Difference between revisions of "1999 AHSME Problems"

Latest revision as of 17:28, 19 June 2023

1999 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.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30

Problem 1

$1 - 2 + 3 -4 + \cdots - 98 + 99 =$

$\mathrm{(A) \ -50 } \qquad \mathrm{(B) \ -49 } \qquad \mathrm{(C) \ 0 } \qquad \mathrm{(D) \ 49 } \qquad \mathrm{(E) \ 50 }$

Solution

Problem 2

Which of the following statements is false?

$\text{(A) All equilateral triangles are congruent to each other.}$

$\text{(B) All equilateral triangles are convex.}$

$\text{(C) All equilateral triangles are equiangular.}$

$\text{(D) All equilateral triangles are regular polygons.}$

$\text{(E) All equilateral triangles are similar to each other.}$

Solution

Problem 3

The number halfway between $1/8$ and $1/10$ is

$\mathrm{(A) \ } \frac 1{80} \qquad \mathrm{(B) \ } \frac 1{40} \qquad \mathrm{(C) \ } \frac 1{18} \qquad \mathrm{(D) \ } \frac 1{9} \qquad \mathrm{(E) \ } \frac 9{80}$

Solution

Problem 4

Find the sum of all prime numbers between $1$ and $100$ that are simultaneously $1$ greater than a multiple of $4$ and $1$ less than a multiple of $5$ .

$\mathrm{(A) \ } 118 \qquad \mathrm{(B) \ }137 \qquad \mathrm{(C) \ } 158 \qquad \mathrm{(D) \ } 187 \qquad \mathrm{(E) \ } 245$

Solution

Problem 5

The marked price of a book was $30 \%$ less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay?

$\mathrm{(A) \ }25 \% \qquad \mathrm{(B) \ }30 \% \qquad \mathrm{(C) \ }35 \% \qquad \mathrm{(D) \ }60 \% \qquad \mathrm{(E) \ }65 \%$

Solution

Problem 6

What is the sum of the digits of the decimal form of the product $2^{1999} \cdot 5^{2001}$ ?

$\mathrm{(A) \ }2 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \ }7 \qquad \mathrm{(E) \ }10$

Solution

Problem 7

What is the largest number of acute angles that a convex hexagon can have?

$\mathrm{(A) \ }2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ }4 \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ }6$

Solution

Problem 8

At the end of 1994 Walter was half as old as his grandmother. The sum of the years in which they were born is 3838. How old will Walter be at the end of 1999?

$\mathrm{(A) \ } 48 \qquad \mathrm{(B) \ }49 \qquad \mathrm{(C) \ }53 \qquad \mathrm{(D) \ }55 \qquad \mathrm{(E) \ } 101$

Solution

Problem 9

Before Ashley started a three-hour drive, her car's odometer reading was 29792, a palindrome. (A palindrome is a number that reads the same way from left to right as it does from right to left). At her destination, the odometer reading was another palindrome. If Ashley never exceeded the speed limit of 75 miles per hour, which of the following was her greatest possible average speed?

$\mathrm{(A) \ } 33\frac 13 \qquad \mathrm{(B) \ }53\frac 13 \qquad \mathrm{(C) \ }66\frac 23 \qquad \mathrm{(D) \ }70\frac 13 \qquad \mathrm{(E) \ } 74\frac 13$

Solution

Problem 10

A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false.

I. The digit is 1.

II. the digit is not 2.

III. The digit is 3.

IV. The digit is not 4.

Which one of the following must necessarily be correct?

$\mathrm{(A) \ I\ is\ true} \qquad \mathrm{(B) \ I\ is\ false} \qquad \mathrm{(C) \ II\ is\ true} \qquad \mathrm{(D) \ III\ is\ true} \qquad \mathrm{(E) \ IV\ is\ false}$

Solution

Problem 11

The student lockers at Olympic High are numbered consecutively beginning with locker number $1$ . The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four cents to label locker number $10$ . If it costs $137.94$ to label all the lockers, how many lockers are there at the school?

$\mathrm{(A) \ }2001 \qquad \mathrm{(B) \ }2010 \qquad \mathrm{(C) \ }2100 \qquad \mathrm{(D) \ }2726 \qquad \mathrm{(E) \ }6897$

Solution

Problem 12

What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $y = p(x)$ and $y = q(x)$ , each with leading coefficient $1$ ?

