1999 AMC 12

$$
\begin{aligned} &1 - 2 + 3 - 4 + \cdots - 98 \\ &\quad {}+ 99 = \, ? \end{aligned}
$$

Which one of the following statements is false?

The number halfway between $\tfrac18$ and $\tfrac{1}{10}$ is

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 $5.$

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?

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

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

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 $1999?$

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?

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?

The student lockers at Olympic High are numbered consecutively beginning with locker number $1.$ The plastic digits used to number the lockers cost $2$ cents apiece. Thus, it costs $2$ cents to label locker number $9$ and $4$ cents to label locker number $10.$ If it costs $\$137.94$ to label all the lockers, how many lockers are there at the school?

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 $1?$

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

Four girls — Mary, Alina, Tina, and Hanna — 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?

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

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

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 $(x - 19)(x - 99)?$

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

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

The sequence $a_1,$ $a_2,$ $a_3,$ $\ldots$ satisfies $a_1 = 19,$ $a_9 = 99,$ and, for all $n \ge 3,$ $a_n$ is the arithmetic mean of the first $n - 1$ terms. Find $a_2.$