Difference between revisions of "1985 AHSME Problems"
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Which terms must be removed from the sum | Which terms must be removed from the sum | ||
− | < | + | <cmath>\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+\frac{1}{12}</cmath> |
− | if the sum of the remaining terms is to equal <math> 1 </math>? | + | if the sum of the remaining terms is to equal <math>1</math>? |
<math> \mathrm{(A)\ } \frac{1}{4}\text{ and }\frac{1}{8} \qquad \mathrm{(B) \ }\frac{1}{4}\text{ and }\frac{1}{12} \qquad \mathrm{(C) \ } \frac{1}{8}\text{ and }\frac{1}{12} \qquad \mathrm{(D) \ } \frac{1}{6}\text{ and }\frac{1}{10} \qquad \mathrm{(E) \ }\frac{1}{8}\text{ and }\frac{1}{10} </math> | <math> \mathrm{(A)\ } \frac{1}{4}\text{ and }\frac{1}{8} \qquad \mathrm{(B) \ }\frac{1}{4}\text{ and }\frac{1}{12} \qquad \mathrm{(C) \ } \frac{1}{8}\text{ and }\frac{1}{12} \qquad \mathrm{(D) \ } \frac{1}{6}\text{ and }\frac{1}{10} \qquad \mathrm{(E) \ }\frac{1}{8}\text{ and }\frac{1}{10} </math> | ||
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[[1985 AHSME Problems/Problem 5|Solution]] | [[1985 AHSME Problems/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
− | One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is <math> \frac{2}{3} </math> of the | + | One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is <math>\frac{2}{3}</math> of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is |
<math> \mathrm{(A)\ } \frac{1}{3} \qquad \mathrm{(B) \ }\frac{2}{5} \qquad \mathrm{(C) \ } \frac{1}{2} \qquad \mathrm{(D) \ } \frac{3}{5} \qquad \mathrm{(E) \ }\frac{2}{3} </math> | <math> \mathrm{(A)\ } \frac{1}{3} \qquad \mathrm{(B) \ }\frac{2}{5} \qquad \mathrm{(C) \ } \frac{1}{2} \qquad \mathrm{(D) \ } \frac{3}{5} \qquad \mathrm{(E) \ }\frac{2}{3} </math> | ||
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[[1985 AHSME Problems/Problem 6|Solution]] | [[1985 AHSME Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
− | In some computer languages (such as APL), when there are no parentheses in an algebraic expression, the operations are grouped from right to left. Thus, <math> a\times b-c </math> in such languages means the same as <math> a(b-c) </math> in ordinary algebraic notation. If <math> a\div b-c+d </math> is evaluated in such a language, the result in ordinary algebraic notation would be | + | In some computer languages (such as APL), when there are no parentheses in an algebraic expression, the operations are grouped from right to left. Thus, <math>a \times b - c</math> in such languages means the same as <math>a(b-c)</math> in ordinary algebraic notation. If <math>a\div b-c+d</math> is evaluated in such a language, the result in ordinary algebraic notation would be |
<math> \mathrm{(A)\ } \frac{a}{b}-c+d \qquad \mathrm{(B) \ }\frac{a}{b}-c-d \qquad \mathrm{(C) \ } \frac{d+c-b}{a} \qquad \mathrm{(D) \ } \frac{a}{b-c+d} \qquad \mathrm{(E) \ }\frac{a}{b-c-d} </math> | <math> \mathrm{(A)\ } \frac{a}{b}-c+d \qquad \mathrm{(B) \ }\frac{a}{b}-c-d \qquad \mathrm{(C) \ } \frac{d+c-b}{a} \qquad \mathrm{(D) \ } \frac{a}{b-c+d} \qquad \mathrm{(E) \ }\frac{a}{b-c-d} </math> | ||
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[[1985 AHSME Problems/Problem 7|Solution]] | [[1985 AHSME Problems/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
− | Let <math> a, a', b, | + | Let <math>a,a',b,b'</math> be real numbers with <math>a</math> and <math>a'</math> nonzero. The solution to <math>ax+b=0</math> is less than the solution to <math>a'x+b'=0</math> if and only if |
− | <math> \mathrm{(A)\ } a'b<ab' \qquad \mathrm{(B) \ }ab'<a'b \qquad \mathrm{(C) \ } ab<a'b' \qquad \mathrm{(D) \ } \frac{b}{a}<\frac{b'}{a'} \qquad | + | <math> \mathrm{(A)\ } a'b < ab' \qquad \mathrm{(B) \ }ab' < a'b \qquad \mathrm{(C) \ } ab < a'b' \qquad \mathrm{(D) \ } \frac{b}{a} < \frac{b'}{a'} \qquad \mathrm{(E) \ }\frac{b'}{a'} < \frac{b}{a} </math> |
− | |||
− | |||
