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Difference between revisions of "1994 AHSME Problems"

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1 4^4 \cdot 9^4 \cdot 4^9 \cdot 9^9=
+
{{AHSME Problems
 +
|year = 1994
 +
}}
 +
== Problem 1 ==
 +
<math>4^4 \cdot 9^4 \cdot 4^9 \cdot 9^9=</math>
  
−
\textbf{(A)}\ 13^{13} \qquad\textbf{(B)}\ 13^{36} \qquad\textbf{(C)}\ 36^{13} \qquad\textbf{(D)}\ 36^{36} \qquad\textbf{(E)}\...
+
<math> \textbf{(A)}\ 13^{13} \qquad\textbf{(B)}\ 13^{36} \qquad\textbf{(C)}\ 36^{13} \qquad\textbf{(D)}\ 36^{36} \qquad\textbf{(E)}\ 1296^{26} </math>
−
 
−
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?
 
−
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("?&...
 
  
−
\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 25
+
[[1994 AHSME Problems/Problem 1|Solution]]
−
 
−
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}
+
== Problem 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)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4
+
<math> \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 25 </math>
−
 
−
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
+
[[1994 AHSME Problems/Problem 2|Solution]]
−
 
−
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...
+
== Problem 3 ==
−
+
How many of the following are equal to <math>x^x+x^x</math> for all <math>x>0</math>?
−
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
+
<math>\textbf{I:}\ 2x^x \qquad\textbf{II:}\ x^{2x} \qquad\textbf{III:}\ (2x)^x \qquad\textbf{IV:}\ (2x)^{2x}</math>
−
 
−
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\...
+
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 </math>
−
 
−
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
+
[[1994 AHSME Problems/Problem 3|Solution]]
−
 
−
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}{...
+
== Problem 4 ==
−
+
In the <math>xy</math>-plane, the segment with endpoints <math>(-5,0)</math> and <math>(25,0)</math> is the diameter of a circle. If the point <math>(x,15)</math> is on the circle, then <math>x=</math>
−
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\...
 
−
 
−
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
+
<math> \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12.5 \qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 17.5 \qquad\textbf{(E)}\ 20 </math>
−
 
−
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
+
[[1994 AHSME Problems/Problem 4|Solution]]
−
 
−
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-...
+
== Problem 5 ==
−
+
Pat intended to multiply a number by <math>6</math> but instead divided by <math>6</math>. Pat then meant to add <math>14</math> but instead subtracted <math>14</math>. After these mistakes, the result was <math>16</math>. If the correct operations had been used, the value produced would have been
−
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
+
<math> \textbf{(A)}\ \text{less than 400} \qquad\textbf{(B)}\ \text{between 400 and 600} \qquad\textbf{(C)}\ \text{between 600 and 800} \\
−
+
\textbf{(D)}\ \text{between 800 and 1000} \qquad\textbf{(E)}\ \text{greater than 1000}</math>
−
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...
+
[[1994 AHSME Problems/Problem 5|Solution]]
−
 
−
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
+
== Problem 6 ==
−
+
In the sequence
−
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
+
<cmath> ..., a, b, c, d, 0, 1, 1, 2, 3, 5, 8,... </cmath>
−
Label l;l.p=fontsize(6);xaxis("<math>x</math>",0,6,Ticks(l,1.0,0.5),EndArrow);yaxis("<math>y</math>",0,4,Ticks(l,1.0,0.5),EndAr...
+
each term is the sum of the two terms to its left. Find <math>a</math>.
−
\textbf{(A)}\ \frac{2}{7} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{2}{3} \qquad\textbf{(D)}\ \frac{3}{4} \qq...
 
−
 
−
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
+
<math> \textbf{(A)}\ -3 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 3 </math>
−
 
−
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
+
[[1994 AHSME Problems/Problem 6|Solution]]
−
 
−
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...
+
== Problem 7 ==
−
+
Squares <math>ABCD</math> and <math>EFGH</math> are congruent, <math>AB=10</math>, and <math>G</math> is the center of square <math>ABCD</math>. The area of the region in the plane covered by these squares is
−
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)?
+
<asy>
 +
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("B", (10,0), E);
 +
label("C", (10,10), NE);
 +
label("D", (0,10), NW);
 +
label("G", (5,5), N);
 +
label("F", (12,-2), E);
 +
label("E", (5,-9), S);
 +
label("H", (-2,-2), W);
 +
dot((-2,-2));
 +
dot((5,-9));
 +
dot((12,-2));
 +
dot((0,0));
 +
dot((10,0));
 +
dot((10,10));
 +
dot((0,10));
 +
dot((5,5));
 +
</asy>
 +
<math> \textbf{(A)}\ 75 \qquad\textbf{(B)}\ 100 \qquad\textbf{(C)}\ 125 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 175 </math>
  
−
\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4
+
[[1994 AHSME Problems/Problem 7|Solution]]
−
 
−
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...
 
