1994 AHSME Problems

1 $4^4 \cdot 9^4 \cdot 4^9 \cdot 9^9=$

$\textbf{(A)}\ 13^{13} \qquad\textbf{(B)}\ 13^{36} \qquad\textbf{(C)}\ 36^{13} \qquad\textbf{(D)}\ 36^{36} \qquad\textbf{(E)}\...$ (Error compiling LaTeX. Unknown error_msg)

2 A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle? [asy] draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); draw((0,5)--(10,5)); draw((3,0)--(3,7)); label("6", (1.5,6)); label("?", (1.5,2.5)); label("14", (6.5,6)); label("35", (6.5,2.5));[/asy]

$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 25$

3 How many of the following are equal to x^x+x^x for all x>0?

$\textbf{I:}\ 2x^x \qquad\textbf{II:}\ x^{2x} \qquad\textbf{III:}\ (2x)^x \qquad\textbf{IV:}\ (2x)^{2x}$

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

4 In the xy-plane, the segment with endpoints (-5,0) and (25,0) is the diameter of a circle. If the point (x,15) is on the circle, then x=

$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12.5 \qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 17.5 \qquad\textbf{(E)}\ 20$

5 Pat intended to multiply a number by 6 but instead divided by 6. Pat then meant to add 14 but instead subtracted 14. After these mistakes, the result was 16. If the correct operations had been used, the value produced would have been

$\textbf{(A)}\ \text{less than 400} \qquad\textbf{(B)}\ \text{between 400 and 600} \qquad\textbf{(C)}\ \text{between 600 and 8...$ (Error compiling LaTeX. Unknown error_msg)

6 In the sequence ..., a, b, c, d, 0, 1, 1, 2, 3, 5, 8,... each term is the sum of the two terms to its left. Find a.

$\textbf{(A)}\ -3 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 3$

7 Squares ABCD and EFGH are congruent, AB=10, and G is the center of square ABCD. The area of the region in the plane covered by these squares is draw((0,0)--(10,0)--(10,10)--(0,10)--cycle);draw((5,5)--(12,-2)--(5,-9)--(-2,-2)--cycle);label("A", (0,0), W);label... $\textbf{(A)}\ 75 \qquad\textbf{(B)}\ 100 \qquad\textbf{(C)}\ 125 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 175$

8 In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is 56. The area of the region bounded by the polygon is draw((0,0)--(1,0)--(1,-1)--(2,-1)--(2,-2)--(3,-2)--(3,-3)--(4,-3)--(4,-2)--(5,-2)--(5,-1)--(6,-1)--(6,0)--(7,0)--(7,1)--(6,1)... $\textbf{(A)}\ 84 \qquad\textbf{(B)}\ 96 \qquad\textbf{(C)}\ 100 \qquad\textbf{(D)}\ 112 \qquad\textbf{(E)}\ 196$

9 If \angle A is four times \angle B, and the complement of \angle B is four times the complement of \angle A, then \angle B=

$\textbf{(A)}\ 10^{\circ} \qquad\textbf{(B)}\ 12^{\circ} \qquad\textbf{(C)}\ 15^{\circ} \qquad\textbf{(D)}\ 18^{\circ} \qquad\...$ (Error compiling LaTeX. Unknown error_msg)

10 For distinct real numbers x and y, let M(x,y) be the larger of x and y and let m(x,y) be the smaller of x and y. If a<b<c<d<e, then M(M(a,m(b,c)),m(d,m(a,e)))= $\textbf{(A)}\ a \qquad\textbf{(B)}\ b \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ d \qquad\textbf{(E)}\ e$

11 Three cubes of volume 1, 8 and 27 are glued together at their faces. The smallest possible surface area of the resulting configuration is

$\textbf{(A)}\ 36 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 70 \qquad\textbf{(D)}\ 72 \qquad\textbf{(E)}\ 74$

12 If i^2=-1, then (i-i^{-1})^{-1}=

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ -2i \qquad\textbf{(C)}\ 2i \qquad\textbf{(D)}\ -\frac{i}{2} \qquad\textbf{(E)}\ \frac{i}{...$ (Error compiling LaTeX. Unknown error_msg)

13 In triangle ABC, AB=AC. If there is a point P strictly between A and B such that AP=PC=CB, then \angle A = draw((0,0)--(8,0)--(4,12)--cycle);draw((8,0)--(1.6,4.8));label("A", (4,12), N);label("B", (0,0), W);label... $\textbf{(A)}\ 30^{\circ} \qquad\textbf{(B)}\ 36^{\circ} \qquad\textbf{(C)}\ 48^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\...$ (Error compiling LaTeX. Unknown error_msg)

14 Find the sum of the arithmetic series 20+20\frac{1}{5}+20\frac{2}{5}+\cdots+40 $\textbf{(A)}\ 3000 \qquad\textbf{(B)}\ 3030 \qquad\textbf{(C)}\ 3150 \qquad\textbf{(D)}\ 4100 \qquad\textbf{(E)}\ 6000$

15 For how many n in \{1, 2, 3, ..., 100 \} is the tens digit of n^2 odd?

$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 30 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 50$

16 Some marbles in a bag are red and the rest are blue. If one red marble is removed, then one-seventh of the remaining marbles are red. If two blue marbles are removed instead of one red, then one-fifth of the remaining marbles are red. How many marbles were in the bag originally?

