View source for University of South Carolina High School Math Contest/1993 Exam/Problems

== 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?

<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]]

== Problem 2 ==
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]]

== Problem 3 ==
If 3 circles of radius 1 are mutually tangent as shown, what is the area of the gap they enclose?

<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]]

== Problem 4 ==
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]]

== Problem 5 ==
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]]

== Problem 6 ==
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]]

== 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?

<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]]

== Problem 8 ==
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]]

== Problem 9 ==
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]]

== Problem 10 ==
<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]]

== 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?

<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]]

== Problem 12 ==
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]]

== Problem 13 ==
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]]

== 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.

<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]]

== Problem 15 ==
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

<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]]

== Problem 16 ==
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>

<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]]

== Problem 17 ==
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]]

== Problem 18 ==
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 

<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]]

== Problem 19 ==
In the figure below, there are 4 distinct dots <math>A, B, C,</math> and <math>D</math>, 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?

<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]]

== Problem 20 ==
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]]

== 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?

<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]]

== Problem 22 ==
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

<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?

<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]]

== Problem 23 ==
The relation between the sets

<cmath> M = \{ 12 m + 8 n + 4 l: m,n,l \rm{ \ are \ } \rm{integers}\} </cmath>

and

<cmath> N= \{ 20 p + 16q + 12r: p,q,r \rm{ \ are \ } \rm{integers}\} </cmath>

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>


[[University of South Carolina High School Math Contest/1993 Exam/Problem 23|Solution]]

== Problem 24 ==
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]]

== Problem 25 ==
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]]

== Problem 26 ==
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]]

== Problem 27 ==
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>

[[University of South Carolina High School Math Contest/1993 Exam/Problem 27|Solution]]

== Problem 28 ==
Suppose <math>\triangle ABC</math> is a triangle with 3 acute angles <math>A, B,</math> and <math>C</math>.  Then the point <math>( \cos B - \sin A, \sin B - \cos A)</math>

(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

[[University of South Carolina High School Math Contest/1993 Exam/Problem 28|Solution]]

== Problem 29 ==
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
\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]]

== 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]]

== See also ==
* [[University of South Carolina High School Math Contest/1993 Exam]]