Difference between revisions of "1993 AHSME Problems"

Latest revision as of 07:15, 27 June 2021

1993 AHSME (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions This is a 30-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. You will receive 5 points for each correct answer, 2 points for each problem left unanswered, and 0 points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers. Figures are not necessarily drawn to scale. You will have 90 minutes working time to complete the test.
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Problem 1

For integers $a, b$ and $c$ , define $\boxed{a,b,c}$ to mean $a^b-b^c+c^a$ . Then $\boxed{1,-1,2}$ equals

$\text{(A)} \ -4 \qquad \text{(B)} \ -2 \qquad \text{(C)} \ 0 \qquad \text{(D)} \ 2 \qquad \text{(E)} \ 4$

Solution

Problem 2

In $\triangle ABC$ , $\angle A=55^\circ$ , $\angle C=75^\circ$ , $D$ is on side $\overline{AB}$ and $E$ is on side $\overline{BC}$ If $DB=BE$ , then $\angle BED=$

$\text{(A)}\ 50^\circ \qquad \text{(B)}\ 55^\circ \qquad \text{(C)}\ 60^\circ \qquad \text{(D)}\ 65^\circ \qquad \text{(E)}\ 70^\circ$

Solution

Problem 3

$\frac{15^{30}}{45^{15}} =$

$\text{(A) } \left(\frac{1}{3}\right)^{15}\quad \text{(B) } \left(\frac{1}{3}\right)^{2}\quad \text{(C) } 1\quad \text{(D) } 3^{15}\quad \text{(E) } 5^{15}$

Solution

Problem 4

Define the operation " $\circ$ " by $x\circ y=4x-3y+xy$ , for all real numbers $x$ and $y$ . For how many real numbers $y$ does $3\circ y=12$ ?

$\text{(A) } 0\quad \text{(B) } 1\quad \text{(C) } 3\quad \text{(D) } 4\quad \text{(E) more than 4}$

Solution

Problem 5

Last year a bicycle cost $160 and a cycling helmet $40. This year the cost of the bicycle increased by $5\%$ , and the cost of the helmet increased by $10\%$ . The percent increase in the combined cost of the bicycle and the helmet is:

$\text{(A) } 6\%\quad \text{(B) } 7\%\quad \text{(C) } 7.5\%\quad \text{(D) } 8\%\quad \text{(E) } 15\%$

Solution

Problem 6

$\sqrt{\frac{8^{10}+4^{10}}{8^4+4^{11}}}=$

$\text{(A) } \sqrt{2}\quad \text{(B) } 16\quad \text{(C) } 32\quad \text{(D) } (12)^{\tfrac{2}{3}}\quad \text{(E) } 512.5$

Solution

Problem 7

The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111,R_5=11111$ , etc. When $R_{24}$ is divided by $R_4$ , the quotient $Q=R_{24}/R_4$ is an integer whose base-ten representation is a sequence containing only ones and zeroes. The number of zeros in $Q$ is:

$\text{(A) } 10\quad \text{(B) } 11\quad \text{(C) } 12\quad \text{(D) } 13\quad \text{(E) } 15$

Solution

Problem 8

Let $C_1$ and $C_2$ 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 $C_1$ and $C_2$ ?

$\text{(A) } 2\quad \text{(B) } 4\quad \text{(C) } 5\quad \text{(D) } 6\quad \text{(E) } 8$

Solution

Problem 9

Country $A$ has $c\%$ of the world's population and $d\%$ of the worlds wealth. Country $B$ has $e\%$ of the world's population and $f\%$ of its wealth. Assume that the citizens of $A$ share the wealth of $A$ equally,and assume that those of $B$ share the wealth of $B$ equally. Find the ratio of the wealth of a citizen of $A$ to the wealth of a citizen of $B$ .

$\text{(A) } \frac{cd}{ef}\quad \text{(B) } \frac{ce}{ef}\quad \text{(C) } \frac{cf}{de}\quad \text{(D) } \frac{de}{cf}\quad \text{(E) } \frac{df}{ce}$

Solution

Problem 10

Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a,b>0$ . If $r$ equals the product of $a^b$ and $x^b$ where $x>0$ , then $x=$

$\text{(A) } 3\quad \text{(B) } 3a^2\quad \text{(C) } 27a^2\quad \text{(D) } 2a^{3b}\quad \text{(E) } 3a^{2b}$

Solution

Problem 11

If $\log_2(\log_2(\log_2(x)))=2$ , then how many digits are in the base-ten representation for $x$ ?