−
 
−
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
+
== Problem 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 <math>56</math>. The area of the region bounded by the polygon is
 +
<asy>
 +
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)--(6,2)--(5,2)--(5,3)--(4,3)--(4,4)--(3,4)--(3,3)--(2,3)--(2,2)--(1,2)--(1,1)--(0,1)--cycle);
 +
</asy>
 +
<math> \textbf{(A)}\ 84 \qquad\textbf{(B)}\ 96 \qquad\textbf{(C)}\ 100 \qquad\textbf{(D)}\ 112 \qquad\textbf{(E)}\ 196 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 8|Solution]]
 +
 
 +
== Problem 9 ==
 +
If <math>\angle A</math> is four times <math>\angle B</math>, and the complement of <math>\angle B</math> is four times the complement of <math>\angle A</math>, then <math>\angle B=</math>
 +
 
 +
<math> \textbf{(A)}\ 10^{\circ} \qquad\textbf{(B)}\ 12^{\circ} \qquad\textbf{(C)}\ 15^{\circ} \qquad\textbf{(D)}\ 18^{\circ} \qquad\textbf{(E)}\ 22.5^{\circ} </math>
 +
 
 +
[[1994 AHSME Problems/Problem 9|Solution]]
 +
 
 +
== Problem 10 ==
 +
For distinct real numbers <math>x</math> and <math>y</math>, let <math>M(x,y)</math> be the larger of <math>x</math> and <math>y</math> and let <math>m(x,y)</math> be the smaller of <math>x</math> and <math>y</math>. If <math>a<b<c<d<e</math>, then
 +
<cmath> M(M(a,m(b,c)),m(d,m(a,e)))= </cmath>
 +
<math> \textbf{(A)}\ a \qquad\textbf{(B)}\ b \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ d \qquad\textbf{(E)}\ e </math>
 +
 
 +
[[1994 AHSME Problems/Problem 10|Solution]]
 +
 
 +
== Problem 11 ==
 +
Three cubes of volume <math>1, 8</math> and <math>27</math> are glued together at their faces. The smallest possible surface area of the resulting configuration is
 +
 
 +
<math> \textbf{(A)}\ 36 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 70 \qquad\textbf{(D)}\ 72 \qquad\textbf{(E)}\ 74 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 11|Solution]]
 +
 
 +
== Problem 12 ==
 +
If <math>i^2=-1</math>, then <math>(i-i^{-1})^{-1}=</math>
 +
 
 +
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ -2i \qquad\textbf{(C)}\ 2i \qquad\textbf{(D)}\ -\frac{i}{2} \qquad\textbf{(E)}\ \frac{i}{2}</math>
 +
 
 +
[[1994 AHSME Problems/Problem 12|Solution]]
 +
 
 +
== Problem 13 ==
 +
In triangle <math>ABC</math>, <math>AB=AC</math>. If there is a point <math>P</math> strictly between <math>A</math> and <math>B</math> such that <math>AP=PC=CB</math>, then <math>\angle A =</math>
 +
<asy>
 +
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("C", (8,0), E);
 +
label("P", (1.6,4.8), NW);
 +
dot((0,0));
 +
dot((4,12));
 +
dot((8,0));
 +
dot((1.6,4.8));
 +
</asy>
 +
<math> \textbf{(A)}\ 30^{\circ} \qquad\textbf{(B)}\ 36^{\circ} \qquad\textbf{(C)}\ 48^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 72^{\circ} </math>
 +
 
 +
[[1994 AHSME Problems/Problem 13|Solution]]
 +
 
 +
== Problem 14 ==
 +
Find the sum of the arithmetic series
 +
<cmath> 20+20\frac{1}{5}+20\frac{2}{5}+\cdots+40 </cmath>
 +
<math> \textbf{(A)}\ 3000 \qquad\textbf{(B)}\ 3030 \qquad\textbf{(C)}\ 3150 \qquad\textbf{(D)}\ 4100 \qquad\textbf{(E)}\ 6000 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 14|Solution]]
 +
 