[[1985 AHSME Problems/Problem 8|Solution]] | [[1985 AHSME Problems/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
− | The odd positive integers <math> 1, 3, 5, 7, \ | + | The odd positive integers <math>1, 3, 5, 7, \ldots</math>, are arranged into five columns continuing with the pattern shown on the right. Counting from the left, the column in which <math>1985</math> appears is the |
<asy> | <asy> | ||
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==Problem 10== | ==Problem 10== | ||
− | An arbitrary | + | An arbitrary circle can intersect the graph of <math>y = \sin x</math> in |
− | <math> \mathrm{(A)\ } \text{at most }2\text{ points} \qquad \mathrm{(B)\ }\text{at most }4\text{ points} \qquad \mathrm{(C) \ } \text{at most }6\text{ points} \qquad \mathrm{(D) \ } \text{at most }8\text{ points} | + | <math> \mathrm{(A)\ } \text{at most }2\text{ points} \qquad \mathrm{(B)\ }\text{at most }4\text{ points} \qquad \mathrm{(C) \ } \text{at most }6\text{ points} \qquad \mathrm{(D) \ } \text{at most }8\text{ points}</math> |
+ | <math>\mathrm{(E) \ }\text{more than }16\text{ points} </math> | ||
[[1985 AHSME Problems/Problem 10|Solution]] | [[1985 AHSME Problems/Problem 10|Solution]] | ||
==Problem 11== | ==Problem 11== | ||
− | How many distinguishable rearrangements of the letters in <math>CONTEST</math> have both the vowels first? (For instance, <math>OETCNST</math> is one such arrangement but <math>OTETSNC</math> is not.) | + | How many distinguishable rearrangements of the letters in <math>CONTEST</math> have both the vowels first? (For instance, <math>OETCNST</math> is one such arrangement, but <math>OTETSNC</math> is not.) |
<math> \mathrm{(A)\ } 60 \qquad \mathrm{(B) \ }120 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 720 \qquad \mathrm{(E) \ }2520 </math> | <math> \mathrm{(A)\ } 60 \qquad \mathrm{(B) \ }120 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 720 \qquad \mathrm{(E) \ }2520 </math> | ||
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[[1985 AHSME Problems/Problem 11|Solution]] | [[1985 AHSME Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
− | Let <math> p, q </math> and <math> r </math> be distinct | + | Let <math>p</math>, <math>q</math> and <math>r</math> be distinct prime numbers, where <math>1</math> is not considered a prime. Which of the following is the smallest positive perfect cube having <math>n = pq^2r^4</math> as a divisor? |
− | <math> \mathrm{(A)\ } p^8q^8r^8 \qquad \mathrm{(B) \ | + | <math> \mathrm{(A)\ } p^8q^8r^8 \qquad \mathrm{(B) }\left(pq^2r^2\right)^3 \qquad \mathrm{(C) } \left(p^2q^2r^2\right)^3 \qquad \mathrm{(D) } \left(pqr^2\right)^3 \qquad \mathrm{(E) \ }4p^3q^3r^3 </math> |
[[1985 AHSME Problems/Problem 12|Solution]] | [[1985 AHSME Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
− | Pegs are put in a board <math> 1 </math> unit apart both horizontally and vertically. A rubber band is stretched over <math> 4 </math> pegs as shown in the figure, forming a | + | Pegs are put in a board <math>1</math> unit apart both horizontally and vertically. A rubber band is stretched over <math>4</math> pegs as shown in the figure, forming a quadrilateral. Its area in square units is |
<asy> | <asy> | ||
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[[1985 AHSME Problems/Problem 13|Solution]] | [[1985 AHSME Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
− | Exactly three of the interior angles of a convex | + | Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon? |
<math> \mathrm{(A)\ } 4 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ }8 </math> | <math> \mathrm{(A)\ } 4 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ }8 </math> | ||
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[[1985 AHSME Problems/Problem 14|Solution]] | [[1985 AHSME Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
− | If <math> a </math> and <math> b </math> are positive numbers such that <math> a^b=b^a </math> and <math> b=9a </math>, then the value of <math> a </math> is | + | If <math>a</math> and <math>b</math> are positive numbers such that <math>a^b = b^a</math> and <math>b = 9a</math>, then the value of <math>a</math> is |