$\textbf{(A)}\ 8 \qquad\textbf{(B)}\ 22 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 57 \qquad\textbf{(E)}\ 71$

17 An 8 by 2\sqrt{2} rectangle has the same center as a circle of radius 2. The area of the region common to both the rectangle and the circle is

$\textbf{(A)}\ 2\pi \qquad\textbf{(B)}\ 2\pi+2 \qquad\textbf{(C)}\ 4\pi-4 \qquad\textbf{(D)}\ 2\pi+4 \qquad\textbf{(E)}\ 4\pi-...$

18 Triangle ABC is inscribed in a circle, and \angle B = \angle C = 4\angle A. If B and C are adjacent vertices of a regular polygon of n sides inscribed in this circle, then n= draw(Circle((0,0), 5));draw((0,5)--(3,-4)--(-3,-4)--cycle);label("A", (0,5), N);label("B", (-3,-4), SW);l... $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 18$

19 Label one disk "1", two disks "2", three disks "3", ..., fifty disks "50". Put these 1+2+3+ \cdots+50=1275 labeled disks in a box. Disks are then drawn from the box at random without replacement. The minimum number of disks that must be drawn to guarantee drawing at least ten disks with the same label is

$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 51 \qquad\textbf{(C)}\ 415 \qquad\textbf{(D)}\ 451 \qquad\textbf{(E)}\ 501$

20 Suppose x,y,z is a geometric sequence with common ratio r and x \neq y. If x, 2y, 3z is an arithmetic sequence, then r is

$\textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ 2 \qquad\textbf...$

21 Find the number of counter examples to the statement: ``\text{If N is an odd positive integer the sum of whose digits is 4 and none of whose digits is 0, then N is prime}." $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

22 Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta and Gamma choose their chairs?

$\textbf{(A)}\ 12 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ 630$

23 In the xy-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at (0,0), (0,3), (3,3), (3,1), (5,1) and (5,0). The slope of the line through the origin that divides the area of this region exactly in half is Label l;l.p=fontsize(6);xaxis(" $x$ ",0,6,Ticks(l,1.0,0.5),EndArrow);yaxis(" $y$ ",0,4,Ticks(l,1.0,0.5),EndAr... $\textbf{(A)}\ \frac{2}{7} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{2}{3} \qquad\textbf{(D)}\ \frac{3}{4} \qq...$ (Error compiling LaTeX. Unknown error_msg)

24 A sample consisting of five observations has an arithmetic mean of 10 and a median of 12. The smallest value that the range (largest observation minus smallest) can assume for such a sample is

$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 10$

25 If x and y are non-zero real numbers such that |x|+y=3 \qquad \text{and} \qquad |x|y+x^3=0, then the integer nearest to x-y is

$\textbf{(A)}\ -3 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 5$

26 A regular polygon of m sides is exactly enclosed (no overlaps, no gaps) by m regular polygons of n sides each. (Shown here for m=4, n=8.) If m=10, what is the value of n? size(200);defaultpen(linewidth(0.8));draw(unitsquare);path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/... $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 26$

27 A bag of popping corn contains \frac{2}{3} white kernels and \frac{1}{3} yellow kernels. Only \frac{1}{2} of the white kernels will pop, whereas \frac{2}{3} of the yellow ones will pop. A kernel is selected at random from the bag, and pops when placed in the popper. What is the probability that the kernel selected was white?

$\textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{5}{9} \qquad\textbf{(C)}\ \frac{4}{7} \qquad\textbf{(D)}\ \frac{3}{5} \qq...$ (Error compiling LaTeX. Unknown error_msg)

28 In the xy-plane, how many lines whose x-intercept is a positive prime number and whose y-intercept is a positive integer pass through the point (4,3)?

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

29 Points A, B and C on a circle of radius r are situated so that AB=AC, AB>r, and the length of minor arc BC is r. If angles are measured in radians, then AB/BC= draw(Circle((0,0), 13));draw((-13,0)--(12,5)--(12,-5)--cycle);dot((-13,0));dot((12,5));dot((12,-5));label("A", (-13... $\textbf{(A)}\ \frac{1}{2}\csc{\frac{1}{4}} \qquad\textbf{(B)}\ 2\cos{\frac{1}{2}} \qquad\textbf{(C)}\ 4\sin{\frac{1}{2}} \qqu...$ (Error compiling LaTeX. Unknown error_msg)

30 When n standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of S. The smallest possible value of S is

$\textbf{(A)}\ 333 \qquad\textbf{(B)}\ 335 \qquad\textbf{(C)}\ 337 \qquad\textbf{(D)}\ 339 \qquad\textbf{(E)}\ 341$

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1994 AHSME Problems