AHSME 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

$\mathrm{(A) \ \text{remain the same} } \mathrm{(B) \ \text{increase by 1} } \mathrm{(C) \ \text{increase by 2} } \mathrm{(D) \ \text{increase by 3} } \mathrm{(E) \ \text{increase by 4} }$

If $\sqrt {2 + \sqrt {x}} = 3$, then $x =$ $\mathrm{(A) \ 1 } \mathrm{(B) \ \sqrt{7} } \mathrm{(C) \ 7 } \mathrm{(D) \ 49 } \mathrm{(E) \ 121 }$

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

$\mathrm{(A) \ \frac {3}{250} } \mathrm{(B) \ \frac {3}{25} } \mathrm{(C) \ 1 } \mathrm{(D) \ \frac {6}{5} } \mathrm{(E) \ \frac {4}{3} }$

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 through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is

$\mathrm{(A) \ \text{500 thousand} } \mathrm{(B) \ \text{5 million} } \mathrm{(C) \ \text{50 million} } \mathrm{(D) \ \text{500 million} } \mathrm{(E) \ \text{5 billion} }$

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?

$\mathrm{(A) \ A } \mathrm{(B) \ B } \mathrm{(C) \ C } \mathrm{(D) \ D } \mathrm{(E) \ E }$

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 neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is:

$\mathrm{(A) \ 8 } \mathrm{(B) \ 25 } \mathrm{(C) \ 50 } \mathrm{(D) \ 75 } \mathrm{(E) \ 100 }$

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 $BD =$ $\mathrm{(A) \ 5 } \mathrm{(B) \ \frac {16}{3} } \mathrm{(C) \ \frac {20}{3} } \mathrm{(D) \ \frac {15}{2} } \mathrm{(E) \ 8 }$

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 $\mathrm{(A) \ 10 } \mathrm{(B) \ 12 } \mathrm{(C) \ 14 } \mathrm{(D) \ 16 } \mathrm{(E) \ 18 }$

The area of the triangle bounded by the lines $y = x, y = - x$ and $y = 6$ is $\mathrm{(A) \ 12 } \mathrm{(B) \ 12\sqrt{2} } \mathrm{(C) \ 24 } \mathrm{(D) \ 24\sqrt{2} } \mathrm{(E) \ 36 }$

How many base 10 four-digit numbers, $N = \underline{a} \underline{b} \underline{c} \underline{d}$, satisfy all three of the following conditions? (i) $4,000 \leq N < 6,000;$ (ii) $N$ is a multiple of 5; (iii) $3 \leq b < c \leq 6$.

$\mathrm{(A) \ 10 } \mathrm{(B) \ 18 } \mathrm{(C) \ 24 } \mathrm{(D) \ 36 } \mathrm{(E) \ 48 }$

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

$\mathrm{(A) \ f(0) 5 }$

If $f(x) = ax^4 - bx^2 + x + 5$ and $f( - 3) = 2$, then $f(3) =$ $\mathrm{(A) \ -5 } \mathrm{(B) \ -2 } \mathrm{(C) \ 1 } \mathrm{(D) \ 3 } \mathrm{(E) \ 8 }$

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 $\mathrm{(A) \ 1 } \mathrm{(B) \ 2 } \mathrm{(C) \ 3 } \mathrm{(D) \ 4 } \mathrm{(E) \ 5 }$

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 note 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 $\mathrm{(A) \ 10000 } \mathrm{(B) \ 11000 } \mathrm{(C) \ 20000 } \mathrm{(D) \ 21000 } \mathrm{(E) \ 22000 }$

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 $AD$ is

$\mathrm{(A) \ 72 } \mathrm{(B) \ 108 } \mathrm{(C) \ 120 } \mathrm{(D) \ 135 } \mathrm{(E) \ 144 }$

Two rays with common endpoint $O$ forms 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 $\mathrm{(A) \ 1 } \mathrm{(B) \ \frac {1 + \sqrt {3}}{\sqrt 2} } \mathrm{(C) \ \sqrt{3} } \mathrm{(D) \ 2 } \mathrm{(E) \ \frac{4}{\sqrt{3}} }$

Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $\overline{DE} \perp \overline{BC}$. The ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$ is $\mathrm{(A) \ \frac {1}{6} } \mathrm{(B) \ \frac {1}{4} } \mathrm{(C) \ \frac {1}{3} } \mathrm{(D) \ \frac {2}{5} } \mathrm{(E) \ \frac {1}{2} }$

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

$\mathrm{(A) \ \frac {2}{5} } \mathrm{(B) \ \frac {59}{125} } \mathrm{(C) \ \frac {1}{2} } \mathrm{(D) \ \frac {64}{125} } \mathrm{(E) \ \frac {3}{5} }$

Two nonadjacent vertices of a rectangle are $(4,3)$ and $(-4,-3)$, and the coordinates of the other two vertices are integers. The number of such rectangles is $\mathrm{(A) \ 1 } \mathrm{(B) \ 2 } \mathrm{(C) \ 3 } \mathrm{(D) \ 4 } \mathrm{(E) \ 5 }$

A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13, 19, 20, 25$ and $31$, although this is not necessarily their order around the pentagon. The area of the pentagon is $\mathrm{(A) \ 459 } \mathrm{(B) \ 600 } \mathrm{(C) \ 680 } \mathrm{(D) \ 720 } \mathrm{(E) \ 745 }$

The sides of a triangle have lengths $11,15,$ and $k$, where $k$ is an integer. For how many values of $k$ is the triangle obtuse? $\mathrm{(A) \ 5 } \mathrm{(B) \ 7 } \mathrm{(C) \ 12 } \mathrm{(D) \ 13 } \mathrm{(E) \ 14 }$

There exist positive integers $A,B$ and $C$, with no common factor greater than $1$, such that $$A \log_{200} 5 + B \log_{200} 2 = C.$$ What is $A + B + C$?

$\mathrm{(A) \ 6 } \mathrm{(B) \ 7 } \mathrm{(C) \ 8 } \mathrm{(D) \ 9 } \mathrm{(E) \ 10 }$

A list of five positive integers has mean $12$ and range $18$. The mode and median are both $8$. How many different values are possible for the second largest element of the list? $\mathrm{(A) \ 4 } \mathrm{(B) \ 6 } \mathrm{(C) \ 8 } \mathrm{(D) \ 10 } \mathrm{(E) \ 12 }$

In the figure, $\overline{AB}$ and $\overline{CD}$ are diameters of the circle with center $O$, $\overline{AB} \perp \overline{CD}$, and chord $\overline{DF}$ intersects $\overline{AB}$ at $E$. If $DE = 6$ and $EF = 2$, then the area of the circle is

$\mathrm{(A) \ 23 \pi } \mathrm{(B) \ \frac {47}{2} \pi } \mathrm{(C) \ 24 \pi } \mathrm{(D) \ \frac {49}{2} \pi } \mathrm{(E) \ 25 \pi }$

For how many three-element sets of distinct positive integers $\{a,b,c\}$ is it true that $a \times b \times c = 2310$?

$\mathrm{(A) \ 32 } \mathrm{(B) \ 36 } \mathrm{(C) \ 40 } \mathrm{(D) \ 43 } \mathrm{(E) \ 45 }$

A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is $\mathrm{(A) \ 16 } \mathrm{(B) \ 17 } \mathrm{(C) \ 18 } \mathrm{(D) \ 19 } \mathrm{(E) \ 20 }$