<math> \mathrm{(A)\ } 9 \qquad \mathrm{(B) \ }\frac{1}{9} \qquad \mathrm{(C) \ } \sqrt[9]{9} \qquad \mathrm{(D) \ } \sqrt[3]{9} \qquad \mathrm{(E) \ }\sqrt[4]{3} </math> | <math> \mathrm{(A)\ } 9 \qquad \mathrm{(B) \ }\frac{1}{9} \qquad \mathrm{(C) \ } \sqrt[9]{9} \qquad \mathrm{(D) \ } \sqrt[3]{9} \qquad \mathrm{(E) \ }\sqrt[4]{3} </math> | ||
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[[1985 AHSME Problems/Problem 15|Solution]] | [[1985 AHSME Problems/Problem 15|Solution]] | ||
==Problem 16== | ==Problem 16== | ||
− | If <math> A=20 | + | If <math>\usepackage{gensymb} A = 20 \degree</math> and <math>\usepackage{gensymb} B = 25 \degree</math>, then the value of <math>\left(1+\tan A\right)\left(1+\tan B\right)</math> is |
− | <math> \mathrm{(A)\ } \sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 1+\sqrt{2} \qquad \mathrm{(D) \ } 2(\tan A+\tan B) \qquad \mathrm{(E) \ }\text{none of these} </math> | + | <math> \mathrm{(A)\ } \sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 1+\sqrt{2} \qquad \mathrm{(D) \ } 2\left(\tan A+\tan B\right) \qquad \mathrm{(E) \ }\text{none of these} </math> |
[[1985 AHSME Problems/Problem 16|Solution]] | [[1985 AHSME Problems/Problem 16|Solution]] | ||
==Problem 17== | ==Problem 17== | ||
− | + | Diagonal <math>DB</math> of rectangle <math>ABCD</math> is divided into three segments of length <math>1</math> by parallel lines <math>L</math> and <math>L'</math> that pass through <math>A</math> and <math>C</math> and are perpendicular to <math>DB</math>. The area of <math>ABCD</math>, rounded to the one decimal place, is | |
<asy> | <asy> | ||
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[[1985 AHSME Problems/Problem 17|Solution]] | [[1985 AHSME Problems/Problem 17|Solution]] | ||
==Problem 18== | ==Problem 18== | ||
− | Six bags of marbles contain <math> 18, 19, 21, 23, 25 </math> and <math> 34 </math> marbles, respectively. One bag contains chipped marbles only. The other <math> 5 </math> bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, how many chipped marbles are there? | + | Six bags of marbles contain <math>18</math>, <math>19</math>, <math>21</math>, <math>23</math>, <math>25</math> and <math>34</math> marbles, respectively. One bag contains chipped marbles only. The other <math>5</math> bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, how many chipped marbles are there? |
<math> \mathrm{(A)\ } 18 \qquad \mathrm{(B) \ }19 \qquad \mathrm{(C) \ } 21 \qquad \mathrm{(D) \ } 23 \qquad \mathrm{(E) \ }25 </math> | <math> \mathrm{(A)\ } 18 \qquad \mathrm{(B) \ }19 \qquad \mathrm{(C) \ } 21 \qquad \mathrm{(D) \ } 23 \qquad \mathrm{(E) \ }25 </math> | ||
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==Problem 19== | ==Problem 19== | ||
− | Consider the graphs <math> y=Ax^2 </math> and <math> y^2+3=x^2+4y </math>, where <math> A </math> is a positive constant and <math> x </math> and <math> y </math> are real variables. In how many points do the two graphs intersect? | + | Consider the graphs of <math>y = Ax^2</math> and <math>y^2+3 = x^2+4y</math>, where <math>A</math> is a positive constant and <math>x</math> and <math>y</math> are real variables. In how many points do the two graphs intersect? |
<math> \mathrm{(A) \ }\text{exactly }4 \qquad \mathrm{(B) \ }\text{exactly }2 \qquad </math> | <math> \mathrm{(A) \ }\text{exactly }4 \qquad \mathrm{(B) \ }\text{exactly }2 \qquad </math> | ||
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[[1985 AHSME Problems/Problem 19|Solution]] | [[1985 AHSME Problems/Problem 19|Solution]] | ||
==Problem 20== | ==Problem 20== | ||
− | A wooden | + | A wooden cube with edge length <math>n</math> units (where <math>n</math> is an integer <math>>2</math>) is painted black all over. By slices parallel to its faces, the cube is cut into <math>n^3</math> smaller cubes each of unit edge length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is <math>n</math>? |
