作者： tony

2013年AIME I 真题及答案

2013年AIME I 真题：

Problem 1

The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?

Problem 2

Find the number of five-digit positive integers, $n$, that satisfy the following conditions:

(a) the number n is divisible by 5,

    (b) the first and last digits of n are equal, and
    (c) the sum of the digits of n is divisible by 5.

Problem 3

Let $ABCD$ be a square, and let $E$ and $F$ be points on $\overline{AB}$ and $\overline{BC},$ respectively. The line through $E$ parallel to $\overline{BC}$ and the line through $F$ parallel to $\overline{AB}$ divide $ABCD$ into two squares and two nonsquare rectangles. The sum of the areas of the two squares is $\frac{9}{10}$ of the area of square $ABCD.$ Find $\frac{AE}{EB} + \frac{EB}{AE}.$

Problem 4

In the array of $13$ squares shown below, $8$ squares are colored red, and the remaining $5$ squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated $90^{\circ}$ around the central square is $\frac{1}{n}$ , where $n$ is a positive integer. Find $n$.

[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0)); draw((2,0)--(2,2)--(3,2)--(3,0)--(3,1)--(2,1)--(4,1)--(4,0)--(2,0)); draw((1,2)--(1,4)--(0,4)--(0,2)--(0,3)--(1,3)--(-1,3)--(-1,2)--(1,2)); draw((-1,1)--(-3,1)--(-3,0)--(-1,0)--(-2,0)--(-2,1)--(-2,-1)--(-1,-1)--(-1,1)); draw((0,-1)--(0,-3)--(1,-3)--(1,-1)--(1,-2)--(0,-2)--(2,-2)--(2,-1)--(0,-1)); size(100);[/asy]

Problem 5

The real root of the equation $8x^3-3x^2-3x-1=0$ can be written in the form $\frac{\sqrt[3]{a}+\sqrt[3]{b}+1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.

Problem 6

Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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1989年AJHSME 真题及答案

1989年AJHSME 真题：

Problem 1

$(1+11+21+31+41)+(9+19+29+39+49)=$

$\text{(A)}\ 150 \qquad \text{(B)}\ 199 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 249 \qquad \text{(E)}\ 250$

Problem 2

$\frac{2}{10}+\frac{4}{100}+\frac{6}{1000} =$

$\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246$

Problem 3

Which of the following numbers is the largest?

$\text{(A)}\ .99 \qquad \text{(B)}\ .9099 \qquad \text{(C)}\ .9 \qquad \text{(D)}\ .909 \qquad \text{(E)}\ .9009$

Problem 4

Estimate to determine which of the following numbers is closest to $\frac{401}{.205}$.

$\text{(A)}\ .2 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 200 \qquad \text{(E)}\ 2000$

Problem 5

$-15+9\times (6\div 3) =$

$\text{(A)}\ -48 \qquad \text{(B)}\ -12 \qquad \text{(C)}\ -3 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 12$

Problem 6

If the markings on the number line are equally spaced, what is the number $\text{y}$?

[asy] draw((-4,0)--(26,0),Arrows); for(int a=0; a<6; ++a) { draw((4a,-1)--(4a,1)); } label("0",(0,-1),S); label("20",(20,-1),S); label("y",(12,-1),S); [/asy]

$\text{(A)}\ 3 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 16$

Problem 7

If the value of $20$ quarters and $10$ dimes equals the value of $10$ quarters and $n$ dimes, then $n=$

$\text{(A)}\ 10 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 35 \qquad \text{(E)}\ 45$

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1988年AJHSME 真题及答案

1988年AJHSME 真题：

Problem 1

The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of

[asy] draw((-3,0)..(0,3)..(3,0)); draw((-3.5,0)--(-2.5,0)); draw((0,2.5)--(0,3.5)); draw((2.5,0)--(3.5,0)); draw((1.8,1.8)--(2.5,2.5)); draw((-1.8,1.8)--(-2.5,2.5)); draw((0,0)--3*dir(120),EndArrow); label("$10$",(-2.6,0),E); label("$11$",(2.6,0),W); [/asy]

$\text{(A)}\ 10.05 \qquad \text{(B)}\ 10.15 \qquad \text{(C)}\ 10.25 \qquad \text{(D)}\ 10.3 \qquad \text{(E)}\ 10.6$

Problem 2

The product $8\times .25\times 2\times .125 =$

$\text{(A)}\ \frac18 \qquad \text{(B)}\ \frac14 \qquad \text{(C)}\ \frac12 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 2$

Problem 3

$\frac{1}{10}+\frac{2}{20}+\frac{3}{30} =$

$\text{(A)}\ .1 \qquad \text{(B)}\ .123 \qquad \text{(C)}\ .2 \qquad \text{(D)}\ .3 \qquad \text{(E)}\ .6$

