Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problems"
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If the width of a particular rectangle is doubled and the length is increased by 3, then the area is tripled. What is the length of the rectangle? | If the width of a particular rectangle is doubled and the length is increased by 3, then the area is tripled. What is the length of the rectangle? | ||
− | < | + | <cmath> \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ } 6 \qquad \mathrm{(E) \ } 9 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 1|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 1|Solution]] | ||
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Suppose the operation <math>\star</math> is defined by <math>a \star b = a+b+ab.</math> If <math>3\star x = 23,</math> then <math>x =</math> | Suppose the operation <math>\star</math> is defined by <math>a \star b = a+b+ab.</math> If <math>3\star x = 23,</math> then <math>x =</math> | ||
− | < | + | <cmath> \mathrm{(A) \ } 2 \qquad \mathrm{(B) \ }3\qquad \mathrm{(C) \ }4 \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ }6 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 2|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 2|Solution]] | ||
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<center>[[Image:Usc93.3.PNG]]</center> | <center>[[Image:Usc93.3.PNG]]</center> | ||
− | < | + | <cmath> \mathrm{(A) \ }\sqrt{3}-\frac{\pi}2 \qquad \mathrm{(B) \ } \frac 16 \qquad \mathrm{(C) \ }\frac 13 \qquad \mathrm{(D) \ } \frac{\sqrt{3}}2 - \frac{\pi}6 \qquad \mathrm{(E) \ } \frac{\pi}6 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 3|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 3|Solution]] | ||
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If <math>(1 + i)^{100}</math> is expanded and written in the form <math>a + bi</math> where <math>a</math> and <math>b</math> are real numbers, then <math>a =</math> | If <math>(1 + i)^{100}</math> is expanded and written in the form <math>a + bi</math> where <math>a</math> and <math>b</math> are real numbers, then <math>a =</math> | ||
− | < | + | <cmath> \mathrm{(A) \ } -2^{50} \qquad \mathrm{(B) \ } 20^{50} - \frac{100!}{50!50!} \qquad \mathrm{(C) \ } \frac{100!}{(25!)^2 50!} \qquad \mathrm{(D) \ } 100! \left(-\frac 1{50!50!} + \frac 1{25!75!}\right) \qquad \mathrm{(E) \ } 0</cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 4|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 4|Solution]] | ||
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Suppose that <math>f</math> is a function with the property that for all <math>x</math> and <math>y, f(x + y) = f(x) + f(y) + 1</math> and <math>f(1) = 2.</math> What is the value of <math>f(3)</math>? | Suppose that <math>f</math> is a function with the property that for all <math>x</math> and <math>y, f(x + y) = f(x) + f(y) + 1</math> and <math>f(1) = 2.</math> What is the value of <math>f(3)</math>? | ||
− | < | + | <cmath> \mathrm{(A) \ }4 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ }6 \qquad \mathrm{(D) \ }7 \qquad \mathrm{(E) \ }8 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 5|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
− | After a <math>p%</math> price reduction, what increase does it take to restore the original price? | + | After a <math>p \%</math> price reduction, what increase does it take to restore the original price? |
− | < | + | <cmath> \mathrm{(A) \ }p\% \qquad \mathrm{(B) \ }\frac p{1-p}\% \qquad \mathrm{(C) \ } (100-p)\% \qquad \mathrm{(D) \ } \frac{100p}{100+p}\% \qquad \mathrm{(E) \ } \frac{100p}{100-p}\% </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 6|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 6|Solution]] | ||
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<center>[[Image:Usc93.7.PNG]]</center> | <center>[[Image:Usc93.7.PNG]]</center> | ||
