AHSME 1996

Each day Walter gets $3$ dollars for doing his chores or $5$ dollars for doing them exceptionally well. After $10$ days of doing his chores daily, Walter has received a total of $36$ dollars. On how many days did Walter do them exceptionally well? $\text{(A)}\ 3 \text{(B)}\ 4 \text{(C)}\ 5 \text{(D)}\ 6 \text{(E)}\ 7$

$\frac{(3!)!}{3!}=$ $\text{(A)}\ 1 \text{(B)}\ 2 \text{(C)}\ 6 \text{(D)}\ 40 \text{(E)}\ 120$

Six numbers from a list of nine integers are $7,8,3,5,9$ and $5$. The largest possible value of the median of all nine numbers in this list is $\text{(A)}\ 5 \text{(B)}\ 6 \text{(C)}\ 7 \text{(D)}\ 8 \text{(E)}\ 9$

Given that $0 < a < b < c < d$, which of the following is the largest? $\text{(A)}\ \frac{a+b}{c+d} \text{(B)}\ \frac{a+d}{b+c} \text{(C)}\ \frac{b+c}{a+d} \text{(D)}\ \frac{b+d}{a+c} \text{(E)}\ \frac{c+d}{a+b}$

If $f(x) = x^{(x+1)}(x+2)^{(x+3)}$, then $f(0)+f(-1)+f(-2)+f(-3) =$ $\text{(A)}\ -\frac{8}{9} \text{(B)}\ 0 \text{(C)}\ \frac{8}{9} \text{(D)}\ 1 \text{(E)}\ \frac{10}{9}$

A father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges $4.95$ for the father and $0.45$ for each year of a child's age, where age is defined as the age at the most recent birthday. If the bill is $9.45$, which of the following could be the age of the youngest child? $\text{(A)}\ 1 \text{(B)}\ 2 \text{(C)}\ 3 \text{(D)}\ 4 \text{(E)}\ 5$

If $3 = k\cdot 2^r$ and $15 = k\cdot 4^r$, then $r =$ $\text{(A)}\ -\log_{2}5 \text{(B)}\ \log_{5}2 \text{(C)}\ \log_{10}5 \text{(D)}\ \log_{2}5 \text{(E)}\ \frac{5}{2}$

Triangle $PAB$ and square $ABCD$ are in perpendicular planes. Given that $PA = 3, PB = 4$ and $AB = 5$, what is $PD$?

$\text{(A)}\ 5 \text{(B)}\ \sqrt{34} \text{(C)}\ \sqrt{41} \text{(D)}\ 2\sqrt{13} \text{(E)}\ 8$

Given a circle of radius $2$, there are many line segments of length $2$ that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments. $\text{(A)}\ \frac{\pi} 4 \text{(B)}\ 4-\pi \text{(C)}\ \frac{\pi} 2 \text{(D)}\ \pi \text{(E)}\ 2\pi$

A function $f$ from the integers to the integers is defined as follows: $$f(n) =\begin{cases}n+3 &\text{if n is odd}\\ \ n/2 &\text{if n is even}\end{cases}$$ Suppose $k$ is odd and $f(f(f(k))) = 27$. What is the sum of the digits of $k$? $\textbf{(A)}\ 3 \textbf{(B)}\ 6 \textbf{(C)}\ 9 \textbf{(D)}\ 12 \textbf{(E)}\ 15$

Let $E(n)$ denote the sum of the even digits of $n$. For example, $E(5681) = 6+8 = 14$. Find $E(1)+E(2)+E(3)+\cdots+E(100)$ $\text{(A)}\ 200 \text{(B)}\ 360 \text{(C)}\ 400 \text{(D)}\ 900 \text{(E)}\ 2250$

Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n=5, m=7$.] What is the ratio of the area of triangle $A$ to the area of triangle $B$?

$\text{(A)}\ 1 \text{(B)}\ m/n \text{(C)}\ n/m \text{(D)}\ 2m/n \text{(E)}\ 2n/m$

A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equal the third, what is the probability that at least one "2" is tossed? $\text{(A)}\ \frac{1}{6} \text{(B)}\ \frac{91}{216} \text{(C)}\ \frac{1}{2} \text{(D)}\ \frac{8}{15} \text{(E)}\ \frac{7}{12}$

In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE=6$ and $AF=2$. Which of the following is closest to the area of the rectangle $ABCD$?