 +
== Problem 15 ==
 +
For how many <math>n</math> in <math>\{1, 2, 3, ..., 100 \}</math> is the tens digit of <math>n^2</math> odd?
 +
 
 +
<math> \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 30 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 50 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 15|Solution]]
 +
 
 +
== Problem 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?
 +
 
 +
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 22 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 57 \qquad\textbf{(E)}\ 71 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 16|Solution]]
 +
 
 +
== Problem 17 ==
 +
An <math>8</math> by <math>2\sqrt{2}</math> rectangle has the same center as a circle of radius <math>2</math>. The area of the region common to both the rectangle and the circle is
 +
 
 +
<math> \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-2 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 17|Solution]]
 +
 
 +
== Problem 18 ==
 +
Triangle <math>ABC</math> is inscribed in a circle, and <math>\angle B = \angle C = 4\angle A</math>. If <math>B</math> and <math>C</math> are adjacent vertices of a regular polygon of <math>n</math> sides inscribed in this circle, then <math>n=</math>
 +
<asy>
 +
draw(Circle((0,0), 5));
 +
draw((0,5)--(3,-4)--(-3,-4)--cycle);
 +
label("A", (0,5), N);
 +
label("B", (-3,-4), SW);
 +
label("C", (3,-4), SE);
 +
dot((0,5));
 +
dot((3,-4));
 +
dot((-3,-4));
 +
</asy>
 +
<math> \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 18 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 18|Solution]]
 +
 
 +
== Problem 19 ==
 +
Label one disk "<math>1</math>", two disks "<math>2</math>", three disks "<math>3</math>"<math>, ...,</math> fifty disks "<math>50</math>". Put these <math>1+2+3+ \cdots+50=1275</math> 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
 +
 
 +
<math> \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 51 \qquad\textbf{(C)}\ 415 \qquad\textbf{(D)}\ 451 \qquad\textbf{(E)}\ 501 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 19|Solution]]
 +
 
 +
== Problem 20 ==
 +
Suppose <math>x,y,z</math> is a geometric sequence with common ratio <math>r</math> and <math>x \neq y</math>. If <math>x, 2y, 3z</math> is an arithmetic sequence, then <math>r</math> is
 +
 
 +
<math> \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4</math>
 +
 
 +
[[1994 AHSME Problems/Problem 20|Solution]]
 +
 
 +
== Problem 21 ==
 +
Find the number of counter examples to the statement:
 +
<cmath>``\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}."</cmath>
 +
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 21|Solution]]
 +
 
 +
== Problem 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?
 +
 
 +
<math> \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ 630 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 22|Solution]]
 +
 
 +
== Problem 23 ==
 +
In the <math>xy</math>-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at <math>(0,0), (0,3), (3,3), (3,1), (5,1)</math> and <math>(5,0)</math>. The slope of the line through the origin that divides the area of this region exactly in half is
 +
<asy>
 +
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),EndArrow);
 +
draw((0,3)--(3,3)--(3,1)--(5,1)--(5,0)--(0,0)--cycle,black+linewidth(2));</asy>
 +
<math> \textbf{(A)}\ \frac{2}{7} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{2}{3} \qquad\textbf{(D)}\ \frac{3}{4} \qquad\textbf{(E)}\ \frac{7}{9} </math>
 +
 
 +
[[1994 AHSME Problems/Problem 23|Solution]]
 +
 
 +
== Problem 24 ==
 +
A sample consisting of five observations has an arithmetic mean of <math>10</math> and a median of <math>12</math>. The smallest value that the range (largest observation minus smallest) can assume for such a sample is
 +
 
 +
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 10 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 24|Solution]]
 +
 
 +
== Problem 25 ==
 +
If <math>x</math> and <math>y</math> are non-zero real numbers such that
 +
<cmath> |x|+y=3 \qquad \text{and} \qquad |x|y+x^3=0, </cmath>
 +
then the integer nearest to <math>x-y</math> is
 +
 
 +
<math> \textbf{(A)}\ -3 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 5 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 25|Solution]]
 +
 
