1989 AHSME Problems

Revision as of 17:14, 5 February 2012 by Joshxiong (talk | contribs) (Problems 9-19)

Problem 1


$\textrm{(A)}\ -7\qquad\textrm{(B)}\ -2\qquad\textrm{(C)}\ 0\qquad\textrm{(D)}\ 1\qquad\textrm{(E)}\ 57$


Problem 2


$\textrm{(A)}\ \frac{1}5\qquad\textrm{(B)}\ \frac{1}4\qquad\textrm{(C)}\ \frac{2}7\qquad\textrm{(D)}\ \frac{5}{12}\qquad\textrm{(E)}\ \frac{7}{12}$


Problem 3

A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is 24, then the area of the original square is

[asy] draw((0,0)--(9,0)--(9,9)--(0,9)--cycle); draw((3,0)--(3,9), dashed); draw((6,0)--(6,9), dashed);[/asy]

$\textrm{(A)}\ 24\qquad\textrm{(B)}\ 36\qquad\textrm{(C)}\ 64\qquad\textrm{(D)}\ 81\qquad\textrm{(E)}\ 96$


Problem 4

In the figure, $ABCD$ is an isosceles trapezoid with side lengths $AD=BC=5$, $AB=4$, and $DC=10$. The point $C$ is on $\overline{DF}$ and $B$ is the midpoint of hypotenuse $\overline{DE}$ in right triangle $DEF$. Then $CF=$

[asy] defaultpen(fontsize(10)); pair D=origin, A=(3,4), B=(7,4), C=(10,0), E=(14,8), F=(14,0); draw(B--C--F--E--B--A--D--B^^C--D, linewidth(0.7)); dot(A^^B^^C^^D^^E^^F); pair point=(5,3); label("$A$", A, N); label("$B$", B, N); label("$C$", C, S); label("$D$", D, S); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); markscalefactor=0.05; draw(rightanglemark(E,F,D), linewidth(0.7));[/asy]

$\textrm{(A)}\ 3.25\qquad\textrm{(B)}\ 3.5\qquad\textrm{(C)}\ 3.75\qquad\textrm{(D)}\ 4.0\qquad\textrm{(E)}\ 4.25$


Problem 5

Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is

[asy] real xscl = 1.2; int[] x = {0,1,2,4,5},y={0,1,3,4,5}; for(int a:x){ for(int b:y) { dot((a*xscl,b)); } } for(int a:x) { pair prev = (a,y[0]); for(int i = 1;i<y.length;++i) { pair p = (a,y[i]); pen pen = linewidth(.7); if(y[i]-prev.y!=1){  pen+=dotted; } draw((xscl*prev.x,prev.y)--(xscl*p.x,p.y),pen); prev = p; } }for(int a:y) { pair prev = (x[0],a); for(int i = 1;i<x.length;++i) { pair p = (x[i],a); pen pen = linewidth(.7); if(x[i]-prev.x!=1){  pen+=dotted; } draw((xscl*prev.x,prev.y)--(p.x*xscl,p.y),pen); prev = p; } } path lblx = (0,-.7)--(5*xscl,-.7); draw(lblx); label("$10$",lblx); path lbly = (5*xscl+.7,0)--(5*xscl+.7,5); draw(lbly); label("$20$",lbly);[/asy]

$\textrm{(A)}\ 30\qquad\textrm{(B)}\ 200\qquad\textrm{(C)}\ 410\qquad\textrm{(D)}\ 420\qquad\textrm{(E)}\ 430$


Problem 6

If $a,\,b>0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by=6$ has area $6$, then $ab=$

$\textrm{(A)}\ 3\qquad\textrm{(B)}\ 6\qquad\textrm{(C)}\ 12\qquad\textrm{(D)}\ 108\qquad\textrm{(E)}\ 432$


Problem 7

In $\triangle ABC$, $\angle A = 100^\circ$, $\angle B = 50^\circ$, $\angle C = 30^\circ$, $\overline{AH}$ is an altitude, and $\overline{BM}$ is a median. Then $\angle MHC=$

[asy] draw((0,0)--(16,0)--(6,6)--cycle); draw((6,6)--(6,0)--(11,3)--(0,0)); dot((6,6)); dot((0,0)); dot((11,3)); dot((6,0)); dot((16,0)); label("A", (6,6), N); label("B", (0,0), W); label("C", (16,0), E); label("H", (6,0), S); label("M", (11,3), NE);[/asy]

$\textrm{(A)}\ 15^\circ\qquad\textrm{(B)}\ 22.5^\circ\qquad\textrm{(C)}\ 30^\circ\qquad\textrm{(D)}\ 40^\circ\qquad\textrm{(E)}\ 45^\circ$


Problem 8

For how many integers $n$ between $1$ and $100$ does $x^{2}+x-n$ factor into the product of two linear factors with integer coefficients?

$\textrm{(A)}\ 0\qquad\textrm{(B)}\ 1\qquad\textrm{(C)}\ 2\qquad\textrm{(D)}\ 9\qquad\textrm{(E)}\ 10$


Problem 9

Mr. and Mrs. Zeta want to name their baby Zeta so that its monogram (first, middle, and last initials) will be in alphabetical order with no letter repeated. How many such monograms are possible?

$\textrm{(A)}\ 276\qquad\textrm{(B)}\ 300\qquad\textrm{(C)}\ 552\qquad\textrm{(D)}\ 600\qquad\textrm{(E)}\ 15600$


Problem 10

Consider the sequence defined recursively by $u_{1}= a$ (any positive integer), and $u_{n+1}=\frac{-1}{u_{n}+1}$, $n = 1,2,3,\cdots$. For which of the following values of $n$ must $u_{n}=a$?

