Difference between revisions of "1993 AHSME Problems"
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+ | {{AHSME Problems | ||
+ | |year = 1993 | ||
+ | }} | ||
== Problem 1 == | == Problem 1 == | ||
For integers <math>a, b</math> and <math>c</math>, define <math>\boxed{a,b,c}</math> to mean <math>a^b-b^c+c^a</math>. Then <math>\boxed{1,-1,2}</math> equals | For integers <math>a, b</math> and <math>c</math>, define <math>\boxed{a,b,c}</math> to mean <math>a^b-b^c+c^a</math>. Then <math>\boxed{1,-1,2}</math> equals | ||
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== Problem 2 == | == Problem 2 == | ||
− | In <math>\triangle ABC</math>, <math>\angle A=55\circ</math>, <math>\angle C=75\circ</math>, <math>D</math> is on side <math>\ | + | In <math>\triangle ABC</math>, <math>\angle A=55^\circ</math>, <math>\angle C=75^\circ</math>, <math>D</math> is on side <math>\overline{AB}</math> and <math>E</math> is on side <math>\overline{BC}</math> If <math>DB=BE</math>, then <math>\angle BED=</math> |
− | <math>\text{(A)}\ 50\circ \qquad | + | <math>\text{(A)}\ 50^\circ \qquad |
− | \text{(B)}\ 55\circ \qquad | + | \text{(B)}\ 55^\circ \qquad |
− | \text{(C)}\ 60\circ \qquad | + | \text{(C)}\ 60^\circ \qquad |
− | \text{(D)}\ 65\circ \qquad | + | \text{(D)}\ 65^\circ \qquad |
− | \text{(E)}\ 70\circ </math> | + | \text{(E)}\ 70^\circ </math> |
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== Problem 3 == | == Problem 3 == | ||
+ | <math>\frac{15^{30}}{45^{15}} =</math> | ||
+ | |||
+ | <math>\text{(A) } \left(\frac{1}{3}\right)^{15}\quad | ||
+ | \text{(B) } \left(\frac{1}{3}\right)^{2}\quad | ||
+ | \text{(C) } 1\quad | ||
+ | \text{(D) } 3^{15}\quad | ||
+ | \text{(E) } 5^{15}</math> | ||
[[1993 AHSME Problems/Problem 3|Solution]] | [[1993 AHSME Problems/Problem 3|Solution]] | ||
== Problem 4 == | == Problem 4 == | ||
+ | Define the operation "<math>\circ</math>" by <math>x\circ y=4x-3y+xy</math>, for all real numbers <math>x</math> and <math>y</math>. For how many real numbers <math>y</math> does <math>3\circ y=12</math>? | ||
+ | |||
+ | <math>\text{(A) } 0\quad | ||
+ | \text{(B) } 1\quad | ||
+ | \text{(C) } 3\quad | ||
+ | \text{(D) } 4\quad | ||
+ | \text{(E) more than 4} </math> | ||
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== Problem 5 == | == Problem 5 == | ||
+ | Last year a bicycle cost \$160 and a cycling helmet \$40. This year the cost of the bicycle increased by <math>5\%</math>, and the cost of the helmet increased by <math>10\%</math>. The percent increase in the combined cost of the bicycle and the helmet is: | ||
+ | |||
+ | <math>\text{(A) } 6\%\quad | ||
+ | \text{(B) } 7\%\quad | ||
+ | \text{(C) } 7.5\%\quad | ||
+ | \text{(D) } 8\%\quad | ||
+ | \text{(E) } 15\%</math> | ||
[[1993 AHSME Problems/Problem 5|Solution]] | [[1993 AHSME Problems/Problem 5|Solution]] | ||
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== Problem 6 == | == Problem 6 == | ||
+ | |||
+ | <math>\sqrt{\frac{8^{10}+4^{10}}{8^4+4^{11}}}=</math> | ||
+ | |||
+ | <math>\text{(A) } \sqrt{2}\quad | ||
+ | \text{(B) } 16\quad | ||
+ | \text{(C) } 32\quad | ||
+ | \text{(D) } (12)^{\tfrac{2}{3}}\quad | ||
+ | \text{(E) } 512.5</math> | ||
[[1993 AHSME Problems/Problem 6|Solution]] | [[1993 AHSME Problems/Problem 6|Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
+ | |||
