Difference between revisions of "1994 AHSME Problems"
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== Problem 1 == | == Problem 1 == | ||
+ | <math>4^4 \cdot 9^4 \cdot 4^9 \cdot 9^9=</math> | ||
+ | <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> | ||
[[1994 AHSME Problems/Problem 1|Solution]] | [[1994 AHSME Problems/Problem 1|Solution]] | ||
== Problem 2 == | == Problem 2 == | ||
− | + | <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> | ||
[[1994 AHSME Problems/Problem 2|Solution]] | [[1994 AHSME Problems/Problem 2|Solution]] | ||
== Problem 3 == | == Problem 3 == | ||
+ | How many of the following are equal to <math>x^x+x^x</math> for all <math>x>0</math>? | ||
+ | <math>\textbf{I:}\ 2x^x \qquad\textbf{II:}\ x^{2x} \qquad\textbf{III:}\ (2x)^x \qquad\textbf{IV:}\ (2x)^{2x}</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 3|Solution]] | [[1994 AHSME Problems/Problem 3|Solution]] | ||
== Problem 4 == | == 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> | ||
+ | <math> \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12.5 \qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 17.5 \qquad\textbf{(E)}\ 20 </math> | ||
[[1994 AHSME Problems/Problem 4|Solution]] | [[1994 AHSME Problems/Problem 4|Solution]] | ||
== Problem 5 == | == 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 | ||
+ | <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> | ||
[[1994 AHSME Problems/Problem 5|Solution]] | [[1994 AHSME Problems/Problem 5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
+ | In the sequence | ||
+ | <cmath> ..., a, b, c, d, 0, 1, 1, 2, 3, 5, 8,... </cmath> | ||
+ | each term is the sum of the two terms to its left. Find <math>a</math>. | ||
+ | <math> \textbf{(A)}\ -3 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 3 </math> | ||
[[1994 AHSME Problems/Problem 6|Solution]] | [[1994 AHSME Problems/Problem 6|Solution]] | ||
== Problem 7 == | == 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 | |
+ | <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> | ||
[[1994 AHSME Problems/Problem 7|Solution]] | [[1994 AHSME Problems/Problem 7|Solution]] | ||
== Problem 8 == | == 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]] | [[1994 AHSME Problems/Problem 8|Solution]] | ||
== Problem 9 == | == 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]] | [[1994 AHSME Problems/Problem 9|Solution]] | ||
== Problem 10 == | == 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]] | [[1994 AHSME Problems/Problem 10|Solution]] | ||
== Problem 11 == | == 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]] | [[1994 AHSME Problems/Problem 11|Solution]] | ||
== Problem 12 == | == 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]] | [[1994 AHSME Problems/Problem 12|Solution]] | ||
== Problem 13 == | == 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]] | [[1994 AHSME Problems/Problem 13|Solution]] | ||
== Problem 14 == | == 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]] | [[1994 AHSME Problems/Problem 14|Solution]] | ||
== Problem 15 == | == 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]] | [[1994 AHSME Problems/Problem 15|Solution]] | ||
== Problem 16 == | == 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]] | [[1994 AHSME Problems/Problem 16|Solution]] | ||
== Problem 17 == | == 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]] | [[1994 AHSME Problems/Problem 17|Solution]] | ||
== Problem 18 == | == 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]] | [[1994 AHSME Problems/Problem 18|Solution]] | ||
== Problem 19 == | == 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]] | [[1994 AHSME Problems/Problem 19|Solution]] | ||
== Problem 20 == | == 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]] | [[1994 AHSME Problems/Problem 20|Solution]] | ||
== Problem 21 == | == 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]] | [[1994 AHSME Problems/Problem 21|Solution]] | ||
== Problem 22 == | == 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]] | [[1994 AHSME Problems/Problem 22|Solution]] | ||
== Problem 23 == | == 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]] | [[1994 AHSME Problems/Problem 23|Solution]] | ||
== Problem 24 == | == 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]] | [[1994 AHSME Problems/Problem 24|Solution]] | ||
== Problem 25 == | == 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]] | [[1994 AHSME Problems/Problem 25|Solution]] | ||
== Problem 26 == | == 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]] | [[1994 AHSME Problems/Problem 26|Solution]] | ||
== Problem 27 == | == 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]] | [[1994 AHSME Problems/Problem 27|Solution]] | ||
== Problem 28 == | == 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]] | [[1994 AHSME Problems/Problem 28|Solution]] | ||
== Problem 29 == | == 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]] | [[1994 AHSME Problems/Problem 29|Solution]] | ||
== Problem 30 == | == 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]] | [[1994 AHSME Problems/Problem 30|Solution]] |
Revision as of 22:09, 27 June 2014
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 See also
Problem 1
Problem 2
Problem 3
How many of the following are equal to for all
?
Problem 4
In the -plane, the segment with endpoints
and
is the diameter of a circle. If the point
is on the circle, then
Problem 5
Pat intended to multiply a number by but instead divided by
. Pat then meant to add
but instead subtracted
. After these mistakes, the result was
. If the correct operations had been used, the value produced would have been
Problem 6
In the sequence
each term is the sum of the two terms to its left. Find
.
Problem 7
Squares and
are congruent,
, and
is the center of square
. The area of the region in the plane covered by these squares is
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 . The area of the region bounded by the polygon is
Problem 9
If is four times
, and the complement of
is four times the complement of
, then
Problem 10
For distinct real numbers and
, let
be the larger of
and
and let
be the smaller of
and
. If
, then
Problem 11
Three cubes of volume and
are glued together at their faces. The smallest possible surface area of the resulting configuration is
Problem 12
If , then
Problem 13
In triangle ,
. If there is a point
strictly between
and
such that
, then
Problem 14
Find the sum of the arithmetic series
Problem 15
For how many in
is the tens digit of
odd?
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?
Problem 17
An by
rectangle has the same center as a circle of radius
. The area of the region common to both the rectangle and the circle is
Problem 18
Triangle is inscribed in a circle, and
. If
and
are adjacent vertices of a regular polygon of
sides inscribed in this circle, then
Problem 19
Label one disk "", two disks "
", three disks "
"
fifty disks "
". Put these
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
Problem 20
Suppose is a geometric sequence with common ratio
and
. If
is an arithmetic sequence, then
is
Problem 21
Find the number of counter examples to the statement:
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?
Problem 23
In the -plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at
and
. The slope of the line through the origin that divides the area of this region exactly in half is
Problem 24
A sample consisting of five observations has an arithmetic mean of and a median of
. The smallest value that the range (largest observation minus smallest) can assume for such a sample is
Problem 25
If and
are non-zero real numbers such that
then the integer nearest to
is
Problem 26
A regular polygon of sides is exactly enclosed (no overlaps, no gaps) by
regular polygons of
sides each. (Shown here for
.) If
, what is the value of
?
Problem 27
A bag of popping corn contains white kernels and
yellow kernels. Only
of the white kernels will pop, whereas
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?
Problem 28
In the -plane, how many lines whose
-intercept is a positive prime number and whose
-intercept is a positive integer pass through the point
?
Problem 29
Points and
on a circle of radius
are situated so that
, and the length of minor arc
is
. If angles are measured in radians, then
Problem 30
When 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
. The smallest possible value of
is
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.