$\text{(A) } 5\quad \text{(B) } 7\quad \text{(C) } 9\quad \text{(D) } 11\quad \text{(E) } 13$

Solution

Problem 12

If $f(2x)=\frac{2}{2+x}$ for all $x>0$ , then $2f(x)=$

$\text{(A) } \frac{2}{1+x}\quad \text{(B) } \frac{2}{2+x}\quad \text{(C) } \frac{4}{1+x}\quad \text{(D) } \frac{4}{2+x}\quad \text{(E) } \frac{8}{4+x}$

Solution

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?

$\text{(A) } \sqrt{58}\quad \text{(B) } \frac{7\sqrt{5}}{2}\quad \text{(C) } 8\quad \text{(D) } \sqrt{65}\quad \text{(E) } 5\sqrt{3}$

Solution

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 $ABCDE$ has $\angle{A}=\angle{B}=120^\circ,EA=AB=BC=2$ and $CD=DE=4$ . What is the area of ABCDE?

$\text{(A) } 10\quad \text{(B) } 7\sqrt{3}\quad \text{(C) } 15\quad \text{(D) } 9\sqrt{3}\quad \text{(E) } 12\sqrt{5}$

Solution

Problem 15

For how many values of $n$ will an $n$ -sided regular polygon have interior angles with integral measures?

$\text{(A) } 16\quad \text{(B) } 18\quad \text{(C) } 20\quad \text{(D) } 22\quad \text{(E) } 24$

Solution

Problem 16

Consider the non-decreasing sequence of positive integers $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,\cdots$ in which the $n^{th}$ positive integer appears $n$ times. The remainder when the $1993^{rd}$ term is divided by $5$ is

$\text{(A) } 0\quad \text{(B) } 1\quad \text{(C) } 2\quad \text{(D) } 3\quad \text{(E) } 4$

Solution

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 $t$ is the area of one of the eight triangular regions such as that between 12 o'clock and 1 o'clock, and $q$ is the area of one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock, then $\frac{q}{t}=$

$\text{(A) } 2\sqrt{3}-2\quad \text{(B) } \frac{3}{2}\quad \text{(C) } \frac{\sqrt{5}+1}{2}\quad \text{(D) } \sqrt{3}\quad \text{(E) } 2$

Solution

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?

$\text{(A) } 48\quad \text{(B) } 50\quad \text{(C) } 72\quad \text{(D) } 75\quad \text{(E) } 100$

Solution

Problem 19

How many ordered pairs $(m,n)$ of positive integers are solutions to $\frac{4}{m}+\frac{2}{n}=1?$

$\text{(A) } 1\quad \text{(B) } 2\quad \text{(C) } 3\quad \text{(D) } 4\quad \text{(E) } \text{more than }6$

Solution

Problem 20

Consider the equation $10z^2-3iz-k=0$ , where $z$ is a complex variable and $i^2=-1$ . Which of the following statements is true?

$\text{(A) For all positive real numbers k, both roots are pure imaginary} \quad\\ \text{(B) For all negative real numbers k, both roots are pure imaginary} \quad\\ \text{(C) For all pure imaginary numbers k, both roots are real and rational} \quad\\ \text{(D) For all pure imaginary numbers k, both roots are real and irrational} \quad\\ \text{(E) For all complex numbers k, neither root is real}$

Solution

Problem 21

Let $a_1,a_2,\cdots,a_k$ be a finite arithmetic sequence with $a_4 +a_7+a_{10} = 17$ and $a_4+a_5+\cdots+a_{13} +a_{14} = 77$ .

If $a_k = 13$ , then $k =$

$\text{(A) } 16\quad \text{(B) } 18\quad \text{(C) } 20\quad \text{(D) } 22\quad \text{(E) } 24$