− | < | + | <cmath> \mathrm{(A) \ }4.2 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ }5.6 \qquad \mathrm{(D) \ }6.2 \qquad \mathrm{(E) \ }6.8 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 7|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 7|Solution]] | ||
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What is the coefficient of <math>x^3</math> in the expansion of | What is the coefficient of <math>x^3</math> in the expansion of | ||
− | < | + | <cmath>(1 + x + x^2 + x^3 + x^4 + x^5 )^6? </cmath> |
− | < | + | <cmath> \mathrm{(A) \ } 40 \qquad \mathrm{(B) \ }48 \qquad \mathrm{(C) \ }56 \qquad \mathrm{(D) \ }62 \qquad \mathrm{(E) \ } 64 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 8|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 8|Solution]] | ||
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Suppose that <math>x</math> and <math>y</math> are integers such that <math>y > x > 1</math> and <math>y^2 - x^2 = 187</math>. Then one possible value of <math>xy</math> is | Suppose that <math>x</math> and <math>y</math> are integers such that <math>y > x > 1</math> and <math>y^2 - x^2 = 187</math>. Then one possible value of <math>xy</math> is | ||
− | < | + | <cmath> \mathrm{(A) \ }30 \qquad \mathrm{(B) \ }36 \qquad \mathrm{(C) \ }40 \qquad \mathrm{(D) \ }42 \qquad \mathrm{(E) \ }54 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 9|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
− | <math>\ | + | <math>\arcsin(1/3) + \arccos(1/3) + \arctan(1/3) + arccot(1/3) =</math> |
− | < | + | <cmath> \mathrm{(A) \ }\pi \qquad \mathrm{(B) \ }\pi/2 \qquad \mathrm{(C) \ }\pi/3 \qquad \mathrm{(D) \ }2\pi/3 \qquad \mathrm{(E) \ }3/\pi/4 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 10|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 10|Solution]] | ||
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Suppose that 4 cards labeled 1 to 4 are placed randomly into 4 boxes also labeled 1 to 4, one card per box. What is the probability that no card gets placed into a box having the same label as the card? | Suppose that 4 cards labeled 1 to 4 are placed randomly into 4 boxes also labeled 1 to 4, one card per box. What is the probability that no card gets placed into a box having the same label as the card? | ||
− | < | + | <cmath> \mathrm{(A) \ } 1/3 \qquad \mathrm{(B) \ }3/8 \qquad \mathrm{(C) \ }5/12 \qquad \mathrm{(D) \ } 1/2 \qquad \mathrm{(E) \ }9/16 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 11|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 11|Solution]] | ||
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If the equations <math> (1) x^2 + ax + b = 0</math> and <math> (2) x^2 + cx + d = 0 </math> have exactly one root in common, and <math> abcd\ne 0,</math> then the other root of equation <math> (2) </math> is | If the equations <math> (1) x^2 + ax + b = 0</math> and <math> (2) x^2 + cx + d = 0 </math> have exactly one root in common, and <math> abcd\ne 0,</math> then the other root of equation <math> (2) </math> is | ||
− | < | + | <cmath> \mathrm{(A) \ }\frac{c-a}{b-d}d \qquad \mathrm{(B) \ }\frac{a+c}{b+d}d \qquad \mathrm{(C) \ }\frac{b+c}{a+d}c \qquad \mathrm{(D) \ }\frac{a-c}{b-d} \qquad \mathrm{(E) \ }\frac{a+c}{b-d}c </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 12|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 12|Solution]] | ||
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Suppose that <math>x</math> and <math>y</math> are numbers such that <math>\sin(x+y) = 0.3</math> and <math>\sin(x-y) = 0.5</math>. Then <math> \sin (x)\cdot \cos (y) = </math> | Suppose that <math>x</math> and <math>y</math> are numbers such that <math>\sin(x+y) = 0.3</math> and <math>\sin(x-y) = 0.5</math>. Then <math> \sin (x)\cdot \cos (y) = </math> | ||
− | < | + | <cmath> \mathrm{(A) \ }0.1 \qquad \mathrm{(B) \ }0.3 \qquad \mathrm{(C) \ }0.4 \qquad \mathrm{(D) \ }0.5 \qquad \mathrm{(E) \ }0.6 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 13|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 13|Solution]] | ||
