1999 AHSME Problems

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?

$\mathrm{(A) \ All\ equilateral\ triangles\ are\ congruent\ to\ each\ other.}$ $\mathrm{(B) \ All\ equilateral\ triangles\ are\ convex.}$ $\mathrm{(C) \ All\ equilateral\ triangles\ are\ equiangular.}$ $\mathrm{(D) \ All\ equilateral\ triangles\ are\ regular\ polygons.}$ $\mathrm{(E) \ All\ equilateral\ triangles\ are\ similar\ to\ each\ other.}$

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 Olymmpic 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 $[[1999 AHSME Problems/Problem 11|Solution]]

== Problem 12 == What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions$ (Error compiling LaTeX. Unknown error_msg)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$[[1999 AHSME Problems/Problem 12|Solution]]

== Problem 13 == Define a sequence of real numbers$ (Error compiling LaTeX. Unknown error_msg)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} $[[1999 AHSME Problems/Problem 13|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?$ (Error compiling LaTeX. Unknown error_msg) \mathrm{(A) \ 7 } \qquad \mathrm{(B) \ 8 } \qquad \mathrm{(C) \ 9 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 11 } $[[1999 AHSME Problems/Problem 14|Solution]]

== Problem 15 == Let$ (Error compiling LaTeX. Unknown error_msg)x $be a real number such that$ \sec x - \tan x = 2 $. Then$ \sec x + \tan x =$$ (Error compiling LaTeX. Unknown error_msg) \mathrm{(A) \ } 0.1 \qquad \mathrm{(B) \ } 0.2 \qquad \mathrm{(C) \ } 0.3 \qquad \mathrm{(D) \ } 0.4 \qquad \mathrm{(E) \ } 0.5$[[1999 AHSME Problems/Problem 15|Solution]]

== Problem 16 == What is the radius of a circle inscribed in a rhombus with diagonals of length$ (Error compiling LaTeX. Unknown error_msg)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 $[[1999 AHSME Problems/Problem 16|Solution]]

== Problem 17 == Let$ (Error compiling LaTeX. Unknown error_msg)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$[[1999 AHSME Problems/Problem 17|Solution]]

== Problem 18 == How many zeros does$ (Error compiling LaTeX. Unknown error_msg)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}$[[1999 AHSME Problems/Problem 18|Solution]]

== Problem 19 ==

Consider all triangles$ (Error compiling LaTeX. Unknown error_msg)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