Question bank AMC12 1997

If $a$ and $b$ are digits for which $$
\begin{array}{r} 2a\\[-2pt] {}\times b3\\ \hline 69\\ 92\phantom0\\ \hline 989 \end{array}
$$ then $a+b=$

The adjacent sides of the decagon shown meet at right angles. What is its perimeter?

If $x,$ $y,$ and $z$ are real numbers such that $$
\begin{aligned} (x-3)^2+(y-4)^2\\ {}+(z-5)^2&=0, \end{aligned}
$$ then $x+y+z=$

If $a$ is $50\%$ larger than $c,$ and $b$ is $25\%$ larger than $c,$ then $a$ is what percent larger than $b?$

A rectangle with perimeter $176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles?

Consider the sequence $$
1,-2,3,-4,5,-6,\ldots,
$$ whose $n$th term is $(-1)^{n+1}n.$ What is the average of the first $200$ terms of the sequence?

The sum of seven integers is $-1.$ What is the maximum number of the seven integers that can be larger than $13?$

Mientka Publishing Company prices its best seller *Where’s Walter?* as follows: $$
C(n)= \begin{cases} 12n, & 1\le n\le24,\\ 11n, & 25\le n\le48,\\ 10n, & 49\le n, \end{cases}
$$ where $n$ is the number of books ordered, and $C(n)$ is the cost in dollars of $n$ books. Notice that $25$ books cost less than $24$ books. For how many values of $n$ is it cheaper to buy more than $n$ books than to buy exactly $n$ books?

In the figure, $ABCD$ is a $2$ by $2$ square, $E$ is the midpoint of $\overline{AD},$ and $F$ is on $\overline{BE}.$ If $\overline{CF}$ is perpendicular to $\overline{BE},$ then the area of quadrilateral $CDEF$ is

Two six-sided dice are fair in the sense that each face is equally likely to turn up. However, one of the dice has the $4$ replaced by $3$ and the other die has the $3$ replaced by $4.$ When these dice are rolled, what is the probability that the sum is odd?

In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored $23,$ $14,$ $11,$ and $20$ points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than $18,$ what is the least number of points she could have scored in the tenth game?

If $m$ and $b$ are real numbers and $mb\gt0,$ then the line whose equation is $y=mx+b$ cannot contain the point

How many two-digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square?

The number of geese in a flock increases so that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1.$ If the populations in the years $1994,$ $1995,$ and $1997$ were $39,$ $60,$ and $123,$ respectively, then the population in $1996$ was

Medians $\overline{BD}$ and $\overline{CE}$ of triangle $ABC$ are perpendicular, $BD=8,$ and $CE=12.$ The area of triangle $ABC$ is

The three row sums and the three column sums of the array $$
\begin{bmatrix} 4&9&2\\ 8&1&6\\ 3&5&7 \end{bmatrix}
$$ are the same. What is the least number of entries that must be altered to make all six sums different from one another?

A line $x=k$ intersects the graph of $y=\log_5x$ and the graph of $y=\log_5(x+4).$ The distance between the points of intersection is $0.5.$ Given that $k=a+\sqrt b,$ where $a$ and $b$ are integers, what is $a+b?$

A list of integers has mode $32$ and mean $22.$ The smallest number in the list is $10.$ The median $m$ of the list is a member of the list. If the list member $m$ were replaced by $m+10,$ the mean and median of the new list would be $24$ and $m+10,$ respectively. If $m$ were instead replaced by $m-8,$ the median of the new list would be $m-4.$ What is $m?$

A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown, where $AB=1.$ To the nearest hundredth, what is the radius of the circle?

Which one of the following integers can be expressed as the sum of $100$ consecutive positive integers?