1996 AMC 12

The addition below is incorrect. What is the largest digit that can be changed to make the addition correct? $$
\begin{array}{r} 641\\ 852\\ {}+973\\ \hline 2456 \end{array}
$$

Each day Walter gets $\$3$ for doing his chores or $\$5$ for doing them exceptionally well. After $10$ days of doing his chores daily, Walter has received a total of $\$36.$ On how many days did Walter do them exceptionally well?

$\displaystyle\frac{(3!)!}{3!}=$

Six numbers from a list of nine integers are $7,$ $8,$ $3,$ $5,$ $9,$ and $5.$ The largest possible value of the median of all nine numbers in this list is

Given that $0\lt a\lt b\lt c\lt d,$ which of the following is the largest?

If $f(x)=x^{x+1}(x+2)^{x+3},$ then $$
\begin{gathered} f(0)+f(-1)\\ {}+f(-2)+f(-3)= \end{gathered}
$$

A father takes his twins and a younger child out to dinner on the twins’ birthday. The restaurant charges $\$4.95$ for the father and $\$0.45$ for each year of a child’s age, where age is defined as the age at the most recent birthday. If the bill is $\$9.45,$ which of the following could be the age of the youngest child?

If $3=k\cdot2^r$ and $15=k\cdot4^r,$ then $r=$

Triangle $PAB$ and square $ABCD$ are in perpendicular planes. Given that $PA=3,$ $PB=4,$ and $AB=5,$ what is $PD?$

How many line segments have both their endpoints located at the vertices of a given cube?

Given a circle of radius $2,$ there are many line segments of length $2$ that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.

A function $f$ from the integers to the integers is defined as follows: $$
f(n)= \begin{cases} n+3 & \text{if \(n\) is odd},\\ \frac{n}{2} & \text{if \(n\) is even}. \end{cases}
$$ Suppose $k$ is odd and $f(f(f(k)))=27.$ What is the sum of the digits of $k?$

Sunny runs at a steady rate, and Moonbeam runs $m$ times as fast, where $m$ is a number greater than $1.$ If Moonbeam gives Sunny a head start of $h$ meters, how many meters must Moonbeam run to overtake Sunny?

Let $E(n)$ denote the sum of the even digits of $n.$ For example, $E(5681)=6+8=14.$ Find $$
\begin{gathered} E(1)+E(2)+E(3)+\cdots\\ {}+E(100). \end{gathered}
$$

Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A.$ The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B.$ [See figure for $n=5,$ $m=7.$] What is the ratio of the area of triangle $A$ to the area of triangle $B?$

A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equals the third, what is the probability that at least one $2$ is tossed?

In rectangle $ABCD,$ angle $C$ is trisected by $\overline{CF}$ and $\overline{CE},$ where $E$ is on $\overline{AB},$ $F$ is on $\overline{AD},$ $BE=6,$ and $AF=2.$ Which of the following is closest to the area of the rectangle $ABCD?$

A circle of radius $2$ has center at $(2,0).$ A circle of radius $1$ has center at $(5,0).$ A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the $y$-intercept of the line?

The midpoints of the sides of a regular hexagon $ABCDEF$ are joined to form a smaller hexagon. What fraction of the area of $ABCDEF$ is enclosed by the smaller hexagon?

In the $xy$-plane, what is the length of the shortest path from $(0,0)$ to $(12,16)$ that does not go inside the circle $(x-6)^2+(y-8)^2=25?$