 +
== Problem 26 ==
 +
A regular polygon of <math>m</math> sides is exactly enclosed (no overlaps, no gaps) by <math>m</math> regular polygons of <math>n</math> sides each. (Shown here for <math>m=4, n=8</math>.) If <math>m=10</math>, what is the value of <math>n</math>?
 +
<asy>
 +
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)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle;
 +
draw(p);
 +
draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p);
 +
draw(shift((0,-2-sqrt(2)))*p);
 +
draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);</asy>
 +
<math> \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 26 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 26|Solution]]
 +
 
 +
== Problem 27 ==
 +
A bag of popping corn contains <math>\frac{2}{3}</math> white kernels and <math>\frac{1}{3}</math> yellow kernels. Only <math>\frac{1}{2}</math> of the white kernels will pop, whereas <math>\frac{2}{3}</math> 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?
 +
 
 +
<math> \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{5}{9} \qquad\textbf{(C)}\ \frac{4}{7} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{3} </math>
 +
 
 +
[[1994 AHSME Problems/Problem 27|Solution]]
 +
 
 +
== Problem 28 ==
 +
In the <math>xy</math>-plane, how many lines whose <math>x</math>-intercept is a positive prime number and whose <math>y</math>-intercept is a positive integer pass through the point <math>(4,3)</math>?
 +
 
 +
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4</math>
 +
 
 +
[[1994 AHSME Problems/Problem 28|Solution]]
 +
 
 +
== Problem 29 ==
 +
Points <math>A, B</math> and <math>C</math> on a circle of radius <math>r</math> are situated so that <math>AB=AC, AB>r</math>, and the length of minor arc <math>BC</math> is <math>r</math>. If angles are measured in radians, then <math>AB/BC=</math>
 +
<asy>
 +
draw(Circle((0,0), 13));
 +
draw((-13,0)--(12,5)--(12,-5)--cycle);
 +
dot((-13,0));
 +
dot((12,5));
 +
dot((12,-5));
 +
label("A", (-13,0), W);
 +
label("B", (12,5), NE);
 +
label("C", (12,-5), SE);
 +
</asy>
 +
<math> \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}} \qquad\textbf{(D)}\ \csc{\frac{1}{2}} \qquad\textbf{(E)}\ 2\sec{\frac{1}{2}} </math>
 +
 
 +
[[1994 AHSME Problems/Problem 29|Solution]]
 +
 
 +
== Problem 30 ==
 +
When <math>n</math> 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 <math>S</math>. The smallest possible value of <math>S</math> is
 +
 
 +
<math> \textbf{(A)}\ 333 \qquad\textbf{(B)}\ 335 \qquad\textbf{(C)}\ 337 \qquad\textbf{(D)}\ 339 \qquad\textbf{(E)}\ 341 </math>
 +
 
 +
[[1994 AHSME Problems/Problem 30|Solution]]
 +
 
 +
== See also ==
 +
 
 +
* [[AMC 12 Problems and Solutions]]
 +
* [[Mathematics competition resources]]
 +
 
 +
{{AHSME box|year=1994|before=[[1993 AHSME]]|after=[[1995 AHSME]]}} 
 +
 
 +
{{MAA Notice}}

Latest revision as of 12:39, 19 February 2020

1994 AHSME (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 30-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 5 points for each correct answer, 2 points for each problem left unanswered, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers.
  4. Figures are not necessarily drawn to scale.
  5. You will have 90 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30

Problem 1

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

$\textbf{(A)}\ 13^{13} \qquad\textbf{(B)}\ 13^{36} \qquad\textbf{(C)}\ 36^{13} \qquad\textbf{(D)}\ 36^{36} \qquad\textbf{(E)}\ 1296^{26}$

Solution

Problem 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]

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

Solution

Problem 3

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

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

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

Solution

Problem 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 $x=$

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

Solution

Problem 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

$\textbf{(A)}\ \text{less than 400} \qquad\textbf{(B)}\ \text{between 400 and 600} \qquad\textbf{(C)}\ \text{between 600 and 800} \\ \textbf{(D)}\ \text{between 800 and 1000} \qquad\textbf{(E)}\ \text{greater than 1000}$

Solution

Problem 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 $a$.