Solution

Problem 22

$[asy] size((400)); draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0), linewidth(1)); draw((5,0)--(10,0)--(15,0)--(20,0)--(20,5)--(15,5)--(10,5)--(5,5)--(6,7)--(11,7)--(16,7)--(21,7)--(21,2)--(20,0), linewidth(1)); draw((10,0)--(10,5)--(11,7), linewidth(1)); draw((15,0)--(15,5)--(16,7), linewidth(1)); draw((20,0)--(20,5)--(21,7), linewidth(1)); draw((0,5)--(1,7)--(6,7), linewidth(1)); draw((3.5,7)--(4.5,9)--(9.5,9)--(14.5,9)--(19.5,9)--(18.5,7)--(19.5,9)--(19.5,7), linewidth(1)); draw((8.5,7)--(9.5,9), linewidth(1)); draw((13.5,7)--(14.5,9), linewidth(1)); draw((7,9)--(8,11)--(13,11)--(18,11)--(17,9)--(18,11)--(18,9), linewidth(1)); draw((12,9)--(13,11), linewidth(1)); draw((10.5,11)--(11.5,13)--(16.5,13)--(16.5,11)--(16.5,13)--(15.5,11), linewidth(1)); draw((25,0)--(30,0)--(30,5)--(25,5)--(25,0), dashed); draw((30,0)--(35,0)--(40,0)--(45,0)--(45,5)--(40,5)--(35,5)--(30,5)--(31,7)--(36,7)--(41,7)--(46,7)--(46,2)--(45,0), dashed); draw((35,0)--(35,5)--(36,7), dashed); draw((40,0)--(40,5)--(41,7), dashed); draw((45,0)--(45,5)--(46,7), dashed); draw((25,5)--(26,7)--(31,7), dashed); draw((28.5,7)--(29.5,9)--(34.5,9)--(39.5,9)--(44.5,9)--(43.5,7)--(44.5,9)--(44.5,7), dashed); draw((33.5,7)--(34.5,9), dashed); draw((38.5,7)--(39.5,9), dashed); draw((32,9)--(33,11)--(38,11)--(43,11)--(42,9)--(43,11)--(43,9), dashed); draw((37,9)--(38,11), dashed); draw((35.5,11)--(36.5,13)--(41.5,13)--(41.5,11)--(41.5,13)--(40.5,11), dashed); draw((50,0)--(55,0)--(55,5)--(50,5)--(50,0), dashed); draw((55,0)--(60,0)--(65,0)--(70,0)--(70,5)--(65,5)--(60,5)--(55,5)--(56,7)--(61,7)--(66,7)--(71,7)--(71,2)--(70,0), dashed); draw((60,0)--(60,5)--(61,7), dashed); draw((65,0)--(65,5)--(66,7), dashed); draw((70,0)--(70,5)--(71,7), dashed); draw((50,5)--(51,7)--(56,7), dashed); draw((53.5,7)--(54.5,9)--(59.5,9)--(64.5,9)--(69.5,9)--(68.5,7)--(69.5,9)--(69.5,7), dashed); draw((58.5,7)--(59.5,9), dashed); draw((63.5,7)--(64.5,9), dashed); draw((57,9)--(58,11)--(63,11)--(68,11)--(67,9)--(68,11)--(68,9), dashed); draw((62,9)--(63,11), dashed); draw((60.5,11)--(61.5,13)--(66.5,13)--(66.5,11)--(66.5,13)--(65.5,11), dashed); draw((75,0)--(80,0)--(80,5)--(75,5)--(75,0), dashed); draw((80,0)--(85,0)--(90,0)--(95,0)--(95,5)--(90,5)--(85,5)--(80,5)--(81,7)--(86,7)--(91,7)--(96,7)--(96,2)--(95,0), dashed); draw((85,0)--(85,5)--(86,7), dashed); draw((90,0)--(90,5)--(91,7), dashed); draw((95,0)--(95,5)--(96,7), dashed); draw((75,5)--(76,7)--(81,7), dashed); draw((78.5,7)--(79.5,9)--(84.5,9)--(89.5,9)--(94.5,9)--(93.5,7)--(94.5,9)--(94.5,7), dashed); draw((83.5,7)--(84.5,9), dashed); draw((88.5,7)--(89.5,9), dashed); draw((82,9)--(83,11)--(88,11)--(93,11)--(92,9)--(93,11)--(93,9), dashed); draw((87,9)--(88,11), dashed); draw((85.5,11)--(86.5,13)--(91.5,13)--(91.5,11)--(91.5,13)--(90.5,11), dashed); draw((28,6)--(33,6)--(38,6)--(43,6)--(43,11)--(38,11)--(33,11)--(28,11)--(28,6), linewidth(1)); draw((28,11)--(29,13)--(34,13)--(39,13)--(44,13)--(43,11)--(44,13)--(44,8)--(43,6), linewidth(1)); draw((33,6)--(33,11)--(34,13)--(39,13)--(38,11)--(38,6), linewidth(1)); draw((31,13)--(32,15)--(37,15)--(36,13)--(37,15)--(42,15)--(41,13)--(42,15)--(42,13), linewidth(1)); draw((34.5,15)--(35.5,17)--(40.5,17)--(39.5,15)--(40.5,17)--(40.5,15), linewidth(1)); draw((53,6)--(58,6)--(63,6)--(68,6)--(68,11)--(63,11)--(58,11)--(53,11)--(53,6), dashed); draw((53,11)--(54,13)--(59,13)--(64,13)--(69,13)--(68,11)--(69,13)--(69,8)--(68,6), dashed); draw((58,6)--(58,11)--(59,13)--(64,13)--(63,11)--(63,6), dashed); draw((56,13)--(57,15)--(62,15)--(61,13)--(62,15)--(67,15)--(66,13)--(67,15)--(67,13), dashed); draw((59.5,15)--(60.5,17)--(65.5,17)--(64.5,15)--(65.5,17)--(65.5,15), dashed); draw((78,6)--(83,6)--(88,6)--(93,6)--(93,11)--(88,11)--(83,11)--(78,11)--(78,6), dashed); draw((78,11)--(79,13)--(84,13)--(89,13)--(94,13)--(93,11)--(94,13)--(94,8)--(93,6), dashed); draw((83,6)--(83,11)--(84,13)--(89,13)--(88,11)--(88,6), dashed); draw((81,13)--(82,15)--(87,15)--(86,13)--(87,15)--(92,15)--(91,13)--(92,15)--(92,13), dashed); draw((84.5,15)--(85.5,17)--(90.5,17)--(89.5,15)--(90.5,17)--(90.5,15), dashed); draw((56,12)--(61,12)--(66,12)--(66,17)--(61,17)--(56,17)--(56,12), linewidth(1)); draw((61,12)--(61,17)--(62,19)--(57,19)--(56,17)--(57,19)--(67,19)--(66,17)--(67,19)--(67,14)--(66,12), linewidth(1)); draw((59.5,19)--(60.5,21)--(65.5,21)--(64.5,19)--(65.5,21)--(65.5,19), linewidth(1)); draw((81,12)--(86,12)--(91,12)--(91,17)--(86,17)--(81,17)--(81,12), dashed); draw((86,12)--(86,17)--(87,19)--(82,19)--(81,17)--(82,19)--(92,19)--(91,17)--(92,19)--(92,14)--(91,12), dashed); draw((84.5,19)--(85.5,21)--(90.5,21)--(89.5,19)--(90.5,21)--(90.5,19), dashed); draw((84,18)--(89,18)--(89,23)--(84,23)--(84,18)--(84,23)--(85,25)--(90,25)--(89,23)--(90,25)--(90,20)--(89,18), linewidth(1));[/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.

