Question bank AMC12 1994

$4^4\cdot9^4\cdot4^9\cdot9^9=$

A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle? $6$ $14$ ? $35$

How many of the following are equal to $x^x+x^x$ for all $x\gt0?$ $$
\begin{aligned} \mathrm{I:}\quad&2x^x\\ \mathrm{II:}\quad&x^{2x}\\ \mathrm{III:}\quad&(2x)^x\\ \mathrm{IV:}\quad&(2x)^{2x} \end{aligned}
$$

In the $xy$-plane, the segment with endpoints $(-5,0)$ and $(25,0)$ is the diameter of a circle. If the point $(x,15)$ is on the circle, then $x=$

Pat intended to multiply a number by $6$ but instead divided by $6.$ Pat then meant to add $14$ but instead subtracted $14.$ After these mistakes, the result was $16.$ If the correct operations had been used, the value produced would have been

In the sequence $$
\ldots,a,b,c,d,0,1,1,2,3,5,8,\ldots
$$ each term is the sum of the two terms to its left. Find $a.$

Squares $ABCD$ and $EFGH$ are congruent, $AB=10,$ and $G$ is the center of square $ABCD.$ The area of the region in the plane covered by these squares is

In the polygon shown, each side is perpendicular to its adjacent sides, and all $28$ of the sides are congruent. The perimeter of the polygon is $56.$ The area of the region bounded by the polygon is

If $\angle A$ is four times $\angle B,$ and the complement of $\angle B$ is four times the complement of $\angle A,$ then $\angle B=$

For distinct real numbers $x$ and $y,$ let $M(x,y)$ be the larger of $x$ and $y$ and let $m(x,y)$ be the smaller of $x$ and $y.$ If $$
a\lt b\lt c\lt d\lt e,
$$ then $$
\begin{aligned} &M\bigl(M(a,m(b,c)),\\ &\qquad m(d,m(a,e))\bigr)= \end{aligned}
$$

Three cubes of volume $1,$ $8$ and $27$ are glued together at their faces. The smallest possible surface area of the resulting configuration is

If $i^2=-1,$ then $$
\left(i-i^{-1}\right)^{-1}=
$$

In triangle $ABC,$ $AB=AC.$ If there is a point $P$ strictly between $A$ and $B$ such that $AP=PC=CB,$ then $\angle A=$

Find the sum of the arithmetic series $$
20+20\frac15+20\frac25+\cdots+40.
$$

For how many $n$ in $\{1,2,3,\ldots,100\}$ is the tens digit of $n^2$ odd?

Some marbles in a bag are red and the rest are blue. If one red marble is removed, then one-seventh of the remaining marbles are red. If two blue marbles are removed instead of one red, then one-fifth of the remaining marbles are red. How many marbles were in the bag originally?

An $8$ by $2\sqrt2$ rectangle has the same center as a circle of radius $2.$ The area of the region common to both the rectangle and the circle is

Triangle $ABC$ is inscribed in a circle, and $\angle B=\angle C=4\angle A.$ If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, then $n=$

Label one disk “$1$,” two disks “$2$,” three disks “$3$,” $\ldots,$ fifty disks “$50$.” Put these $1+2+3+\cdots+50=1275$ labeled disks in a box. Disks are then drawn from the box at random without replacement. The minimum number of disks that must be drawn to guarantee drawing at least ten disks with the same label is

Suppose $x,$ $y,$ $z$ is a geometric sequence with common ratio $r$ and $x\ne y.$ If $x,$ $2y,$ $3z$ is an arithmetic sequence, then $r$ is