.5),NW);MP("A",(3,2.5),N);MP("P",(3.5,2.5),N);MP("B",(4,2.5),N); | ||
+ | MP("D'",(0,-2.5),NW);MP("B'",(1,-2.5),NW);MP("P'",(2.25,-2.5),N);MP("A'",(5,-2.5),NE); | ||
+ | MP("0",(0,2.5),SE);MP("1",(1,2.5),SE);MP("2",(2,2.5),SE);MP("3",(3,2.5),SE);MP("4",(4,2.5),SE);MP("5",(5,2.5),SE); | ||
+ | MP("0",(0,-2.5),SE);MP("1",(1,-2.5),SE);MP("2",(2,-2.5),SE);MP("3",(3,-2.5),SE);MP("4",(4,-2.5),SE);MP("5",(5,-2.5),SE); | ||
+ | |||
+ | </asy> | ||
+ | |||
+ | <math> \textbf{(A)}\ 13a\qquad \textbf{(B)}\ 17a - 51\qquad \textbf{(C)}\ 17 - 3a\qquad \textbf{(D)}\ \frac {17 - 3a}{4}\qquad \textbf{(E)}\ 12a - 34</math> | ||
[[1958 AHSME Problems/Problem 50|Solution]] | [[1958 AHSME Problems/Problem 50|Solution]] | ||
== See also == | == See also == | ||
− | + | ||
− | * [[ | + | * [[AMC 12 Problems and Solutions]] |
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{AHSME 50p box|year=1958|before=[[1957 AHSME|1957 AHSC]]|after=[[1959 AHSME|1959 AHSC]]}} | ||
+ | |||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 14:21, 20 February 2020
1958 AHSC (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
| ||
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 |
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 Problem 41
- 42 Problem 42
- 43 Problem 43
- 44 Problem 44
- 45 Problem 45
- 46 Problem 46
- 47 Problem 47
- 48 Problem 48
- 49 Problem 49
- 50 Problem 50
- 51 See also
Problem 1
The value of is:
Problem 2
If , then
equals:
Problem 3
Of the following expressions the one equal to is:
Problem 4
In the expression each
is replaced by
. The resulting expression, evaluated for
, equals:
Problem 5
The expression equals:
Problem 6
The arithmetic mean between and
, when
, is:
Problem 7
A straight line joins the points and
. Its
-intercept is:
Problem 8
Which of these four numbers , is (are) rational:
Problem 9
A value of satisfying the equation
is:
Problem 10
For what real values of , other than
, does the equation
have real roots?
Problem 11
The number of roots satisfying the equation is:
Problem 12
If then
equals:
Problem 13
The sum of two numbers is ; their product is
. The sum of their reciprocals is:
Problem 14
At a dance party a group of boys and girls exchange dances as follows: one boy dances with girls, a second boy dances with
girls, and so on, the last boy dancing with all the girls. If
represents the number of boys and
the number of girls, then:
Problem 15
A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:
Problem 16
The area of a circle inscribed in a regular hexagon is . The area of hexagon is:
Problem 17
If is positive and
, then:
Problem 18
The area of a circle is doubled when its radius is increased by
. Then
equals:
Problem 19
The sides of a right triangle are and
and the hypotenuse is
. A perpendicular from the vertex divides
into segments
and
, adjacent respectively to
and
. If
, then the ratio of
to
is:
Problem 20
If , then
equals:
Problem 21
In the accompanying figure and
are equal chords of a circle with center
. Arc
is a quarter-circle. Then the ratio of the area of triangle
to the area of triangle
is:
Problem 22
A particle is placed on the parabola at a point
whose
-coordinate is
. It is allowed to roll along the parabola until it reaches the nearest point
whose
-coordinate is
. The horizontal distance traveled by the particle (the numerical value of the difference in the
-coordinates of
and
) is:
Problem 23
If, in the expression ,
increases or decreases by a positive amount of
, the expression changes by an amount:
Problem 24
A man travels feet due north at
minutes per mile. He returns due south to his starting point at
miles per minute. The average rate in miles per hour for the entire trip is:
Problem 25
If , then
equals:
Problem 26
A set of numbers has the sum
. Each number of the set is increased by
, then multiplied by
, and then decreased by
. The sum of the numbers in the new set thus obtained is:
Problem 27
The points ,
, and
are on the same straight line. The value(s) of
is (are):
Problem 28
A -quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This is done a third and a fourth time. The fractional part of the final mixture that is water is:
Problem 29
In a general triangle (as shown) lines
and
are drawn. Which of the following angle relations is true?
Problem 30
If and
, then
equals:
Problem 31
The altitude drawn to the base of an isosceles triangle is , and the perimeter
. The area of the triangle is:
Problem 32
With $ a rancher is to buy steers at $
each and cows at $
each. If the number of steers
and the number of cows
are both positive integers, then:
Problem 33
For one root of to be double the other, the coefficients
must be related as follows:
Problem 34
The numerator of a fraction is , then denominator is
, and
can have any value between
and
, both included. The values of
for which the numerator is greater than the denominator are:
Problem 35
A triangle is formed by joining three points whose coordinates are integers. If the -coordinate and the
-coordinate each have a value of
, then the area of the triangle, in square units:
Problem 36
The sides of a triangle are ,
, and
units. If an altitude is dropped upon the side of length
, the larger segment cut off on this side is:
Problem 37
The first term of an arithmetic series of consecutive integers is . The sum of
terms of this series may be expressed as:
Problem 38
Let be the distance from the origin to a point
with coordinates
and
. Designate the ratio
by
and the ratio
by
. Then the values of
are limited to the numbers:
Problem 39
We may say concerning the solution of that:
Problem 40
Given ,
, and the general relation
for
. Then
equals:
Problem 41
The roots of are
and
. For the roots of
to be and
,
must equal:
Problem 42
In a circle with center , chord
equals chord
. Chord
cuts
in
. If
and
, then
equals:
Problem 43
is the hypotenuse of a right triangle
. Median
has length
and median
has length
. The length of
is:
Problem 44
Given the true statements: (1) If is greater than
, then
is greater than
(2) If
is less than
, then
is greater than
. A valid conclusion is:
Problem 45
A check is written for dollars and
cents,
and
both two-digit numbers. In error it is cashed for
dollars and
cents, the incorrect amount exceeding the correct amount by
. Then:
Problem 46
For values of less than
but greater than
, the expression
has:
Problem 47
is a rectangle (see the accompanying diagram) with
any point on
.
and
.
and
. Then
is equal to:
Problem 48
Diameter of a circle with center
is
units.
is a point
units from
, and on
.
is a point
units from
, and on
.
is any point on the circle. Then the broken-line path from
to
to
:
Problem 49
In the expansion of there are
dissimilar terms. The number of dissimilar terms in the expansion of
is:
Problem 50
In this diagram a scheme is indicated for associating all the points of segment with those of segment
, and reciprocally. To described this association scheme analytically, let
be the distance from a point
on
to
and let
be the distance from the associated point
of
to
. Then for any pair of associated points, if
equals:
See also
1958 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1957 AHSC |
Followed by 1959 AHSC | |
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.