$\text{(A) } 55\quad \text{(B) } 83\quad \text{(C) } 114\quad \text{(D) } 137\quad \text{(E) } 144$

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 $A,B,C$ and $D$ are on a circle of diameter $1$ , and $X$ is on diameter $\overline{AD}.$

If $BX=CX$ and $3\angle{BAC}=\angle{BXC}=36^\circ$ , then $AX=$

$\text{(A) } \cos(6^\circ)\cos(12^\circ)\sec(18^\circ)\quad\\ \text{(B) } \cos(6^\circ)\sin(12^\circ)\csc(18^\circ)\quad\\ \text{(C) } \cos(6^\circ)\sin(12^\circ)\sec(18^\circ)\quad\\ \text{(D) } \sin(6^\circ)\sin(12^\circ)\csc(18^\circ)\quad\\ \text{(E) } \sin(6^\circ)\sin(12^\circ)\sec(18^\circ)$

Solution

Problem 24

A box contains $3$ shiny pennies and $4$ dull pennies. One by one, pennies are drawn at random from the box and not replaced. If the probability is $a/b$ that it will take more than four draws until the third shiny penny appears and $a/b$ is in lowest terms, then $a+b=$

$\text{(A) } 11\quad \text{(B) } 20\quad \text{(C) } 35\quad \text{(D) } 58\quad \text{(E) } 66$