<math> \mathrm{(A)\ } 5 \qquad \mathrm{(B) \ }6 \qquad \mathrm{(C) \ } 7 \qquad \mathrm{(D) \ } 8 \qquad \mathrm{(E) \ }\text{none of these} </math> | <math> \mathrm{(A)\ } 5 \qquad \mathrm{(B) \ }6 \qquad \mathrm{(C) \ } 7 \qquad \mathrm{(D) \ } 8 \qquad \mathrm{(E) \ }\text{none of these} </math> | ||
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[[1985 AHSME Problems/Problem 20|Solution]] | [[1985 AHSME Problems/Problem 20|Solution]] | ||
==Problem 21== | ==Problem 21== | ||
− | How many integers <math> x </math> satisfy the equation < | + | How many integers <math>x</math> satisfy the equation <cmath>\left(x^2-x-1\right)^{x+2} = 1?</cmath> |
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ }\text{none of these} </math> | <math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ }\text{none of these} </math> | ||
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[[1985 AHSME Problems/Problem 21|Solution]] | [[1985 AHSME Problems/Problem 21|Solution]] | ||
==Problem 22== | ==Problem 22== | ||
− | In a circle with center <math> O </math>, <math> AD </math> is a | + | In a circle with center <math>O</math>, <math>AD</math> is a diameter, <math>ABC</math> is a chord, <math>BO = 5</math> and <math>\angle ABO = \ \stackrel{\frown}{CD} \ = 60^{\circ}</math>. Then the length of <math>BC</math> is |
<asy> | <asy> | ||
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[[1985 AHSME Problems/Problem 22|Solution]] | [[1985 AHSME Problems/Problem 22|Solution]] | ||
==Problem 23== | ==Problem 23== | ||
− | If < | + | If <cmath>x = \frac{-1+i\sqrt{3}}{2} \qquad\text{and}\qquad y = \frac{-1-i\sqrt{3}}{2},</cmath> where <math>i^2 = -1</math>, then which of the following is not correct? |
− | <math> \mathrm{(A)\ } x^5+y^5=-1 \qquad \mathrm{(B) \ }x^7+y^7=-1 \qquad \mathrm{(C) \ } x^9+y^9=-1 \qquad </math> | + | <math> \mathrm{(A)\ } x^5+y^5 = -1 \qquad \mathrm{(B) \ }x^7+y^7 = -1 \qquad \mathrm{(C) \ } x^9+y^9 = -1 \qquad </math> |
− | <math> \mathrm{(D) \ } x^{11}+y^{11}=-1 \qquad \mathrm{(E) \ }x^{13}+y^{13}=-1 </math> | + | <math> \mathrm{(D) \ } x^{11}+y^{11} = -1 \qquad \mathrm{(E) \ }x^{13}+y^{13} = -1 </math> |
[[1985 AHSME Problems/Problem 23|Solution]] | [[1985 AHSME Problems/Problem 23|Solution]] | ||
==Problem 24== | ==Problem 24== | ||
− | A non-zero | + | A non-zero digit is chosen in such a way that the probability of choosing digit <math>d</math> is <math>\log_{10}{(d+1)}-\log_{10}{d}</math>. The probability that the digit <math>2</math> is chosen is exactly <math>1/2</math> the probability that the digit chosen is in the set |
− | <math> \mathrm{(A)\ } \{2, 3\} \qquad \mathrm{(B) \ }\{3, 4\} \qquad \mathrm{(C) \ } \{4, 5, 6, 7, 8\} \qquad \mathrm{(D) \ } \{5, 6, 7, 8, 9\} \qquad \mathrm{(E) \ }\{4, 5, 6, 7, 8, 9\} </math> | + | <math> \mathrm{(A)\ } \{2,3\} \qquad \mathrm{(B) \ }\{3,4\} \qquad \mathrm{(C) \ } \{4,5,6,7,8\} \qquad \mathrm{(D) \ } \{5,6,7,8,9\} \qquad \mathrm{(E) \ }\{4,5,6,7,8,9\} </math> |
[[1985 AHSME Problems/Problem 24|Solution]] | [[1985 AHSME Problems/Problem 24|Solution]] | ||
==Problem 25== | ==Problem 25== | ||
− | The | + | The volume of a certain rectangular solid is <math>8</math> cm<sup>3</sup>, its total surface area is <math>32</math> cm<sup>2</sup>, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is |
<math> \mathrm{(A)\ } 28 \qquad \mathrm{(B) \ }32 \qquad \mathrm{(C) \ } 36 \qquad \mathrm{(D) \ } 40 \qquad \mathrm{(E) \ }44 </math> | <math> \mathrm{(A)\ } 28 \qquad \mathrm{(B) \ }32 \qquad \mathrm{(C) \ } 36 \qquad \mathrm{(D) \ } 40 \qquad \mathrm{(E) \ }44 </math> | ||
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[[1985 AHSME Problems/Problem 25|Solution]] | [[1985 AHSME Problems/Problem 25|Solution]] | ||
==Problem 26== | ==Problem 26== | ||
− | Find the least | + | Find the least positive integer <math>n</math> for which <math>\frac{n-13}{5n+6}</math> is a non-zero reducible fraction. |