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For example, 8 1 5 7 2 3 9 4 6 would be such a permutation. | For example, 8 1 5 7 2 3 9 4 6 would be such a permutation. | ||
− | < | + | <cmath> \mathrm{(A) \ }9\cdot 7! \qquad \mathrm{(B) \ } 8! \qquad \mathrm{(C) \ }5!4! \qquad \mathrm{(D) \ }8!4! \qquad \mathrm{(E) \ }8!+6!+4! </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 14|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 14|Solution]] | ||
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If we express the sum | If we express the sum | ||
− | < | + | <cmath> \frac 1{3\cdot 5\cdot 7\cdot 11} + \frac 1{3\cdot 5\cdot 7\cdot 13} + \frac 1{3\cdot 5\cdot 11\cdot 13} + \frac 1{3\cdot 7\cdot 11\cdot 13} + \frac 1{5\cdot 7\cdot 11\cdot 13} </cmath> |
as a rational number in reduced form, then the denominator will be | as a rational number in reduced form, then the denominator will be | ||
− | < | + | <cmath> \mathrm{(A) \ }15015 \qquad \mathrm{(B) \ }5005 \qquad \mathrm{(C) \ }455 \qquad \mathrm{(D) \ }385 \qquad \mathrm{(E) \ }91 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 15|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 15|Solution]] | ||
== Problem 16 == | == Problem 16 == | ||
− | In the triangle below, <math> | + | In the triangle below, <math>M, N, </math> and <math>P</math> are the midpoints of <math>BC, AB,</math> and <math>AC</math> respectively. <math>CN</math> and <math>AM</math> intersect at <math>O</math>. If the length of <math>CQ</math> is 4, then what is the length of <math>OQ</math>? |
<center>[[Image:Usc93.16.PNG]]</center> | <center>[[Image:Usc93.16.PNG]]</center> | ||
− | < | + | <cmath> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }4/3 \qquad \mathrm{(C) \ }\sqrt{2} \qquad \mathrm{(D) \ }3/2 \qquad \mathrm{(E) \ }2 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 16|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 16|Solution]] | ||
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Let <math>[x]</math> represent the greatest integer that is less than or equal to <math>x</math>. For example, <math>[2.769]=2</math> and <math>[\pi]=3</math>. Then what is the value of | Let <math>[x]</math> represent the greatest integer that is less than or equal to <math>x</math>. For example, <math>[2.769]=2</math> and <math>[\pi]=3</math>. Then what is the value of | ||
− | < | + | <cmath> [\log_2 2] + [\log_2 3] + [\log_2 4] + \cdots + [\log_2 99] + [\log_2 100] ?</cmath> |
− | < | + | <cmath> \mathrm{(A) \ } 480 \qquad \mathrm{(B) \ }481 \qquad \mathrm{(C) \ }482 \qquad \mathrm{(D) \ }483 \qquad \mathrm{(E) \ }484 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 17|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 17|Solution]] | ||
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The minimum value of the function | The minimum value of the function | ||
− | < | + | <cmath>f(x) = \frac{\sin (x)}{\sqrt{1 - \cos^2 (x)}} + \frac{\cos(x)}{\sqrt{1 - \sin^2 (x) }} + \frac{\tan(x)}{\sqrt{\sec^2 (x) - 1}} + \frac{\cot (x)}{\sqrt{\csc^2 (x) - 1}}</cmath> |
as <math>x</math> varies over all numbers in the largest possible domain of <math>f</math>, is | as <math>x</math> varies over all numbers in the largest possible domain of <math>f</math>, is | ||
− | < | + | <cmath> \mathrm{(A) \ }-4 \qquad \mathrm{(B) \ }-2 \qquad \mathrm{(C) \ }0 \qquad \mathrm{(D) \ }2 \qquad \mathrm{(E) \ }4 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 18|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 18|Solution]] | ||
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<center>[[Image:Usc93.19.PNG]]</center> | <center>[[Image:Usc93.19.PNG]]</center> | ||
− | < | + | <cmath> \mathrm{(A) \ }24 \qquad \mathrm{(B) \ }72 \qquad \mathrm{(C) \ }84 \qquad \mathrm{(D) \ }96 \qquad \mathrm{(E) \ }108 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 19|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 19|Solution]] | ||