Solution

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 $S$ be the set of points on the rays forming the sides of a $120^{\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$ . (Points $Q$ and $R$ may be on the same ray, and switching the names of $Q$ and $R$ does not create a distinct triangle.) There are

$\text{(A) exactly 2 such triangles} \quad\\ \text{(B) exactly 3 such triangles} \quad\\ \text{(C) exactly 7 such triangles} \quad\\ \text{(D) exactly 15 such triangles} \quad\\ \text{(E) more than 15 such triangles}$

Solution

Problem 26

Find the largest positive value attained by the function $f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48} ,\quad x \text{ a real number}$

$\text{(A) } \sqrt{7}-1\quad \text{(B) } 3\quad \text{(C) } 2\sqrt{3}\quad \text{(D) } 4\quad \text{(E) } \sqrt{55}-\sqrt{5}$

Solution

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 $\triangle ABC$ have lengths $6,8,$ and $10$ . A circle with center $P$ and radius $1$ rolls around the inside of $\triangle ABC$ , always remaining tangent to at least one side of the triangle. When $P$ first returns to its original position, through what distance has $P$ traveled?

$\text{(A) } 10\quad \text{(B) } 12\quad \text{(C) } 14\quad \text{(D) } 15\quad \text{(E) } 17$

Solution

Problem 28

How many triangles with positive area are there whose vertices are points in the $xy$ -plane whose coordinates are integers $(x,y)$ satisfying $1\le x\le 4$ and $1\le y\le 4$ ?

$\text{(A) } 496\quad \text{(B) } 500\quad \text{(C) } 512\quad \text{(D) } 516\quad \text{(E) } 560$

Solution

Problem 29

Which of the following could NOT be the lengths of the external diagonals of a right regular prism [a "box"]? (An $\textit{external diagonal}$ is a diagonal of one of the rectangular faces of the box.)

$\text{(A) }\{4,5,6\} \quad \text{(B) } \{4,5,7\} \quad \text{(C) } \{4,6,7\} \quad \text{(D) } \{5,6,7\} \quad \text{(E) } \{5,7,8\}$

Solution

Problem 30

Given $0\le x_0<1$ , let $x_n=\left\{ \begin{array}{ll} 2x_{n-1} &\text{ if }2x_{n-1}<1 \\ 2x_{n-1}-1 &\text{ if }2x_{n-1}\ge 1 \end{array}\right.$ for all integers $n>0$ . For how many $x_0$ is it true that $x_0=x_5$ ?

$\text{(A) 0} \quad \text{(B) 1} \quad \text{(C) 5} \quad \text{(D) 31} \quad \text{(E) }\infty$

Solution

1993 AHSME (Problems • Answer Key • Resources)
Preceded by 1992 AHSME	Followed by 1994 AHSME
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Difference between revisions of "1993 AHSME Problems"