+ | The symbol <math>R_k</math> stands for an integer whose base-ten representation is a sequence of <math>k</math> ones. For example, <math>R_3=111,R_5=11111</math>, etc. When <math>R_{24}</math> is divided by <math>R_4</math>, the quotient <math>Q=R_{24}/R_4</math> is an integer whose base-ten representation is a sequence containing only ones and zeroes. The number of zeros in <math>Q</math> is: | ||
+ | |||
+ | <math>\text{(A) } 10\quad | ||
+ | \text{(B) } 11\quad | ||
+ | \text{(C) } 12\quad | ||
+ | \text{(D) } 13\quad | ||
+ | \text{(E) } 15</math> | ||
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== Problem 8 == | == Problem 8 == | ||
+ | Let <math>C_1</math> and <math>C_2</math> be circles of radius 1 that are in the same plane and tangent to each other. How many circles of radius 3 are in this plane and tangent to both <math>C_1</math> and <math>C_2</math>? | ||
+ | |||
+ | <math>\text{(A) } 2\quad | ||
+ | \text{(B) } 4\quad | ||
+ | \text{(C) } 5\quad | ||
+ | \text{(D) } 6\quad | ||
+ | \text{(E) } 8</math> | ||
[[1993 AHSME Problems/Problem 8|Solution]] | [[1993 AHSME Problems/Problem 8|Solution]] | ||
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== Problem 9 == | == Problem 9 == | ||
+ | |||
+ | Country <math>A</math> has <math>c\%</math> of the world's population and <math>d\%</math> of the worlds wealth. Country <math>B</math> has <math>e\%</math> of the world's population and <math>f\%</math> of its wealth. Assume that the citizens of <math>A</math> share the wealth of <math>A</math> equally,and assume that those of <math>B</math> share the wealth of <math>B</math> equally. Find the ratio of the wealth of a citizen of <math>A</math> to the wealth of a citizen of <math>B</math>. | ||
+ | |||
+ | <math>\text{(A) } \frac{cd}{ef}\quad | ||
+ | \text{(B) } \frac{ce}{ef}\quad | ||
+ | \text{(C) } \frac{cf}{de}\quad | ||
+ | \text{(D) } \frac{de}{cf}\quad | ||
+ | \text{(E) } \frac{df}{ce}</math> | ||
[[1993 AHSME Problems/Problem 9|Solution]] | [[1993 AHSME Problems/Problem 9|Solution]] | ||
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== Problem 10 == | == Problem 10 == | ||
+ | |||
+ | Let <math>r</math> be the number that results when both the base and the exponent of <math>a^b</math> are tripled, where <math>a,b>0</math>. If <math>r</math> equals the product of <math>a^b</math> and <math>x^b</math> where <math>x>0</math>, then <math>x=</math> | ||
+ | |||
+ | <math>\text{(A) } 3\quad | ||
+ | \text{(B) } 3a^2\quad | ||
+ | \text{(C) } 27a^2\quad | ||
+ | \text{(D) } 2a^{3b}\quad | ||
+ | \text{(E) } 3a^{2b}</math> | ||
[[1993 AHSME Problems/Problem 10|Solution]] | [[1993 AHSME Problems/Problem 10|Solution]] | ||
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== Problem 11 == | == Problem 11 == | ||
+ | |||
+ | If <math>\log_2(\log_2(\log_2(x)))=2</math>, then how many digits are in the base-ten representation for <math>x</math>? | ||
+ | |||
+ | <math>\text{(A) } 5\quad | ||
+ | \text{(B) } 7\quad | ||
+ | \text{(C) } 9\quad | ||
+ | \text{(D) } 11\quad | ||
+ | \text{(E) } 13</math> | ||
[[1993 AHSME Problems/Problem 11|Solution]] | [[1993 AHSME Problems/Problem 11|Solution]] | ||
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== Problem 12 == | == Problem 12 == | ||
+ | If <math>f(2x)=\frac{2}{2+x}</math> for all <math>x>0</math>, then <math>2f(x)=</math> | ||
+ | |||
+ | <math>\text{(A) } \frac{2}{1+x}\quad | ||
+ | \text{(B) } \frac{2}{2+x}\quad | ||
+ | \text{(C) } \frac{4}{1+x}\quad | ||
+ | \text{(D) } \frac{4}{2+x}\quad | ||
+ | \text{(E) } \frac{8}{4+x}</math> | ||