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Let <math>A_1, A_2, \ldots , A_{63}</math> be the 63 nonempty subsets of <math>\{ 1,2,3,4,5,6 \}</math>. For each of these sets <math>A_i</math>, let <math>\pi(A_i)</math> denote the product of all the elements in <math>A_i</math>. Then what is the value of <math>\pi(A_1)+\pi(A_2)+\cdots+\pi(A_{63})</math>? | Let <math>A_1, A_2, \ldots , A_{63}</math> be the 63 nonempty subsets of <math>\{ 1,2,3,4,5,6 \}</math>. For each of these sets <math>A_i</math>, let <math>\pi(A_i)</math> denote the product of all the elements in <math>A_i</math>. Then what is the value of <math>\pi(A_1)+\pi(A_2)+\cdots+\pi(A_{63})</math>? | ||
− | < | + | <cmath> \mathrm{(A) \ }5003 \qquad \mathrm{(B) \ }5012 \qquad \mathrm{(C) \ }5039 \qquad \mathrm{(D) \ }5057 \qquad \mathrm{(E) \ }5093 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 20|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 20|Solution]] | ||
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Suppose that each pair of eight tennis players either played exactly one game last week or did not play at all. Each player participated in all but 12 games. How many games were played among the eight players? | Suppose that each pair of eight tennis players either played exactly one game last week or did not play at all. Each player participated in all but 12 games. How many games were played among the eight players? | ||
− | < | + | <cmath> \mathrm{(A) \ }10 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ }14 \qquad \mathrm{(D) \ }16 \qquad \mathrm{(E) \ }18 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 21|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 21|Solution]] | ||
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Let | Let | ||
− | < | + | <cmath> A = \left( 1 + \frac 12 + \frac 14 + \frac 18 + \frac 1{16} \right) \left( 1 + \frac 13 + \frac 19\right) \left( 1 + \frac 15\right) \left( 1 + \frac 17\right) \left( 1 + \frac 1{11} \right) \left( 1 + \frac 1{13}\right), </cmath> |
− | < | + | <cmath>B = \left( 1 - \frac 12\right)^{-1} \left( 1 - \frac 13 \right)^{-1} \left(1 - \frac 15\right)^{-1} \left(1 - \frac 17\right)^{-1} \left(1-\frac 1{11}\right)^{-1} \left(1 - \frac 1{13}\right)^{-1}, </cmath> |
and | and | ||
− | < | + | <cmath> C = 1 + \frac 12 + \frac 13 + \frac 14 + \frac 15 + \frac 16 + \frac 17 + \frac 18 + \frac 19 + \frac 1{10} + \frac 1{11} + \frac 1{12} + \frac 1{13} + \frac 1{14} + \frac 1{15} +\frac 1{16}. </cmath> |
Then which of the following inequalities is true? | Then which of the following inequalities is true? | ||
− | < | + | <cmath> \mathrm{(A) \ } A > B > C \qquad \mathrm{(B) \ } B > A > C \qquad \mathrm{(C) \ } C > B > A \qquad \mathrm{(D) \ } C > A > B \qquad \mathrm{(E) \ } B > C > A </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 22|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 22|Solution]] | ||
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The relation between the sets | The relation between the sets | ||
− | < | + | <cmath> M = \{ 12 m + 8 n + 4 l: m,n,l \rm{ \ are \ } \rm{integers}\} </cmath> |
and | and | ||
− | < | + | <cmath> N= \{ 20 p + 16q + 12r: p,q,r \rm{ \ are \ } \rm{integers}\} </cmath> |
is | is | ||
− | < | + | <cmath> \mathrm{(A) \ } M\subset N \qquad \mathrm{(B) \ } N\subset M \qquad \mathrm{(C) \ } M\cup N = \{0\} \qquad \mathrm{(D) \ }60244 \rm{ \ is \ } \rm{in \ } M \rm{ \ but \ } \rm{not \ } \rm{in \ } N \qquad \mathrm{(E) \ } M=N </cmath> |
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If <math>f(x) = \frac{1 + x}{1 - 3x}, f_1(x) = f(f(x)), f_2(x) = f(f_1(x)),</math> and in general <math>f_n(x) = f(f_{n-1}(x)),</math> then <math>f_{1993}(3)=</math> | If <math>f(x) = \frac{1 + x}{1 - 3x}, f_1(x) = f(f(x)), f_2(x) = f(f_1(x)),</math> and in general <math>f_n(x) = f(f_{n-1}(x)),</math> then <math>f_{1993}(3)=</math> | ||
− | < | + | <cmath> \mathrm{(A) \ }3 \qquad \mathrm{(B) \ }1993 \qquad \mathrm{(C) \ }\frac 12 \qquad \mathrm{(D) \ }\frac 15 \qquad \mathrm{(E) \ } -2^{-1993} </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 24|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 24|Solution]] | ||