Latest revision as of 07:15, 27 June 2021

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

See also

@@ Line 1: / Line 1: @@
+{{AHSME Problems
+|year = 1993
+}}
 == 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
@@ Line 8: / Line 11: @@
 == 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>\overbar{AB}</math> and <math>E</math> is on side <math>\overbar{BC}</math> If <math>DB=BE</math>, then <math>\angle BED=</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>
@@ Line 21: / Line 24: @@
 == 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]]
 == 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>
@@ Line 31: / Line 48: @@
 == 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]]
@@ Line 36: / Line 60: @@
 == 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]]
 == 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>
@@ Line 46: / Line 86: @@
 == 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]]
@@ Line 51: / Line 98: @@
 == 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]]
@@ Line 56: / Line 111: @@
 == 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]]
@@ Line 61: / Line 124: @@
 == 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]]
@@ Line 66: / Line 137: @@
 == 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]]
 == 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>
@@ Line 75: / Line 161: @@
 == 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]]
@@ Line 81: / Line 179: @@
 == 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]]
 == 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]]
@@ Line 91: / Line 205: @@
 == 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]]
 == 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]]
@@ Line 101: / Line 238: @@
 == 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]]
@@ Line 106: / Line 251: @@
 == 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]]
 == 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>
@@ Line 116: / Line 278: @@
 == Problem 22 ==
+<asy>
+size((400));
+draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0), linewidth(1));
+draw((5,0)--(10,0)--(15,0)--(20,0)--(20,5)--(15,5)--(10,5)--(5,5)--(6,7)--(11,7)--(16,7)--(21,7)--(21,2)--(20,0), linewidth(1));
+draw((10,0)--(10,5)--(11,7), linewidth(1));
+draw((15,0)--(15,5)--(16,7), linewidth(1));
+draw((20,0)--(20,5)--(21,7), linewidth(1));
+draw((0,5)--(1,7)--(6,7), linewidth(1));
+draw((3.5,7)--(4.5,9)--(9.5,9)--(14.5,9)--(19.5,9)--(18.5,7)--(19.5,9)--(19.5,7), linewidth(1));
+draw((8.5,7)--(9.5,9), linewidth(1));
+draw((13.5,7)--(14.5,9), linewidth(1));
+draw((7,9)--(8,11)--(13,11)--(18,11)--(17,9)--(18,11)--(18,9), linewidth(1));
+draw((12,9)--(13,11), linewidth(1));
+draw((10.5,11)--(11.5,13)--(16.5,13)--(16.5,11)--(16.5,13)--(15.5,11), linewidth(1));
+draw((25,0)--(30,0)--(30,5)--(25,5)--(25,0), dashed);
+draw((30,0)--(35,0)--(40,0)--(45,0)--(45,5)--(40,5)--(35,5)--(30,5)--(31,7)--(36,7)--(41,7)--(46,7)--(46,2)--(45,0), dashed);
+draw((35,0)--(35,5)--(36,7), dashed);
+draw((40,0)--(40,5)--(41,7), dashed);
+draw((45,0)--(45,5)--(46,7), dashed);
+draw((25,5)--(26,7)--(31,7), dashed);
+draw((28.5,7)--(29.5,9)--(34.5,9)--(39.5,9)--(44.5,9)--(43.5,7)--(44.5,9)--(44.5,7), dashed);
+draw((33.5,7)--(34.5,9), dashed);
+draw((38.5,7)--(39.5,9), dashed);
+draw((32,9)--(33,11)--(38,11)--(43,11)--(42,9)--(43,11)--(43,9), dashed);
+draw((37,9)--(38,11), dashed);
+draw((35.5,11)--(36.5,13)--(41.5,13)--(41.5,11)--(41.5,13)--(40.5,11), dashed);
+draw((50,0)--(55,0)--(55,5)--(50,5)--(50,0), dashed);
+draw((55,0)--(60,0)--(65,0)--(70,0)--(70,5)--(65,5)--(60,5)--(55,5)--(56,7)--(61,7)--(66,7)--(71,7)--(71,2)--(70,0), dashed);
+draw((60,0)--(60,5)--(61,7), dashed);
+draw((65,0)--(65,5)--(66,7), dashed);
+draw((70,0)--(70,5)--(71,7), dashed);
+draw((50,5)--(51,7)--(56,7), dashed);
+draw((53.5,7)--(54.5,9)--(59.5,9)--(64.5,9)--(69.5,9)--(68.5,7)--(69.5,9)--(69.5,7), dashed);
+draw((58.5,7)--(59.5,9), dashed);
+draw((63.5,7)--(64.5,9), dashed);