$\textrm{(A)} \ 1 \qquad \textrm{(B)} \ 2 \qquad \textrm{(C)} \ 3 \qquad \textrm{(D)} \ 4 \qquad \textrm{(E)} \ 8$

Solution

Problem 13

Define a sequence of real numbers $a_1, a_2, a_3, \ldots$ by $a_1 = 1$ and $a_{n+1}^3 = 99a_n^3$ for all $n \ge 1$ . Then $a_{100}$ equals

$\mathrm{(A) \ } 33^{33} \qquad \mathrm{(B) \ } 33^{99} \qquad \mathrm{(C) \ } 99^{33} \qquad \mathrm{(D) \ }99^{99} \qquad \mathrm{(E) \ none\ of\ the\ above}$

Solution

Problem 14

Four girls - Mary, Aline, Tina, and Hana - sang songs in a concert as trios, with one girl sitting out each time. Hanna sang 7 songs, which was more than any other girl, and Mary sang 4 songs, which was fewer than any other girl. How many songs did these trios sing?

$\mathrm{(A) \ 7 } \qquad \mathrm{(B) \ 8 } \qquad \mathrm{(C) \ 9 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 11 }$

Solution

Problem 15

Let $x$ be a real number such that $\sec x - \tan x = 2$ . Then $\sec x + \tan x =$

$\mathrm{(A) \ } 0.1 \qquad \mathrm{(B) \ } 0.2 \qquad \mathrm{(C) \ } 0.3 \qquad \mathrm{(D) \ } 0.4 \qquad \mathrm{(E) \ } 0.5$

Solution

Problem 16

What is the radius of a circle inscribed in a rhombus with diagonals of length $10$ and $24$ ?

$\mathrm{(A) \ }4 \qquad \mathrm{(B) \ }\frac {58}{13} \qquad \mathrm{(C) \ }\frac{60}{13} \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ }6$

Solution

Problem 17

Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x-19$ , the remainder is $99$ , and when $P(x)$ is divided by $x - 99$ , the remainder is $19$ . What is the remainder when $P(x)$ is divided by $(x-19)(x-99)$ ?

$\mathrm{(A) \ } -x + 80 \qquad \mathrm{(B) \ } x + 80 \qquad \mathrm{(C) \ } -x + 118 \qquad \mathrm{(D) \ } x + 118 \qquad \mathrm{(E) \ } 0$

Solution

Problem 18

How many zeros does $f(x) = \cos(\log x)$ have on the interval $0 < x < 1$ ?

$\mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ } 10 \qquad \mathrm{(E) \ } \text{infinitely\ many}$

Solution

Problem 19

Consider all triangles $ABC$ satisfying in the following conditions: $AB = AC$ , $D$ is a point on $\overline{AC}$ for which $\overline{BD} \perp \overline{AC}$ , $AC$ and $CD$ are integers, and $BD^{2} = 57$ . Among all such triangles, the smallest possible value of $AC$ is

$[asy] pair A,B,C,D; A=(5,12); B=origin; C=(10,0); D=(8.52071005917,3.55029585799); draw(A--B--C--cycle); draw(B--D); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NE); [/asy]$

$\textrm{(A)} \ 9 \qquad \textrm{(B)} \ 10 \qquad \textrm{(C)} \ 11 \qquad \textrm{(D)} \ 12 \qquad \textrm{(E)} \ 13$

Solution

Problem 20

The sequence $a_{1},a_{2},a_{3},\ldots$ satisfies $a_{1} = 19,a_{9} = 99$ , and, for all $n\geq 3$ , $a_{n}$ is the arithmetic mean of the first $n - 1$ terms. Find $a_2$ .

$\textrm{(A)} \ 29 \qquad \textrm{(B)} \ 59 \qquad \textrm{(C)} \ 79 \qquad \textrm{(D)} \ 99 \qquad \textrm{(E)} \ 179$

Solution

Problem 21

A circle is circumscribed about a triangle with sides $20,21,$ and $29,$ thus dividing the interior of the circle into four regions. Let $A,B,$ and $C$ be the areas of the non-triangular regions, with $C$ be the largest. Then

$\mathrm{(A) \ }A+B=C \qquad \mathrm{(B) \ }A+B+210=C \qquad \mathrm{(C) \ }A^2+B^2=C^2 \qquad \mathrm{(D) \ }20A+21B=29C \qquad \mathrm{(E) \ } \frac 1{A^2}+\frac 1{B^2}= \frac 1{C^2}$

Solution

Problem 22

The graphs of $y = -|x-a| + b$ and $y = |x-c| + d$ intersect at points $(2,5)$ and $(8,3)$ . Find $a+c$ .