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What is the center of the circle passing through the point <math>(6,0)</math> and tangent to the circle <math>x^2 + y^2 = 4</math> at <math>(0,2)</math>? (Two circles are tangent at a point <math>P</math> if they intersect at <math>P</math> and at no other point.) | What is the center of the circle passing through the point <math>(6,0)</math> and tangent to the circle <math>x^2 + y^2 = 4</math> at <math>(0,2)</math>? (Two circles are tangent at a point <math>P</math> if they intersect at <math>P</math> and at no other point.) | ||
− | < | + | <cmath> \mathrm{(A) \ }(0,-6) \qquad \mathrm{(B) \ } (1,-9) \qquad \mathrm{(C) \ } (-1,-9) \qquad \mathrm{(D) \ } (0,-9) \qquad \mathrm{(E) \ } \rm{none \ } \rm{of \ } \rm{these} </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 25|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 25|Solution]] | ||
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Let <math>n=1667</math>. Then the first nonzero digit in the decimal expansion of <math>\sqrt{n^2 + 1} - n</math> is | Let <math>n=1667</math>. Then the first nonzero digit in the decimal expansion of <math>\sqrt{n^2 + 1} - n</math> is | ||
− | < | + | <cmath> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }3 \qquad \mathrm{(D) \ }4 \qquad \mathrm{(E) \ }5 </cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 26|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 26|Solution]] | ||
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Suppose <math>\triangle ABC</math> is a triangle with area 24 and that there is a point <math>P</math> inside <math>\triangle ABC</math> which is distance 2 from each of the sides of <math>\triangle ABC</math>. What is the perimeter of <math>\triangle ABC</math>? | Suppose <math>\triangle ABC</math> is a triangle with area 24 and that there is a point <math>P</math> inside <math>\triangle ABC</math> which is distance 2 from each of the sides of <math>\triangle ABC</math>. What is the perimeter of <math>\triangle ABC</math>? | ||
− | < | + | <cmath> \mathrm{(A) \ } 12 \qquad \mathrm{(B) \ }24 \qquad \mathrm{(C) \ }36 \qquad \mathrm{(D) \ }12\sqrt{2} \qquad \mathrm{(E) \ }12\sqrt{3} </cmath> |
− | \mathrm{(A) \ } 12 \qquad \mathrm{(B) \ }24 \qquad \mathrm{(C) \ }36 \qquad \mathrm{(D) \ }12\sqrt{2} \qquad \mathrm{(E) \ }12\sqrt{3} </ | ||
[[University of South Carolina High School Math Contest/1993 Exam/Problem 27|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 27|Solution]] | ||
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If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the circle circumscribing the triangle? | If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the circle circumscribing the triangle? | ||
− | < | + | <cmath>\mathrm{(A)}\quad 2 |
− | \mathrm{(A) \ | + | \quad \mathrm{(B) }\quad 8/\sqrt{15} |
− | \ | + | \quad \mathrm{(C) }\quad 5/2 |
− | \ | + | \quad \mathrm{(D) }\quad \sqrt{6} |
− | \ | + | \quad \mathrm{(E) }\quad (\sqrt{6} + 1)/2</cmath> |
− | \ | ||
− | |||
[[University of South Carolina High School Math Contest/1993 Exam/Problem 29|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 29|Solution]] | ||
== Problem 30 == | == Problem 30 == | ||
− | < | + | <cmath> \frac 1{1\cdot 2\cdot 3\cdot 4} + \frac 1{2\cdot 3\cdot 4\cdot 5} + \frac 1{3\cdot 4\cdot 5\cdot 6} + \cdots + \frac 1{28\cdot 29\cdot 30\cdot 31} = </cmath> |
− | < | + | <cmath> \mathrm{(A) \ }1/18 \qquad \mathrm{(B) \ }1/21 \qquad \mathrm{(C) \ }4/93 \qquad \mathrm{(D) \ }128/2505 \qquad \mathrm{(E) \ } 749/13485</cmath> |
[[University of South Carolina High School Math Contest/1993 Exam/Problem 30|Solution]] | [[University of South Carolina High School Math Contest/1993 Exam/Problem 30|Solution]] |
Latest revision as of 22:03, 26 October 2018
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 See also
Problem 1
If the width of a particular rectangle is doubled and the length is increased by 3, then the area is tripled. What is the length of the rectangle?