Problem 4

The figure consists of alternating light and dark squares. The number of dark squares exceeds the number of light squares by

$\text{(A)}\ 7 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11$

[asy] unitsize(12); //Force a white background in middle even when transparent fill((3,1)--(12,1)--(12,4)--(3,4)--cycle,white); //Black Squares, Gray Border (blends better than white) for(int a=0; a<7; ++a) { filldraw((2a,0)--(2a+1,0)--(2a+1,1)--(2a,1)--cycle,black,gray); } for(int b=7; b<15; ++b) { filldraw((b,14-b)--(b+1,14-b)--(b+1,15-b)--(b,15-b)--cycle,black,gray); } for(int c=1; c<7; ++c) { filldraw((c,c)--(c+1,c)--(c+1,c+1)--(c,c+1)--cycle,black,gray); } filldraw((6,4)--(7,4)--(7,5)--(6,5)--cycle,black,gray); filldraw((7,5)--(8,5)--(8,6)--(7,6)--cycle,black,gray); filldraw((8,4)--(9,4)--(9,5)--(8,5)--cycle,black,gray); //White Squares, Black Border filldraw((7,4)--(8,4)--(8,5)--(7,5)--cycle,white,black); for(int a=0; a<7; ++a) { filldraw((2a+1,0)--(2a+2,0)--(2a+2,1)--(2a+1,1)--cycle,white,black); } for(int b=9; b<15; ++b) { filldraw((b-1,14-b)--(b,14-b)--(b,15-b)--(b-1,15-b)--cycle,white,black); } for(int c=1; c<7; ++c) { filldraw((c+1,c)--(c+2,c)--(c+2,c+1)--(c+1,c+1)--cycle,white,black); } label("same",(6.3,2.45),N); label("pattern here",(7.5,1.4),N); [/asy]

Problem 5

If $\angle \text{CBD}$ is a right angle, then this protractor indicates that the measure of $\angle \text{ABC}$ is approximately

[asy] unitsize(36); pair A,B,C,D; A=3*dir(160); B=origin; C=3*dir(110); D=3*dir(20); draw((1.5,0)..(0,1.5)..(-1.5,0)); draw((2.5,0)..(0,2.5)..(-2.5,0)--cycle); draw(A--B); draw(C--B); draw(D--B); label("O",(-2.5,0),W); label("A",A,W); label("B",B,S); label("C",C,W); label("D",D,E); label("0",(-1.8,0),W); label("20",(-1.7,.5),NW); label("160",(1.6,.5),NE); label("180",(1.7,0),E); [/asy]

$\text{(A)}\ 20^\circ \qquad \text{(B)}\ 40^\circ \qquad \text{(C)}\ 50^\circ \qquad \text{(D)}\ 70^\circ \qquad \text{(E)}\ 120^\circ$

Problem 6

$\frac{(.2)^3}{(.02)^2} =$

$\text{(A)}\ .2 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 20$

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1985年AJHSME 真题及答案

1985年AJHSME 真题：

Problem 1

$\frac{3\times 5}{9\times 11}\times \frac{7\times 9\times 11}{3\times 5\times 7}=$

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 49 \qquad \textbf{(D)} \frac{1}{49}\ \qquad \textbf{(E)}\ 50$

Problem 2

$90+91+92+93+94+95+96+97+98+99=$

$\textbf{(A)}\ 845 \qquad \textbf{(B)}\ 945 \qquad \textbf{(C)}\ 1005 \qquad \textbf{(D)}\ 1025 \qquad \textbf{(E)}\ 1045$

Problem 3

$\frac{10^7}{5\times 10^4}=$

$\textbf{(A)}\ .002 \qquad \textbf{(B)}\ .2 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 200 \qquad \textbf{(E)}\ 2000$

Problem 4

The area of polygon $ABCDEF$, in square units, is

$\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 46 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 74$

[asy] draw((0,9)--(6,9)--(6,0)--(2,0)--(2,4)--(0,4)--cycle); label("$A$",(0,9),NW); label("$B$",(6,9),NE); label("$C$",(6,0),SE); label("$D$",(2,0),SW); label("$E$",(2,4),NE); label("$F$",(0,4),SW); label("6",(3,9),N); label("9",(6,4.5),E); label("4",(4,0),S); label("5",(0,6.5),W); [/asy]