+draw((57,9)--(58,11)--(63,11)--(68,11)--(67,9)--(68,11)--(68,9), dashed);
+draw((62,9)--(63,11), dashed);
+draw((60.5,11)--(61.5,13)--(66.5,13)--(66.5,11)--(66.5,13)--(65.5,11), dashed);
+draw((75,0)--(80,0)--(80,5)--(75,5)--(75,0), dashed);
+draw((80,0)--(85,0)--(90,0)--(95,0)--(95,5)--(90,5)--(85,5)--(80,5)--(81,7)--(86,7)--(91,7)--(96,7)--(96,2)--(95,0), dashed);
+draw((85,0)--(85,5)--(86,7), dashed);
+draw((90,0)--(90,5)--(91,7), dashed);
+draw((95,0)--(95,5)--(96,7), dashed);
+draw((75,5)--(76,7)--(81,7), dashed);
+draw((78.5,7)--(79.5,9)--(84.5,9)--(89.5,9)--(94.5,9)--(93.5,7)--(94.5,9)--(94.5,7), dashed);
+draw((83.5,7)--(84.5,9), dashed);
+draw((88.5,7)--(89.5,9), dashed);
+draw((82,9)--(83,11)--(88,11)--(93,11)--(92,9)--(93,11)--(93,9), dashed);
+draw((87,9)--(88,11), dashed);
+draw((85.5,11)--(86.5,13)--(91.5,13)--(91.5,11)--(91.5,13)--(90.5,11), dashed);
+draw((28,6)--(33,6)--(38,6)--(43,6)--(43,11)--(38,11)--(33,11)--(28,11)--(28,6), linewidth(1));
+draw((28,11)--(29,13)--(34,13)--(39,13)--(44,13)--(43,11)--(44,13)--(44,8)--(43,6), linewidth(1));
+draw((33,6)--(33,11)--(34,13)--(39,13)--(38,11)--(38,6), linewidth(1));
+draw((31,13)--(32,15)--(37,15)--(36,13)--(37,15)--(42,15)--(41,13)--(42,15)--(42,13), linewidth(1));
+draw((34.5,15)--(35.5,17)--(40.5,17)--(39.5,15)--(40.5,17)--(40.5,15), linewidth(1));
+draw((53,6)--(58,6)--(63,6)--(68,6)--(68,11)--(63,11)--(58,11)--(53,11)--(53,6), dashed);
+draw((53,11)--(54,13)--(59,13)--(64,13)--(69,13)--(68,11)--(69,13)--(69,8)--(68,6), dashed);
+draw((58,6)--(58,11)--(59,13)--(64,13)--(63,11)--(63,6), dashed);
+draw((56,13)--(57,15)--(62,15)--(61,13)--(62,15)--(67,15)--(66,13)--(67,15)--(67,13), dashed);
+draw((59.5,15)--(60.5,17)--(65.5,17)--(64.5,15)--(65.5,17)--(65.5,15), dashed);
+draw((78,6)--(83,6)--(88,6)--(93,6)--(93,11)--(88,11)--(83,11)--(78,11)--(78,6), dashed);
+draw((78,11)--(79,13)--(84,13)--(89,13)--(94,13)--(93,11)--(94,13)--(94,8)--(93,6), dashed);
+draw((83,6)--(83,11)--(84,13)--(89,13)--(88,11)--(88,6), dashed);
+draw((81,13)--(82,15)--(87,15)--(86,13)--(87,15)--(92,15)--(91,13)--(92,15)--(92,13), dashed);
+draw((84.5,15)--(85.5,17)--(90.5,17)--(89.5,15)--(90.5,17)--(90.5,15), dashed);
+draw((56,12)--(61,12)--(66,12)--(66,17)--(61,17)--(56,17)--(56,12), linewidth(1));
+draw((61,12)--(61,17)--(62,19)--(57,19)--(56,17)--(57,19)--(67,19)--(66,17)--(67,19)--(67,14)--(66,12), linewidth(1));
+draw((59.5,19)--(60.5,21)--(65.5,21)--(64.5,19)--(65.5,21)--(65.5,19), linewidth(1));
+draw((81,12)--(86,12)--(91,12)--(91,17)--(86,17)--(81,17)--(81,12), dashed);
+draw((86,12)--(86,17)--(87,19)--(82,19)--(81,17)--(82,19)--(92,19)--(91,17)--(92,19)--(92,14)--(91,12), dashed);
+draw((84.5,19)--(85.5,21)--(90.5,21)--(89.5,19)--(90.5,21)--(90.5,19), dashed);
+draw((84,18)--(89,18)--(89,23)--(84,23)--(84,18)--(84,23)--(85,25)--(90,25)--(89,23)--(90,25)--(90,20)--(89,18), linewidth(1));</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]]
+== 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>
-== Problem 23 ==
+<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>
@@ Line 127: / Line 389: @@
 == 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]]
@@ Line 132: / Line 401: @@
 == 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]]
@@ Line 137: / Line 424: @@
 == 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]]
@@ Line 142: / Line 437: @@
 == 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]]
 == 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>
@@ Line 152: / Line 469: @@
 == 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]]
@@ Line 157: / Line 481: @@
 == Problem 30 ==
+Given <math>0\le x_0<1</math>, let
+<cmath>
+x_n=\left\{ \begin{array}{ll}
+x_{n-1} &\text{ if }2x_{n-1}<1 \\
+x_{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]]
 == See also ==
-* [[AHSME]]
-* [[AHSME Problems and Solutions]]
+* [[AMC 12 Problems and Solutions]]
 * [[Mathematics competition resources]]
+{{AHSME box|year=1993|before=[[1992 AHSME]]|after=[[1994 AHSME]]}}
+{{MAA Notice}}