[[1993 AHSME Problems/Problem 12|Solution]] | [[1993 AHSME Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
+ | |||
+ | A square of perimeter 20 is inscribed in a square of perimeter 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square? | ||
+ | |||
+ | <math>\text{(A) } \sqrt{58}\quad | ||
+ | \text{(B) } \frac{7\sqrt{5}}{2}\quad | ||
+ | \text{(C) } 8\quad | ||
+ | \text{(D) } \sqrt{65}\quad | ||
+ | \text{(E) } 5\sqrt{3}</math> | ||
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== Problem 14 == | == Problem 14 == | ||
+ | <asy> | ||
+ | draw((-1,0)--(1,0)--(1+sqrt(2),sqrt(2))--(0,sqrt(2)+sqrt(13-2*sqrt(2)))--(-1-sqrt(2),sqrt(2))--cycle,black+linewidth(.75)); | ||
+ | MP("A",(-1,0),SW);MP("B",(1,0),SE);MP("C",(1+sqrt(2),sqrt(2)),E);MP("D",(0,sqrt(2)+sqrt(13-2*sqrt(2))),N);MP("E",(-1-sqrt(2),sqrt(2)),W); | ||
+ | dot((-1,0));dot((1,0));dot((1+sqrt(2),sqrt(2)));dot((-1-sqrt(2),sqrt(2)));dot((0,sqrt(2)+sqrt(13-2*sqrt(2)))); | ||
+ | </asy> | ||
+ | The convex pentagon <math>ABCDE</math> has <math>\angle{A}=\angle{B}=120^\circ,EA=AB=BC=2</math> and <math>CD=DE=4</math>. What is the area of ABCDE? | ||
+ | |||
+ | <math>\text{(A) } 10\quad | ||
+ | \text{(B) } 7\sqrt{3}\quad | ||
+ | \text{(C) } 15\quad | ||
+ | \text{(D) } 9\sqrt{3}\quad | ||
+ | \text{(E) } 12\sqrt{5}</math> | ||
[[1993 AHSME Problems/Problem 14|Solution]] | [[1993 AHSME Problems/Problem 14|Solution]] | ||
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== Problem 15 == | == Problem 15 == | ||
+ | |||
+ | For how many values of <math>n</math> will an <math>n</math>-sided regular polygon have interior angles with integral measures? | ||
+ | |||
+ | <math>\text{(A) } 16\quad | ||
+ | \text{(B) } 18\quad | ||
+ | \text{(C) } 20\quad | ||
+ | \text{(D) } 22\quad | ||
+ | \text{(E) } 24</math> | ||
[[1993 AHSME Problems/Problem 15|Solution]] | [[1993 AHSME Problems/Problem 15|Solution]] | ||
== Problem 16 == | == Problem 16 == | ||
+ | Consider the non-decreasing sequence of positive integers | ||
+ | <cmath>1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,\cdots</cmath> | ||
+ | in which the <math>n^{th}</math> positive integer appears <math>n</math> times. The remainder when the <math>1993^{rd} </math> term is divided by <math>5</math> is | ||
+ | <math>\text{(A) } 0\quad | ||
+ | \text{(B) } 1\quad | ||
+ | \text{(C) } 2\quad | ||
+ | \text{(D) } 3\quad | ||
+ | \text{(E) } 4</math> | ||
[[1993 AHSME Problems/Problem 16|Solution]] | [[1993 AHSME Problems/Problem 16|Solution]] | ||
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== Problem 17 == | == Problem 17 == | ||
+ | <asy> | ||
+ | draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle, black+linewidth(.75)); | ||
+ | draw((0,-1)--(0,1), black+linewidth(.75)); | ||
+ | draw((-1,0)--(1,0), black+linewidth(.75)); | ||
+ | draw((-1,-1/sqrt(3))--(1,1/sqrt(3)), black+linewidth(.75)); | ||
+ | draw((-1,1/sqrt(3))--(1,-1/sqrt(3)), black+linewidth(.75)); | ||
+ | draw((-1/sqrt(3),-1)--(1/sqrt(3),1), black+linewidth(.75)); | ||
+ | draw((1/sqrt(3),-1)--(-1/sqrt(3),1), black+linewidth(.75)); | ||
+ | </asy> | ||
+ | |||
+ | Amy painted a dartboard over a square clock face using the "hour positions" as boundaries.[See figure.] If <math>t</math> is the area of one of the eight triangular regions such as that between 12 o'clock and 1 o'clock, and <math>q</math> is the area of one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock, then <math>\frac{q}{t}=</math> | ||
+ | |||
+ | <math>\text{(A) } 2\sqrt{3}-2\quad | ||