<math> \mathrm{(A)\ } 45 \qquad \mathrm{(B) \ }68 \qquad \mathrm{(C) \ } 155 \qquad \mathrm{(D) \ } 226 \qquad \mathrm{(E) \ }\text{none of these} </math> | <math> \mathrm{(A)\ } 45 \qquad \mathrm{(B) \ }68 \qquad \mathrm{(C) \ } 155 \qquad \mathrm{(D) \ } 226 \qquad \mathrm{(E) \ }\text{none of these} </math> | ||
[[1985 AHSME Problems/Problem 26|Solution]] | [[1985 AHSME Problems/Problem 26|Solution]] | ||
+ | |||
==Problem 27== | ==Problem 27== | ||
− | Consider a sequence <math> x_1, x_2, x_3, \ | + | Consider a sequence <math>x_1,x_2,x_3,\dotsc</math> defined by: |
− | + | <cmath>\begin{align*}&x_1 = \sqrt[3]{3}, \ &x_2 = \left(\sqrt[3]{3}\right)^{\sqrt[3]{3}},\end{align*}</cmath> | |
− | < | ||
− | |||
− | |||
− | |||
and in general | and in general | ||
+ | <cmath>x_n = \left(x_{n-1}\right)^{\sqrt[3]{3}} \text{ for } n > 1.</cmath> | ||
− | + | What is the smallest value of <math>n</math> for which <math>x_n</math> is an integer? | |
− | |||
− | What is the smallest value of <math> n </math> for which <math> x_n </math> is an | ||
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 9 \qquad \mathrm{(E) \ }27 </math> | <math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 9 \qquad \mathrm{(E) \ }27 </math> | ||
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==Problem 28== | ==Problem 28== | ||
− | In <math> \triangle ABC </math>, we have <math> \angle C=3\angle A, a=27 </math> and <math> c=48 </math>. What is <math> b </math>? | + | In <math>\triangle ABC</math>, we have <math>\angle C = 3\angle A</math>, <math>a = 27</math> and <math>c = 48</math>. What is <math>b</math>? |
<asy> | <asy> | ||
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[[1985 AHSME Problems/Problem 28|Solution]] | [[1985 AHSME Problems/Problem 28|Solution]] | ||
+ | |||
==Problem 29== | ==Problem 29== | ||
− | In their base <math> 10 </math> representations, the integer <math> a </math> consists of a sequence of <math> 1985 </math> eights and the integer <math> b </math> consists of a sequence of <math> 1985 </math> fives. What is the sum of the digits of the base <math> 10 </math> representation of <math> 9ab </math>? | + | In their base <math>10</math> representations, the integer <math>a</math> consists of a sequence of <math>1985</math> eights and the integer <math>b</math> consists of a sequence of <math>1985</math> fives. What is the sum of the digits of the base <math>10</math> representation of the integer <math>9ab</math>? |
<math> \mathrm{(A)\ } 15880 \qquad \mathrm{(B) \ }17856 \qquad \mathrm{(C) \ } 17865 \qquad \mathrm{(D) \ } 17874 \qquad \mathrm{(E) \ }19851 </math> | <math> \mathrm{(A)\ } 15880 \qquad \mathrm{(B) \ }17856 \qquad \mathrm{(C) \ } 17865 \qquad \mathrm{(D) \ } 17874 \qquad \mathrm{(E) \ }19851 </math> | ||
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==Problem 30== | ==Problem 30== | ||
− | Let <math> \lfloor x \rfloor </math> be the greatest integer less than or equal to <math> x </math>. Then the number of real solutions to <math> 4x^2-40\lfloor x \rfloor +51=0 </math> is | + | Let <math>\left\lfloor x\right\rfloor</math> be the greatest integer less than or equal to <math>x</math>. Then the number of real solutions to <math>4x^2-40\left\lfloor x\right\rfloor+51 = 0</math> is |
<math> \mathrm{(A)\ } 0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ } 3 \qquad \mathrm{(E) \ }4 </math> | <math> \mathrm{(A)\ } 0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ } 3 \qquad \mathrm{(E) \ }4 </math> |
Latest revision as of 02:26, 20 March 2024
1985 AHSME (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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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 |
Contents
[hide]- 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 See also
Problem 1
If , then
Problem 2
In an arcade game, the "monster" is the shaded sector of a circle of radius cm, as shown in the figure. The missing piece (the mouth) has central angle . What is the perimeter of the monster in cm?
