Question bank AMC12 1995

Kim earned scores of $87,$ $83$ and $88$ on her first three mathematics examinations. If Kim receives a score of $90$ on the fourth exam, then her average will

If $\sqrt{2+\sqrt{x}}=3,$ then $x=$

The total in-store price for an appliance is $\$99.99.$ A television commercial advertises the same product for three easy payments of $\$29.98$ and a one-time shipping and handling charge of $\$9.98.$ How much is saved by buying the appliance from the television advertiser?

If $M$ is $30\%$ of $Q,$ $Q$ is $20\%$ of $P,$ and $N$ is $50\%$ of $P,$ then $\frac{M}{N}=$

A rectangular field is $300$ feet wide and $400$ feet long. Random sampling indicates that there are, on the average, three ants per square inch throughout the field. [$12$ inches = $1$ foot.] Of the following, the number that most closely approximates the number of ants in the field is

The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked $x?$

The radius of Earth at the equator is approximately $4000$ miles. Suppose a jet flies once around Earth at a speed of $500$ miles per hour relative to Earth. If the flight path is a negligible height above the equator, then, among the following choices, the best estimate of the number of hours of flight is

In triangle $ABC,$ $\angle C=90^\circ,$ $AC=6$ and $BC=8.$ Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC},$ respectively, and $\angle BED=90^\circ.$ If $DE=4,$ then $BD=$

Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is

The area of the triangle bounded by the lines $y=x,$ $y=-x$ and $y=6$ is

How many base $10$ four-digit numbers, $N=\underline{a\,b\,c\,d},$ satisfy all three of the following conditions?

(i) $4{,}000\le N\lt6{,}000;$

(ii) $N$ is a multiple of $5;$

(iii) $3\le b\lt c\le6.$

Let $f$ be a linear function with the properties that $f(1)\le f(2),$ $f(3)\ge f(4),$ and $f(5)=5.$ Which of the following statements is true?

The addition below is incorrect. The display can be made correct by changing one digit $d,$ wherever it occurs, to another digit $e.$ Find the sum of $d$ and $e.$

$$
\begin{array}{rrrrrrr}&7&4&2&5&8&6\\+&8&2&9&4&3&0\\\hline1&2&1&2&0&1&6\end{array}
$$

If $f(x)=ax^4-bx^2+x+5$ and $f(-3)=2,$ then $f(3)=$

Five points on a circle are numbered $1,$ $2,$ $3,$ $4,$ and $5$ in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point $5,$ after $1995$ jumps it will be on point

Anita attends a baseball game in Atlanta and estimates that there are $50{,}000$ fans in attendance. Bob attends a baseball game in Boston and estimates that there are $60{,}000$ fans in attendance. A league official who knows the actual numbers attending the two games notes that:

i. The actual attendance in Atlanta is within $10\%$ of Anita’s estimate.

ii. Bob’s estimate is within $10\%$ of the actual attendance in Boston.

To the nearest $1{,}000,$ the largest possible difference between the numbers attending the two games is

Given regular pentagon $ABCDE,$ a circle can be drawn that is tangent to $\overline{DC}$ at $D$ and to $\overline{AB}$ at $A.$ The number of degrees in minor arc $\overset{\frown}{AD}$ is

Two rays with common endpoint $O$ form a $30^\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB=1.$ The maximum possible length of $OB$ is

Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ as shown with $\overline{DE}\perp\overline{BC}.$ The ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$ is

If $a,$ $b$ and $c$ are three (not necessarily different) numbers chosen randomly and with replacement from the set $\{1,2,3,4,5\},$ the probability that $ab+c$ is even is