$\mathrm{(A) \ } 7 \qquad \mathrm{(B) \ } 8 \qquad \mathrm{(C) \ } 10 \qquad \mathrm{(D) \ } 13\qquad \mathrm{(E) \ } 18$

Solution

Problem 23

The equiangular convex hexagon $ABCDEF$ has $AB = 1, BC = 4, CD = 2,$ and $DE = 4.$ The area of the hexagon is

$\mathrm{(A) \ } \frac {15}2\sqrt{3} \qquad \mathrm{(B) \ }9\sqrt{3} \qquad \mathrm{(C) \ }16 \qquad \mathrm{(D) \ }\frac{39}4\sqrt{3} \qquad \mathrm{(E) \ } \frac{43}4\sqrt{3}$

Solution

Problem 24

Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords form a convex quadrilateral?

$\mathrm{(A) \ } \frac 1{15} \qquad \mathrm{(B) \ } \frac 1{91} \qquad \mathrm{(C) \ } \frac 1{273} \qquad \mathrm{(D) \ } \frac 1{455} \qquad \mathrm{(E) \ } \frac 1{1365}$

Solution

Problem 25

There are unique integers $a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}$ such that

$\frac {5}{7} = \frac {a_{2}}{2!} + \frac {a_{3}}{3!} + \frac {a_{4}}{4!} + \frac {a_{5}}{5!} + \frac {a_{6}}{6!} + \frac {a_{7}}{7!}$

where $0\leq a_{i} < i$ for $i = 2,3,\ldots,7$ . Find $a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7}$ .

$\textrm{(A)} \ 8 \qquad \textrm{(B)} \ 9 \qquad \textrm{(C)} \ 10 \qquad \textrm{(D)} \ 11 \qquad \textrm{(E)} \ 12$

Solution

Problem 26

Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$ . The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\circ}$ . Thus the three polygons form a new polygon with $A$ as an interior point. What is the largest possible perimeter that this polygon can have?

$\mathrm{(A) \ }12 \qquad \mathrm{(B) \ }14 \qquad \mathrm{(C) \ }18 \qquad \mathrm{(D) \ }21 \qquad \mathrm{(E) \ } 24$

Solution

Problem 27

In triangle $ABC$ , $3 \sin A + 4 \cos B = 6$ and $4 \sin B + 3 \cos A = 1$ . Then $\angle C$ in degrees is

$\mathrm{(A) \ }30 \qquad \mathrm{(B) \ }60 \qquad \mathrm{(C) \ }90 \qquad \mathrm{(D) \ }120 \qquad \mathrm{(E) \ }150$

Solution

Problem 28

Let $x_1, x_2, \ldots , x_n$ be a sequence of integers such that

$\text{(i)}$ $-1 \le x_i \le 2$ $\text{for}$ $i = 1,2, \ldots n$

$\text{(ii)}$ $x_1 + \cdots + x_n = 19$ ; $\text{and}$

$\text{(iii)}$ $x_1^2 + x_2^2 + \cdots + x_n^2 = 99$ .

Let $m$ and $M$ be the minimal and maximal possible values of $x_1^3 + \cdots + x_n^3$ , respectively. Then $\frac Mm =$

$\mathrm{(A) \ }3 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \ }6 \qquad \mathrm{(E) \ }7$

Solution

Problem 29

A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $P$ is selected at random inside the circumscribed sphere. The probability that $P$ lies inside one of the five small spheres is closest to

$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }0.1 \qquad \mathrm{(C) \ }0.2 \qquad \mathrm{(D) \ }0.3 \qquad \mathrm{(E) \ }0.4$

Solution

Problem 30

The number of ordered pairs of integers $(m,n)$ for which $mn \ge 0$ and

$m^3 + n^3 + 99mn = 33^3$

is equal to

$\mathrm{(A) \ }2 \qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 33\qquad \mathrm{(D) \ }35 \qquad \mathrm{(E) \ } 99$