Problem 3
In right with legs and , arcs of circles are drawn, one with center and radius , the other with center and radius . They intersect the hypotenuse in and . Then has length
Problem 4
A large bag of coins contains pennies, dimes and quarters. There are twice as many dimes as pennies and three times as many quarters as dimes. An amount of money which could be in the bag is
Problem 5
Which terms must be removed from the sum
if the sum of the remaining terms is to equal ?
Problem 6
One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is
Problem 7
In some computer languages (such as APL), when there are no parentheses in an algebraic expression, the operations are grouped from right to left. Thus, in such languages means the same as in ordinary algebraic notation. If is evaluated in such a language, the result in ordinary algebraic notation would be
Problem 8
Let be real numbers with and nonzero. The solution to is less than the solution to if and only if
Problem 9
The odd positive integers , are arranged into five columns continuing with the pattern shown on the right. Counting from the left, the column in which appears is the
Problem 10
An arbitrary circle can intersect the graph of in
Problem 11
How many distinguishable rearrangements of the letters in have both the vowels first? (For instance, is one such arrangement, but is not.)
Problem 12
Let , and be distinct prime numbers, where is not considered a prime. Which of the following is the smallest positive perfect cube having as a divisor?
Problem 13
Pegs are put in a board unit apart both horizontally and vertically. A rubber band is stretched over pegs as shown in the figure, forming a quadrilateral. Its area in square units is
Problem 14
Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon?
Problem 15
If and are positive numbers such that and , then the value of is
Problem 16
If and , then the value of is
Problem 17
Diagonal of rectangle is divided into three segments of length by parallel lines and that pass through and and are perpendicular to . The area of , rounded to the one decimal place, is
Problem 18
Six bags of marbles contain , , , , and marbles, respectively. One bag contains chipped marbles only. The other bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, how many chipped marbles are there?
Problem 19
Consider the graphs of and , where is a positive constant and and are real variables. In how many points do the two graphs intersect?
Problem 20
A wooden cube with edge length units (where is an integer ) is painted black all over. By slices parallel to its faces, the cube is cut into smaller cubes each of unit edge length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is ?
Problem 21
How many integers satisfy the equation
Problem 22
In a circle with center , is a diameter, is a chord, and . Then the length of is
Problem 23
If where , then which of the following is not correct?
Problem 24
A non-zero digit is chosen in such a way that the probability of choosing digit is . The probability that the digit is chosen is exactly the probability that the digit chosen is in the set
Problem 25
The volume of a certain rectangular solid is cm3, its total surface area is cm2, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is
Problem 26
Find the least positive integer for which is a non-zero reducible fraction.
Problem 27
Consider a sequence defined by: and in general
What is the smallest value of for which is an integer?
Problem 28
In , we have , and . What is ?
Problem 29
In their base representations, the integer consists of a sequence of eights and the integer consists of a sequence of fives. What is the sum of the digits of the base representation of the integer ?
Problem 30
Let be the greatest integer less than or equal to . Then the number of real solutions to is
See also
1986 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1984 AHSME |
Followed by 1986 AHSME | |
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All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.