Problem 2
Suppose the operation is defined by
If
then
Problem 3
If 3 circles of radius 1 are mutually tangent as shown, what is the area of the gap they enclose?
Problem 4
If is expanded and written in the form
where
and
are real numbers, then
Problem 5
Suppose that is a function with the property that for all
and
and
What is the value of
?
Problem 6
After a price reduction, what increase does it take to restore the original price?
Problem 7
Each card below covers up a number. The number written below each card is the sum of all the numbers covered by all of the other cards. What is the sum of all of the hidden numbers?
Problem 8
What is the coefficient of in the expansion of
Problem 9
Suppose that and
are integers such that
and
. Then one possible value of
is
Problem 10
Problem 11
Suppose that 4 cards labeled 1 to 4 are placed randomly into 4 boxes also labeled 1 to 4, one card per box. What is the probability that no card gets placed into a box having the same label as the card?
Problem 12
If the equations and
have exactly one root in common, and
then the other root of equation
is
Problem 13
Suppose that and
are numbers such that
and
. Then
Problem 14
How many permutations of 1, 2, 3, 4, 5, 6, 7, 8, 9 have:
- 1 appearing somewhere to the left of 2,
- 3 somewhere to the left of 4, and
- 5 somewhere to the left of 6?
For example, 8 1 5 7 2 3 9 4 6 would be such a permutation.
Problem 15
If we express the sum
as a rational number in reduced form, then the denominator will be
Problem 16
In the triangle below, and
are the midpoints of
and
respectively.
and
intersect at
. If the length of
is 4, then what is the length of
?
Problem 17
Let represent the greatest integer that is less than or equal to
. For example,
and
. Then what is the value of
Problem 18
The minimum value of the function
as varies over all numbers in the largest possible domain of
, is
Problem 19
In the figure below, there are 4 distinct dots and
, joined by edges. Each dot is to be colored either red, blue, green, or yellow. No two dots joined by an edge are to be colored with the same color. How many completed colorings are possible?
Problem 20
Let be the 63 nonempty subsets of
. For each of these sets
, let
denote the product of all the elements in
. Then what is the value of
?
Problem 21
Suppose that each pair of eight tennis players either played exactly one game last week or did not play at all. Each player participated in all but 12 games. How many games were played among the eight players?
Problem 22
Let
and
Then which of the following inequalities is true?
Problem 23
The relation between the sets
and
is
Problem 24
If and in general
then
Problem 25
What is the center of the circle passing through the point and tangent to the circle
at
? (Two circles are tangent at a point
if they intersect at
and at no other point.)
Problem 26
Let . Then the first nonzero digit in the decimal expansion of
is
Problem 27
Suppose is a triangle with area 24 and that there is a point
inside
which is distance 2 from each of the sides of
. What is the perimeter of
?
Problem 28
Suppose is a triangle with 3 acute angles
and
. Then the point
(A) can be in the 1st quadrant and can be in the 2nd quadrant only
(B) can be in the 3rd quadrant and can be in the 4th quadrant only
(C) can be in the 2nd quadrant and can be in the 3rd quadrant only
(D) can be in the 2nd quadrant only
(E) can be in any of the 4 quadrants
Problem 29
If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the circle circumscribing the triangle?
Problem 30