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

Solution

Problem 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 [asy] 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("B", (10,0), E); label("C", (10,10), NE); label("D", (0,10), NW); label("G", (5,5), N); label("F", (12,-2), E); label("E", (5,-9), S); label("H", (-2,-2), W); dot((-2,-2)); dot((5,-9)); dot((12,-2)); dot((0,0)); dot((10,0)); dot((10,10)); dot((0,10)); dot((5,5)); [/asy] $\textbf{(A)}\ 75 \qquad\textbf{(B)}\ 100 \qquad\textbf{(C)}\ 125 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 175$

Solution

Problem 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 [asy] 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)--(6,2)--(5,2)--(5,3)--(4,3)--(4,4)--(3,4)--(3,3)--(2,3)--(2,2)--(1,2)--(1,1)--(0,1)--cycle); [/asy] $\textbf{(A)}\ 84 \qquad\textbf{(B)}\ 96 \qquad\textbf{(C)}\ 100 \qquad\textbf{(D)}\ 112 \qquad\textbf{(E)}\ 196$

Solution

Problem 9

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

$\textbf{(A)}\ 10^{\circ} \qquad\textbf{(B)}\ 12^{\circ} \qquad\textbf{(C)}\ 15^{\circ} \qquad\textbf{(D)}\ 18^{\circ} \qquad\textbf{(E)}\ 22.5^{\circ}$

Solution

Problem 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 \[M(M(a,m(b,c)),m(d,m(a,e)))=\] $\textbf{(A)}\ a \qquad\textbf{(B)}\ b \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ d \qquad\textbf{(E)}\ e$

Solution

Problem 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

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

Solution

Problem 12

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

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ -2i \qquad\textbf{(C)}\ 2i \qquad\textbf{(D)}\ -\frac{i}{2} \qquad\textbf{(E)}\ \frac{i}{2}$

Solution

Problem 13

In triangle $ABC$, $AB=AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP=PC=CB$, then $\angle A =$ [asy] 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("C", (8,0), E); label("P", (1.6,4.8), NW); dot((0,0)); dot((4,12)); dot((8,0)); dot((1.6,4.8)); [/asy] $\textbf{(A)}\ 30^{\circ} \qquad\textbf{(B)}\ 36^{\circ} \qquad\textbf{(C)}\ 48^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 72^{\circ}$

Solution

Problem 14

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

Solution

Problem 15

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

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

Solution

Problem 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?

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

Solution

Problem 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

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

Solution

Problem 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 $n=$ [asy] draw(Circle((0,0), 5)); draw((0,5)--(3,-4)--(-3,-4)--cycle); label("A", (0,5), N); label("B", (-3,-4), SW); label("C", (3,-4), SE); dot((0,5)); dot((3,-4)); dot((-3,-4)); [/asy] $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 18$

Solution

Problem 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

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

Solution

Problem 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

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

Solution

Problem 21

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

Solution

Problem 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?

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

Solution

Problem 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 [asy] 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),EndArrow); draw((0,3)--(3,3)--(3,1)--(5,1)--(5,0)--(0,0)--cycle,black+linewidth(2));[/asy] $\textbf{(A)}\ \frac{2}{7} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{2}{3} \qquad\textbf{(D)}\ \frac{3}{4} \qquad\textbf{(E)}\ \frac{7}{9}$

Solution

Problem 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

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

Solution

Problem 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

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

Solution

Problem 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 $n$? [asy] 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)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle; draw(p); draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p); draw(shift((0,-2-sqrt(2)))*p); draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);[/asy] $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 26$

Solution

Problem 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?

$\textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{5}{9} \qquad\textbf{(C)}\ \frac{4}{7} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{3}$

Solution

Problem 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 $(4,3)$?

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

Solution

Problem 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 $AB/BC=$ [asy] draw(Circle((0,0), 13)); draw((-13,0)--(12,5)--(12,-5)--cycle); dot((-13,0)); dot((12,5)); dot((12,-5)); label("A", (-13,0), W); label("B", (12,5), NE); label("C", (12,-5), SE); [/asy] $\textbf{(A)}\ \frac{1}{2}\csc{\frac{1}{4}} \qquad\textbf{(B)}\ 2\cos{\frac{1}{2}} \qquad\textbf{(C)}\ 4\sin{\frac{1}{2}} \qquad\textbf{(D)}\ \csc{\frac{1}{2}} \qquad\textbf{(E)}\ 2\sec{\frac{1}{2}}$

Solution

Problem 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

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

Solution

See also

1994 AHSME (Problems • Answer Key • Resources)
Preceded by
1993 AHSME 		Followed by
1995 AHSME
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions


The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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