$\text{(A)}\ 110 \text{(B)}\ 120 \text{(C)}\ 130 \text{(D)}\ 140 \text{(E)}\ 150$

A circle of radius $2$ has center at $(2,0)$. A circle of radius $1$ has center at $(5,0)$. A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the $y$-intercept of the line? $\text{(A)}\ \sqrt{2}/4 \text{(B)}\ 8/3 \text{(C)}\ 1+\sqrt 3 \text{(D)}\ 2\sqrt 2 \text{(E)}\ 3$

The midpoints of the sides of a regular hexagon $ABCDEF$ are joined to form a smaller hexagon. What fraction of the area of $ABCDEF$ is enclosed by the smaller hexagon?

$\text{(A)}\ \frac{1}{2} \text{(B)}\ \frac{\sqrt 3}{3} \text{(C)}\ \frac{2}{3} \text{(D)}\ \frac{3}{4} \text{(E)}\ \frac{\sqrt 3}{2}$

In the xy-plane, what is the length of the shortest path from $(0,0)$ to $(12,16)$ that does not go inside the circle $(x-6)^{2}+(y-8)^{2}= 25$? $\text{(A)}\ 10\sqrt 3 \text{(B)}\ 10\sqrt 5 \text{(C)}\ 10\sqrt 3+\frac{ 5\pi}{3} \text{(D)}\ 40\frac{\sqrt{3}}{3} \text{(E)}\ 10+5\pi$

Triangles $ABC$ and $ABD$ are isosceles with $AB=AC=BD$, and $BD$ intersects $AC$ at $E$. If $BD$ is perpendicular to $AC$, then $\angle C+\angle D$ is

$\text{(A)}\ 115^\circ \text{(B)}\ 120^\circ \text{(C)}\ 130^\circ \text{(D)}\ 135^\circ \text{(E)}\ \text{not uniquely determined}$

Four distinct points, $A$, $B$, $C$, and $D$, are to be selected from $1996$ points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord $AB$ intersects the chord $CD$? $\text{(A)}\ \frac{1}{4} \text{(B)}\ \frac{1}{3} \text{(C)}\ \frac{1}{2} \text{(D)}\ \frac{2}{3} \text{(E)}\ \frac{3}{4}$

The sum of the lengths of the twelve edges of a rectangular box is $140$, and the distance from one corner of the box to the farthest corner is $21$. The total surface area of the box is $\text{(A)}\ 776 \text{(B)}\ 784 \text{(C)}\ 798 \text{(D)}\ 800 \text{(E)}\ 812$

The sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2,\ldots$ consists of $1$'s separated by blocks of $2$'s with $n$ $2$'s in the $n^{th}$ block. The sum of the first $1234$ terms of this sequence is $\text{(A)}\ 1996 \text{(B)}\ 2419 \text{(C)}\ 2429 \text{(D)}\ 2439 \text{(E)}\ 2449$

Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have? $\text{(A)}\ 72 \text{(B)}\ 73 \text{(C)}\ 74 \text{(D)}\ 75 \text{(E)}\ 76$

An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely: (a) the selection of four red marbles; (b) the selection of one white and three red marbles; (c) the selection of one white, one blue, and two red marbles; and (d) the selection of one marble of each color. What is the smallest number of marbles satisfying the given condition? $\text{(A)}\ 19 \text{(B)}\ 21 \text{(C)}\ 46 \text{(D)}\ 69 \text{(E)}\ \text{more than 69}$

Consider two solid spherical balls, one centered at $\left(0, 0,\frac{21}{2}\right)$ with radius $6$, and the other centered at $(0, 0, 1)$ with radius $\frac{9}{2}$. How many points with only integer coordinates (lattice points) are there in the intersection of the balls? $\text{(A)}\ 7 \text{(B)}\ 9 \text{(C)}\ 11 \text{(D)}\ 13 \text{(E)}\ 15$

On a $4\times 4\times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to

$\text{(A)}\ 1.6 \text{(B)}\ 1.9 \text{(C)}\ 2.1 \text{(D)}\ 2.7 \text{(E)}\ 2.9$

If $n$ is a positive integer such that $2n$ has $28$ positive divisors and $3n$ has $30$ positive divisors, then how many positive divisors does $6n$ have? $\text{(A)}\ 32 \text{(B)}\ 34 \text{(C)}\ 35 \text{(D)}\ 36 \text{(E)}\ 38$

A hexagon inscribed in a circle has three consecutive sides each of length 3 and three consecutive sides each of length 5. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length 3 and the other with three sides each of length 5, has length equal to $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. $\textbf{(A)}\ 309 \textbf{(B)}\ 349 \textbf{(C)}\ 369 \textbf{(D)}\ 389 \textbf{(E)}\ 409$