+ | \text{(B) } \frac{3}{2}\quad | ||
+ | \text{(C) } \frac{\sqrt{5}+1}{2}\quad | ||
+ | \text{(D) } \sqrt{3}\quad | ||
+ | \text{(E) } 2</math> | ||
[[1993 AHSME Problems/Problem 17|Solution]] | [[1993 AHSME Problems/Problem 17|Solution]] | ||
== Problem 18 == | == Problem 18 == | ||
+ | Al and Barb start their new jobs on the same day. Al's schedule is 3 work-days followed by 1 rest-day. Barb's schedule is 7 work-days followed by 3 rest-days. On how many of their first 1000 days do both have rest-days on the same day? | ||
+ | <math>\text{(A) } 48\quad | ||
+ | \text{(B) } 50\quad | ||
+ | \text{(C) } 72\quad | ||
+ | \text{(D) } 75\quad | ||
+ | \text{(E) } 100</math> | ||
[[1993 AHSME Problems/Problem 18|Solution]] | [[1993 AHSME Problems/Problem 18|Solution]] | ||
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== Problem 19 == | == Problem 19 == | ||
+ | How many ordered pairs <math>(m,n)</math> of positive integers are solutions to | ||
+ | <cmath>\frac{4}{m}+\frac{2}{n}=1?</cmath> | ||
+ | |||
+ | <math>\text{(A) } 1\quad | ||
+ | \text{(B) } 2\quad | ||
+ | \text{(C) } 3\quad | ||
+ | \text{(D) } 4\quad | ||
+ | \text{(E) } \text{more than }6</math> | ||
[[1993 AHSME Problems/Problem 19|Solution]] | [[1993 AHSME Problems/Problem 19|Solution]] | ||
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== Problem 20 == | == Problem 20 == | ||
+ | Consider the equation <math>10z^2-3iz-k=0</math>, where <math>z</math> is a complex variable and <math>i^2=-1</math>. Which of the following statements is true? | ||
+ | |||
+ | <math>\text{(A) For all positive real numbers k, both roots are pure imaginary} \quad\\ | ||
+ | \text{(B) For all negative real numbers k, both roots are pure imaginary} \quad\\ | ||
+ | \text{(C) For all pure imaginary numbers k, both roots are real and rational} \quad\\ | ||
+ | \text{(D) For all pure imaginary numbers k, both roots are real and irrational} \quad\\ | ||
+ | \text{(E) For all complex numbers k, neither root is real} </math> | ||
[[1993 AHSME Problems/Problem 20|Solution]] | [[1993 AHSME Problems/Problem 20|Solution]] | ||
== Problem 21 == | == Problem 21 == | ||
+ | |||
+ | Let <math>a_1,a_2,\cdots,a_k</math> be a finite arithmetic sequence with <math>a_4 +a_7+a_{10} = 17</math> and <math>a_4+a_5+\cdots+a_{13} +a_{14} = 77</math>. | ||
+ | |||
+ | If <math>a_k = 13</math>, then <math>k =</math> | ||
+ | |||
+ | <math>\text{(A) } 16\quad | ||
+ | \text{(B) } 18\quad | ||
+ | \text{(C) } 20\quad | ||
+ | \text{(D) } 22\quad | ||
+ | \text{(E) } 24</math> | ||
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== Problem 22 == | == Problem 22 == | ||
+ | <asy> | ||
+ | draw((-1,0)--(1,0)--(1,-1)--(-1,-1)--cycle,black+linewidth(.75)); | ||
+ | MP("?",(0,0),S); | ||
+ | </asy> | ||
+ | |||
+ | Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2,3 and 4 is assigned the number which is the sum of numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block. | ||
+ | |||
+ | <math>\text{(A) } 55\quad | ||
+ | \text{(B) } 83\quad | ||
+ | \text{(C) } 114\quad | ||
+ | \text{(D) } 137\quad | ||
+ | \text{(E) } 144</math> | ||
[[1993 AHSME Problems/Problem 22|Solution]] | [[1993 AHSME Problems/Problem 22|Solution]] | ||
+ | == Problem 23 == | ||
+ | |||
+ | <asy> | ||
+ | draw(circle((0,0),10),black+linewidth(.75)); | ||
+ | draw((-10,0)--(10,0),black+linewidth(.75)); | ||
+ | draw((-10,0)--(9,sqrt(19)),black+linewidth(.75)); | ||
+ | draw((-10,0)--(9,-sqrt(19)),black+linewidth(.75)); | ||
