Question bank AMC12 1993

For integers $a,$ $b,$ and $c,$ define $\boxed{a,b,c}$ to mean $a^b-b^c+c^a.$ Then $\boxed{1,-1,2}$ equals

In $\triangle ABC,$ $\angle A=55^\circ,$ $\angle C=75^\circ,$ $D$ is on side $\overline{AB}$ and $E$ is on side $\overline{BC}.$ If $DB=BE,$ then $\angle BED=$

$$
\frac{15^{30}}{45^{15}}=
$$

Define the operation “$\circ$” by $x\circ y=4x-3y+xy,$ for all real numbers $x$ and $y.$ For how many real numbers $y$ does $3\circ y=12?$

Last year a bicycle cost $\$160$ and a cycling helmet cost $\$40.$ This year the cost of the bicycle increased by $5\%,$ and the cost of the helmet increased by $10\%.$ The percent increase in the combined cost of the bicycle and the helmet is

$$
\sqrt{\frac{8^{10}+4^{10}}{8^4+4^{11}}}=
$$

The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111,$ $R_5=11111,$ etc. When $R_{24}$ is divided by $R_4,$ the quotient $Q=\frac{R_{24}}{R_4}$ is an integer whose base-ten representation is a sequence containing only ones and zeros. The number of zeros in $Q$ is

Let $C_1$ and $C_2$ be circles of radius $1$ that are in the same plane and tangent to each other. How many circles of radius $3$ are in this plane and tangent to both $C_1$ and $C_2?$

Country $A$ has $c\%$ of the world’s population and owns $d\%$ of the world’s wealth. Country $B$ has $e\%$ of the world’s population and $f\%$ of its wealth. Assume that the citizens of $A$ share the wealth of $A$ equally, and assume that those of $B$ share the wealth of $B$ equally. Find the ratio of the wealth of a citizen of $A$ to the wealth of a citizen of $B.$

Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a,$ $b\gt0.$ If $r$ equals the product of $a^b$ and $x^b$ where $x\gt0,$ then $x=$

If $\log_2(\log_2(\log_2(x)))=2,$ then how many digits are in the base-ten representation for $x?$

If $f(2x)=\frac{2}{2+x}$ for all $x\gt0,$ then $2f(x)=$

A square of perimeter $20$ is inscribed in a square of perimeter $28.$ What is the greatest distance between a vertex of the inner square and a vertex of the outer square?

The convex pentagon $ABCDE$ has $\angle A=\angle B=120^\circ,$ $EA=AB=BC=2$ and $CD=DE=4.$ What is the area of $ABCDE?$

For how many values of $n$ will an $n$-sided regular polygon have interior angles with integral degree measures?

Consider the non-decreasing sequence of positive integers $$
\begin{aligned} &1,2,2,3,3,3,4,4,4,4,\\ &5,5,5,5,5,\ldots \end{aligned}
$$ in which the $n$th positive integer appears $n$ times. The remainder when the $1993$rd term is divided by $5$ is

Amy painted a dart board over a square clock face using the “hour positions” as boundaries. [See figure.] If $t$ is the area of one of the eight triangular regions such as that between $12$ o’clock and $1$ o’clock, and $q$ is the area of one of the four corner quadrilaterals such as that between $1$ o’clock and $2$ o’clock, then $\frac qt=$

Al and Barb start their new jobs on the same day. Al’s schedule is $3$ work-days followed by $1$ rest-day. Barb’s schedule is $7$ work-days followed by $3$ rest-days. On how many of their first $1000$ days do both have rest-days on the same day?

How many ordered pairs $(m,n)$ of positive integers are solutions to $$
\frac4m+\frac2n=1?
$$

Consider the equation $10z^2-3iz-k=0,$ where $z$ is a complex variable and $i^2=-1.$ Which of the following statements is true?