Question bank AMC12 1977

If $y=2x$ and $z=2y,$ then $x+y+z$ equals

Which one of the following statements is false? All equilateral triangles are

A man has $\$2.73$ in pennies, nickels, dimes, quarters and half dollars. If he has an equal number of coins of each kind, then the total number of coins he has is

In triangle $ABC,$ $AB=AC$ and $\angle A=80^\circ.$ If points $D,$ $E$ and $F$ lie on sides $BC,$ $AC$ and $AB,$ respectively, and $CE=CD$ and $BF=BD,$ then $\angle EDF$ equals

The set of all points $P$ such that the sum of the (undirected) distances from $P$ to two fixed points $A$ and $B$ equals the distance between $A$ and $B$ is

If $x,$ $y$ and $2x+\frac y2$ are not zero, then $$
\left(2x+\frac y2\right)^{-1}\left[(2x)^{-1}+\left(\frac y2\right)^{-1}\right]
$$ equals

If $t=\dfrac1{1-\sqrt[4]{2}},$ then $t$ equals

For every triple $(a,b,c)$ of nonzero real numbers, form the number $$
\frac a{|a|}+\frac b{|b|}+\frac c{|c|}+\frac{abc}{|abc|}.
$$ The set of all numbers formed is

In the adjoining figure $\angle E=40^\circ$ and arc $AB,$ arc $BC$ and arc $CD$ all have equal length. Find the measure of $\angle ACD.$

If $(3x-1)^7=a_7x^7+a_6x^6+\cdots+a_0,$ then $a_7+a_6+\cdots+a_0$ equals

For each real number $x,$ let $[x]$ be the largest integer not exceeding $x$ (i.e., the integer $n$ such that $n\le x\lt n+1$). Which of the following statements is (are) true?

$\mathrm{I}.$ $[x+1]=[x]+1$ for all $x$

$\mathrm{II}.$ $[x+y]=[x]+[y]$ for all $x$ and $y$

$\mathrm{III}.$ $[xy]=[x][y]$ for all $x$ and $y$

Al’s age is $16$ more than the sum of Bob’s age and Carl’s age, and the square of Al’s age is $1632$ more than the square of the sum of Bob’s age and Carl’s age. The sum of the ages of Al, Bob and Carl is

If $a_1,$ $a_2,$ $a_3,$ $\ldots$ is a sequence of positive numbers such that $a_{n+2}=a_na_{n+1}$ for all positive integers $n,$ then the sequence $a_1,$ $a_2,$ $a_3,$ $\ldots$ is a geometric progression

How many pairs $(m,n)$ of integers satisfy the equation $m+n=mn?$

Each of the three circles in the adjoining figure is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius three, then the perimeter of the triangle is

If $i^2=-1,$ then the sum $$
\begin{aligned} &\cos45^\circ+i\cos135^\circ+\cdots\\ &\quad+i^n\cos(45+90n)^\circ+\cdots\\ &\quad+i^{40}\cos3645^\circ \end{aligned}
$$ equals

Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one?

If $$
\begin{aligned} y={}&(\log_2 3)(\log_3 4)\cdots\\ &(\log_n[n+1])\cdots(\log_{31}32), \end{aligned}
$$ then

Let $E$ be the point of intersection of the diagonals of convex quadrilateral $ABCD,$ and let $P,$ $Q,$ $R$ and $S$ be the centers of the circles circumscribing triangles $ABE,$ $BCE,$ $CDE$ and $ADE,$ respectively. Then

For how many paths consisting of a sequence of horizontal and/or vertical line segments, with each segment connecting a pair of adjacent letters in the diagram below, is the word CONTEST spelled out as the path is traversed from beginning to end?