+ | draw((2,0)--(9,sqrt(19)),black+linewidth(.75)); | ||
+ | draw((2,0)--(9,-sqrt(19)),black+linewidth(.75)); | ||
+ | MP("X",(2,0),N);MP("A",(-10,0),W);MP("D",(10,0),E);MP("B",(9,sqrt(19)),E);MP("C",(9,-sqrt(19)),E); | ||
+ | </asy> | ||
+ | |||
+ | Points <math>A,B,C</math> and <math>D</math> are on a circle of diameter <math>1</math>, and <math>X</math> is on diameter <math>\overline{AD}.</math> | ||
+ | |||
+ | If <math>BX=CX</math> and <math>3\angle{BAC}=\angle{BXC}=36^\circ</math>, then <math>AX=</math> | ||
− | + | ||
+ | <math>\text{(A) } \cos(6^\circ)\cos(12^\circ)\sec(18^\circ)\quad\\ | ||
+ | \text{(B) } \cos(6^\circ)\sin(12^\circ)\csc(18^\circ)\quad\\ | ||
+ | \text{(C) } \cos(6^\circ)\sin(12^\circ)\sec(18^\circ)\quad\\ | ||
+ | \text{(D) } \sin(6^\circ)\sin(12^\circ)\csc(18^\circ)\quad\\ | ||
+ | \text{(E) } \sin(6^\circ)\sin(12^\circ)\sec(18^\circ)</math> | ||
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== Problem 24 == | == Problem 24 == | ||
+ | A box contains <math>3</math> shiny pennies and <math>4</math> dull pennies. One by one, pennies are drawn at random from the box and not replaced. If the probability is <math>a/b</math> that it will take more than four draws until the third shiny penny appears and <math>a/b</math> is in lowest terms, then <math>a+b=</math> | ||
+ | |||
+ | <math>\text{(A) } 11\quad | ||
+ | \text{(B) } 20\quad | ||
+ | \text{(C) } 35\quad | ||
+ | \text{(D) } 58\quad | ||
+ | \text{(E) } 66</math> | ||
[[1993 AHSME Problems/Problem 24|Solution]] | [[1993 AHSME Problems/Problem 24|Solution]] | ||
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== Problem 25 == | == Problem 25 == | ||
+ | <asy> | ||
+ | draw((0,0)--(1,sqrt(3)),black+linewidth(.75),EndArrow); | ||
+ | draw((0,0)--(1,-sqrt(3)),black+linewidth(.75),EndArrow); | ||
+ | draw((0,0)--(1,0),dashed+black+linewidth(.75)); | ||
+ | dot((1,0)); | ||
+ | MP("P",(1,0),E); | ||
+ | </asy> | ||
+ | |||
+ | Let <math>S</math> be the set of points on the rays forming the sides of a <math>120^{\circ}</math> angle, and let <math>P</math> be a fixed point inside the angle | ||
+ | on the angle bisector. Consider all distinct equilateral triangles <math>PQR</math> with <math>Q</math> and <math>R</math> in <math>S</math>. | ||
+ | (Points <math>Q</math> and <math>R</math> may be on the same ray, and switching the names of <math>Q</math> and <math>R</math> does not create a distinct triangle.) | ||
+ | There are | ||
+ | |||
+ | <math>\text{(A) exactly 2 such triangles} \quad\\ | ||
+ | \text{(B) exactly 3 such triangles} \quad\\ | ||
+ | \text{(C) exactly 7 such triangles} \quad\\ | ||
+ | \text{(D) exactly 15 such triangles} \quad\\ | ||
+ | \text{(E) more than 15 such triangles} </math> | ||
[[1993 AHSME Problems/Problem 25|Solution]] | [[1993 AHSME Problems/Problem 25|Solution]] | ||
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== Problem 26 == | == Problem 26 == | ||
+ | Find the largest positive value attained by the function | ||
+ | <cmath>f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48} ,\quad x \text{ a real number}</cmath> | ||
+ | |||
+ | <math>\text{(A) } \sqrt{7}-1\quad | ||
+ | \text{(B) } 3\quad | ||
+ | \text{(C) } 2\sqrt{3}\quad | ||
+ | \text{(D) } 4\quad | ||
+ | \text{(E) } \sqrt{55}-\sqrt{5}</math> | ||
[[1993 AHSME Problems/Problem 26|Solution]] | [[1993 AHSME Problems/Problem 26|Solution]] | ||
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== Problem 27 == | == Problem 27 == | ||