Solution

@@ Line 1: / Line 1: @@
+{{AHSME Problems|year=1999}}
 == Problem 1 ==
 <math>1 - 2 + 3 -4 + \cdots - 98 + 99 = </math>
@@ Line 9: / Line 10: @@
 Which of the following statements is false?
-<math> \mathrm{(A) \ All\ equilateral\ triangles\ are\ congruent\ to\ each\ other.}</math>
+<math>\text{(A) All equilateral triangles are congruent to each other.}</math>
-<math>\mathrm{(B) \  All\ equilateral\ triangles\ are\ convex.}</math>
-<math>\mathrm{(C) \  All\ equilateral\ triangles\ are\ equiangular.}</math>
+<math>\text{(B) All equilateral triangles are convex.}</math>
-<math>\mathrm{(D) \  All\ equilateral\ triangles\ are\ regular\ polygons.}</math>
-<math>\mathrm{(E) \  All\ equilateral\ triangles\ are\ similar\ to\ each\ other.}  </math>
+<math>\text{(C) All equilateral triangles are equiangular.}</math>
+<math>\text{(D) All equilateral triangles are regular polygons.}</math>
+<math>\text{(E) All equilateral triangles are similar to each other.}</math>
 [[1999 AHSME Problems/Problem 2|Solution]]
@@ Line 84: / Line 89: @@
 == Problem 11 ==
-The student lockers at Olymmpic High are numbered consecutively beginning with locker number <math>1</math>. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number <math>9</math> and four cents to label locker number <math>10</math>. If it costs <math>\</math> <math> 137.94</math> to label all the lockers, how many lockers are there at the school?
+The student lockers at Olympic High are numbered consecutively beginning with locker number <math>1</math>. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number <math>9</math> and four cents to label locker number <math>10</math>. If it costs <math>137.94</math> to label all the lockers, how many lockers are there at the school?
 <math> \mathrm{(A) \ }2001 \qquad \mathrm{(B) \ }2010 \qquad \mathrm{(C) \ }2100 \qquad \mathrm{(D) \ }2726 \qquad \mathrm{(E) \ }6897 </math>
@@ Line 216: / Line 221: @@
 == Problem 28 ==
 Let <math>x_1, x_2, \ldots , x_n</math> be a sequence of integers such that
-(i) <math>-1 \le x_i \le 2</math> for <math>i = 1,2, \ldots n</math>
-(ii) <math>x_1 + \cdots + x_n = 19</math>; and
+<math>\text{(i)}</math> <math>-1 \le x_i \le 2</math> <math>\text{for}</math> <math>i = 1,2, \ldots n</math>
-(iii) <math>x_1^2 + x_2^2 + \cdots + x_n^2 = 99</math>.
+<math>\text{(ii)}</math> <math>x_1 + \cdots + x_n = 19</math>; <math>\text{and}</math>
+<math>\text{(iii)}</math> <math>x_1^2 + x_2^2 + \cdots + x_n^2 = 99</math>.
 Let <math>m</math> and <math>M</math> be the minimal and maximal possible values of <math>x_1^3 + \cdots + x_n^3</math>, respectively. Then <math>\frac Mm =</math>
@@ Line 226: / Line 235: @@
 == Problem 29 ==
-A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. a point <math>P</math> is selected at random inside the circumscribed sphere. The probability that <math>P</math> lies inside one of the five small spheres is closest to
+A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point <math>P</math> is selected at random inside the circumscribed sphere. The probability that <math>P</math> lies inside one of the five small spheres is closest to
 <math> \mathrm{(A) \ }0 \qquad \mathrm{(B) \ }0.1 \qquad \mathrm{(C) \ }0.2 \qquad \mathrm{(D) \ }0.3 \qquad \mathrm{(E) \ }0.4 </math>
@@ Line 244: / Line 253: @@
-== See also ==
+== See Also ==
-*[[AHSME]]
+* [[AMC 12 Problems and Solutions]]
+* [[Mathematics competition resources]]
+{{AHSME box|year=1999|before=[[1998 AHSME]]|after=Last AHSME, see [[2000 AMC 12]]}}
+{{MAA Notice}}

1999 AHSME (Problems • Answer Key • Resources)
Preceded by 1998 AHSME	Followed by Last AHSME, see 2000 AMC 12
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Difference between revisions of "1999 AHSME Problems"

Latest revision as of 17:28, 19 June 2023

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