$\textrm{(A)}\ 14\qquad\textrm{(B)}\ 15\qquad\textrm{(C)}\ 16\qquad\textrm{(D)}\ 17\qquad\textrm{(E)}\ 18$


Problem 11

Let $a$, $b$, $c$, and $d$ be positive integers with $a < 2b$, $b < 3c$, and $c<4d$. If $d<100$, the largest possible value for $a$ is

$\textrm{(A)}\ 2367\qquad\textrm{(B)}\ 2375\qquad\textrm{(C)}\ 2391\qquad\textrm{(D)}\ 2399\qquad\textrm{(E)}\ 2400$


Problem 12

The traffic on a certain east-west highway moves at a constant speed of 60 miles per hour in both directions. An eastbound driver passes 20 west-bound vehicles in a five-minute interval. Assume vehicles in the westbound lane are equally spaced. Which of the following is closest to the number of westbound vehicles present in a 100-mile section of highway?

$\textrm{(A)}\ 100\qquad\textrm{(B)}\ 120\qquad\textrm{(C)}\ 200\qquad\textrm{(D)}\ 240\qquad\textrm{(E)}\ 400$


Problem 13

Two strips of width 1 overlap at an angle of $\alpha$ as shown. The area of the overlap (shown shaded) is

[asy] pair a = (0,0),b= (6,0),c=(0,1),d=(6,1); transform t = rotate(-45,(3,.5)); pair e = t*a,f=t*b,g=t*c,h=t*d; pair i = intersectionpoint(a--b,e--f),j=intersectionpoint(a--b,g--h),k=intersectionpoint(c--d,e--f),l=intersectionpoint(c--d,g--h); draw(a--b^^c--d^^e--f^^g--h); filldraw(i--j--l--k--cycle,blue); label("$\alpha$",i+(-.5,.2)); //commented out labeling because it doesn't look right. //path lbl1 = (a+(.5,.2))--(c+(.5,-.2)); //draw(lbl1); //label("$1$",lbl1);[/asy]

$\textrm{(A)}\ \sin\alpha\qquad\textrm{(B)}\ \frac{1}{\sin\alpha}\qquad\textrm{(C)}\ \frac{1}{1-\cos\alpha}\qquad\textrm{(D)}\ \frac{1}{\sin^{2}\alpha}\qquad\textrm{(E)}\ \frac{1}{(1-\cos\alpha)^{2}}$


Problem 14

$\cot 10+\tan 5 =$

$\textrm{(A)}\ \csc 5\qquad\textrm{(B)}\ \csc 10\qquad\textrm{(C)}\ \sec 5\qquad\textrm{(D)}\ \sec 10\qquad\textrm{(E)}\ \sin 15$


Problem 15

In $\triangle ABC$, $AB=5$, $BC=7$, $AC=9$, and $D$ is on $\overline{AC}$ with $BD=5$. Find the ratio of $AD:DC$.

[asy] draw((3,4)--(0,0)--(9,0)--(3,4)--(6,0)); dot((0,0)); dot((9,0)); dot((3,4)); dot((6,0)); label("A", (0,0), W); label("B", (3,4), N); label("C", (9,0), E); label("D", (6,0), S);[/asy]

$\textrm{(A)}\ 4:3\qquad\textrm{(B)}\ 7:5\qquad\textrm{(C)}\ 11:6\qquad\textrm{(D)}\ 13:5\qquad\textrm{(E)}\ 19:8$


Problem 16

A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are (3,17) and (48,281)? (Include both endpoints of the segment in your count.)

$\textrm{(A)}\ 2\qquad\textrm{(B)}\ 4\qquad\textrm{(C)}\ 6\qquad\textrm{(D)}\ 16\qquad\textrm{(E)}\ 46$


Problem 17

The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989$ cm. The length of each side of the triangle exceeds the length of each side of the square by $d$ cm. The square has perimeter greater than 0. How many positive integers are NOT a possible value for $d$?

$\textrm{(A)}\ 0\qquad\textrm{(B)}\ 9\qquad\textrm{(C)}\ 221\qquad\textrm{(D)}\ 663\qquad\textrm{(E)}\ \text{infinitely many}$


Problem 18

The set of all numbers x for which $x+\sqrt{x^{2}+1}-\frac{1}{x+\sqrt{x^{2}+1}}$ is a rational number is the set of all:

$\textrm{(A)}\ \text{ integers }x\qquad\textrm{(B)}\ \text{ rational }x\qquad\textrm{(C)}\ \text{ real }x\qquad\textrm{(D)}\ x\text{ for which }\sqrt{x^{2}+1}\text{ is rational}\qquad\textrm{(E)}\ x\text{ for which }x+\sqrt{x^{2}+1}\text{ is rational }$


Problem 19

A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths $3$, $4$, and $5$. What is the area of the triangle?

$\textrm{(A)}\ 6\qquad\textrm{(B)}\ \frac{18}{\pi^{2}}\qquad\textrm{(C)}\ \frac{9}{\pi^{2}}\left(\sqrt{3}-1\right)\qquad\textrm{(D)}\ \frac{9}{\pi^{2}}\left(\sqrt{3}+1\right)\qquad\textrm{(E)}\ \frac{9}{\pi^{2}}\left(\sqrt{3}+3\right)$


Problem 20


Problem 21


Problem 22


Problem 23


Problem 24


Problem 25


Problem 26


Problem 27


Problem 28


Problem 29


Problem 30


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