+ | <asy> | ||
+ | draw(circle((4,1),1),black+linewidth(.75)); | ||
+ | draw((0,0)--(8,0)--(8,6)--cycle,black+linewidth(.75)); | ||
+ | MP("A",(0,0),SW);MP("B",(8,0),SE);MP("C",(8,6),NE);MP("P",(4,1),NW); | ||
+ | MP("8",(4,0),S);MP("6",(8,3),E);MP("10",(4,3),NW); | ||
+ | MP("->",(5,1),E); | ||
+ | dot((4,1)); | ||
+ | </asy> | ||
+ | The sides of <math>\triangle ABC</math> have lengths <math>6,8,</math> and <math>10</math>. A circle with center <math>P</math> and radius <math>1</math> rolls around the inside of <math>\triangle ABC</math>, always remaining tangent to at least one side of the triangle. When <math>P</math> first returns to its original position, through what distance has <math>P</math> traveled? | ||
+ | |||
+ | <math>\text{(A) } 10\quad | ||
+ | \text{(B) } 12\quad | ||
+ | \text{(C) } 14\quad | ||
+ | \text{(D) } 15\quad | ||
+ | \text{(E) } 17</math> | ||
[[1993 AHSME Problems/Problem 27|Solution]] | [[1993 AHSME Problems/Problem 27|Solution]] | ||
== Problem 28 == | == Problem 28 == | ||
+ | How many triangles with positive area are there whose vertices are points in the <math>xy</math>-plane whose coordinates are integers <math>(x,y)</math> satisfying <math>1\le x\le 4</math> and <math>1\le y\le 4</math>? | ||
+ | |||
+ | <math>\text{(A) } 496\quad | ||
+ | \text{(B) } 500\quad | ||
+ | \text{(C) } 512\quad | ||
+ | \text{(D) } 516\quad | ||
+ | \text{(E) } 560</math> | ||
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== Problem 29 == | == Problem 29 == | ||
+ | Which of the following could NOT be the lengths of the external diagonals of a right regular prism [a "box"]? (An <math>\textit{external diagonal}</math> is a diagonal of one of the rectangular faces of the box.) | ||
+ | |||
+ | <math>\text{(A) }\{4,5,6\} \quad | ||
+ | \text{(B) } \{4,5,7\} \quad | ||
+ | \text{(C) } \{4,6,7\} \quad | ||
+ | \text{(D) } \{5,6,7\} \quad | ||
+ | \text{(E) } \{5,7,8\} </math> | ||
[[1993 AHSME Problems/Problem 29|Solution]] | [[1993 AHSME Problems/Problem 29|Solution]] | ||
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== Problem 30 == | == Problem 30 == | ||
+ | Given <math>0\le x_0<1</math>, let | ||
+ | <cmath> | ||
+ | x_n=\left\{ \begin{array}{ll} | ||
+ | 2x_{n-1} &\text{ if }2x_{n-1}<1 \\ | ||
+ | 2x_{n-1}-1 &\text{ if }2x_{n-1}\ge 1 | ||
+ | \end{array}\right. | ||
+ | </cmath> | ||
+ | for all integers <math>n>0</math>. For how many <math>x_0</math> is it true that <math>x_0=x_5</math>? | ||
+ | |||
+ | <math>\text{(A) 0} \quad | ||
+ | \text{(B) 1} \quad | ||
+ | \text{(C) 5} \quad | ||
+ | \text{(D) 31} \quad | ||
+ | \text{(E) }\infty </math> | ||
[[1993 AHSME Problems/Problem 30|Solution]] | [[1993 AHSME Problems/Problem 30|Solution]] | ||
== See also == | == See also == | ||
− | + | ||
− | * [[ | + | * [[AMC 12 Problems and Solutions]] |
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{AHSME box|year=1993|before=[[1992 AHSME]]|after=[[1994 AHSME]]}} | ||
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+ | {{MAA Notice}} |
Latest revision as of 22:38, 22 December 2020
1993 AHSME (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
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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
For integers and , define to mean . Then equals
Problem 2
In , , , is on side and is on side If , then
Problem 3
Problem 4
Define the operation "" by , for all real numbers and . For how many real numbers does ?
Problem 5
Last year a bicycle cost $160 and a cycling helmet $40. This year the cost of the bicycle increased by , and the cost of the helmet increased by . The percent increase in the combined cost of the bicycle and the helmet is:
Problem 6
Problem 7
The symbol stands for an integer whose base-ten representation is a sequence of ones. For example, , etc. When is divided by , the quotient is an integer whose base-ten representation is a sequence containing only ones and zeroes. The number of zeros in is:
Problem 8
Let and be circles of radius 1 that are in the same plane and tangent to each other. How many circles of radius 3 are in this plane and tangent to both and ?
Problem 9
Country has of the world's population and of the worlds wealth. Country has of the world's population and of its wealth. Assume that the citizens of share the wealth of equally,and assume that those of share the wealth of equally. Find the ratio of the wealth of a citizen of to the wealth of a citizen of .
Problem 10
Let be the number that results when both the base and the exponent of are tripled, where . If equals the product of and where , then
Problem 11
If , then how many digits are in the base-ten representation for ?
Problem 12
If for all , then
Problem 13
A square of perimeter 20 is inscribed in a square of perimeter 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?
Problem 14
The convex pentagon has and . What is the area of ABCDE?
Problem 15
For how many values of will an -sided regular polygon have interior angles with integral measures?
Problem 16
Consider the non-decreasing sequence of positive integers in which the positive integer appears times. The remainder when the term is divided by is
Problem 17
Amy painted a dartboard over a square clock face using the "hour positions" as boundaries.[See figure.] If is the area of one of the eight triangular regions such as that between 12 o'clock and 1 o'clock, and is the area of one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock, then
Problem 18
Al and Barb start their new jobs on the same day. Al's schedule is 3 work-days followed by 1 rest-day. Barb's schedule is 7 work-days followed by 3 rest-days. On how many of their first 1000 days do both have rest-days on the same day?
Problem 19
How many ordered pairs of positive integers are solutions to
Problem 20
Consider the equation , where is a complex variable and . Which of the following statements is true?
Problem 21
Let be a finite arithmetic sequence with and .
If , then
Problem 22
Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2,3 and 4 is assigned the number which is the sum of numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.
Problem 23
Points and are on a circle of diameter , and is on diameter
If and , then
Problem 24
A box contains shiny pennies and dull pennies. One by one, pennies are drawn at random from the box and not replaced. If the probability is that it will take more than four draws until the third shiny penny appears and is in lowest terms, then
Problem 25
Let be the set of points on the rays forming the sides of a angle, and let be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles with and in . (Points and may be on the same ray, and switching the names of and does not create a distinct triangle.) There are
Problem 26
Find the largest positive value attained by the function
Problem 27
The sides of have lengths and . A circle with center and radius rolls around the inside of , always remaining tangent to at least one side of the triangle. When first returns to its original position, through what distance has traveled?
Problem 28
How many triangles with positive area are there whose vertices are points in the -plane whose coordinates are integers satisfying and ?
Problem 29
Which of the following could NOT be the lengths of the external diagonals of a right regular prism [a "box"]? (An is a diagonal of one of the rectangular faces of the box.)
Problem 30
Given , let for all integers . For how many is it true that ?
See also
1993 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1992 AHSME |
Followed by 1994 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.