Problem 5

[asy] unitsize(13); draw((0,0)--(20,0)); draw((0,0)--(0,15)); draw((0,3)--(-1,3)); draw((0,6)--(-1,6)); draw((0,9)--(-1,9)); draw((0,12)--(-1,12)); draw((0,15)--(-1,15)); fill((2,0)--(2,15)--(3,15)--(3,0)--cycle,black); fill((4,0)--(4,12)--(5,12)--(5,0)--cycle,black); fill((6,0)--(6,9)--(7,9)--(7,0)--cycle,black); fill((8,0)--(8,9)--(9,9)--(9,0)--cycle,black); fill((10,0)--(10,15)--(11,15)--(11,0)--cycle,black); label("A",(2.5,-.5),S); label("B",(4.5,-.5),S); label("C",(6.5,-.5),S); label("D",(8.5,-.5),S); label("F",(10.5,-.5),S); label("Grade",(15,-.5),S); label("$1$",(-1,3),W); label("$2$",(-1,6),W); label("$3$",(-1,9),W); label("$4$",(-1,12),W); label("$5$",(-1,15),W); [/asy]

The bar graph shows the grades in a mathematics class for the last grading period. If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory?

$\textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{2}{3} \qquad \textbf{(C)}\ \frac{3}{4} \qquad \textbf{(D)}\ \frac{4}{5} \qquad \textbf{(E)}\ \frac{9}{10}$

Problem 6

A stack of paper containing $500$ sheets is $5$ cm thick. Approximately how many sheets of this type of paper would there be in a stack $7.5$ cm high?

$\textbf{(A)}\ 250 \qquad \textbf{(B)}\ 550 \qquad \textbf{(C)}\ 667 \qquad \textbf{(D)}\ 750 \qquad \textbf{(E)}\ 1250$

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扫码免费预约领取更多AMC历年真题＋答案

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1987年AJHSME 真题及答案

1987年AJHSME 真题：

Problem 1

$.4+.02+.006=$

$\text{(A)}\ .012 \qquad \text{(B)}\ .066 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .24 \qquad \text{(E)} .426$

Problem 2

$\frac{2}{25}=$

$\text{(A)}\ .008 \qquad \text{(B)}\ .08 \qquad \text{(C)}\ .8 \qquad \text{(D)} 1.25 \qquad \text{(E)}\ 12.5$

Problem 3

$2(81+83+85+87+89+91+93+95+97+99)=$

$\text{(A)}\ 1600 \qquad \text{(B)}\ 1650 \qquad \text{(C)}\ 1700 \qquad \text{(D)}\ 1750 \qquad \text{(E)}\ 1800$

Problem 4

Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle?

$\text{(A)}\ 90 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 125 \qquad \text{(D)}\ 180 \qquad \text{(E)}\ 250$

Problem 5

The area of the rectangular region is

[asy] draw((0,0)--(4,0)--(4,2.2)--(0,2.2)--cycle,linewidth(.5 mm)); label(".22 m",(4,1.1),E); label(".4 m",(2,0),S); [/asy]

$\text{(A)}\ \text{.088 m}^2 \qquad \text{(B)}\ \text{.62 m}^2 \qquad \text{(C)}\ \text{.88 m}^2 \qquad \text{(D)}\ \text{1.24 m}^2 \qquad \text{(E)}\ \text{4.22 m}^2$

Problem 6

The smallest product one could obtain by multiplying two numbers in the set $\{ -7, -5, -1, 1, 3 \}$ is

$\text{(A)}\ -35 \qquad \text{(B)}\ -21 \qquad \text{(C)}\ -15 \qquad \text{(D)}\ -1 \qquad \text{(E)}\ 3$

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1986年AJHSME 真题及答案

1986年AJHSME 真题：

Problem 1

In July 1861, $366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month?

$\text{(A)}\ \frac{366}{31\times 24} \qquad \text{(B)}\ \frac{366\times 31}{24}\qquad \text{(C)}\ \frac{366\times 24}{31}\qquad \text{(D)}\ \frac{31\times 24}{366}\qquad \text{(E)}\ 366\times 31\times 24$

Problem 2

Which of the following numbers has the largest reciprocal?

$\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{2}{5} \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 1986$

Problem 3

The smallest sum one could get by adding three different numbers from the set $\{ 7,25,-1,12,-3 \}$ is

$\text{(A)}\ -3 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 21$

Problem 4

The product $(1.8)(40.3+.07)$ is closest to

$\text{(A)}\ 7 \qquad \text{(B)}\ 42 \qquad \text{(C)}\ 74 \qquad \text{(D)}\ 84 \qquad \text{(E)}\ 737$

Problem 5

A contest began at noon one day and ended $1000$ minutes later. At what time did the contest end?

$\text{(A)}\ \text{10:00 p.m.} \qquad \text{(B)}\ \text{midnight} \qquad \text{(C)}\ \text{2:30 a.m.} \qquad \text{(D)}\ \text{4:40 a.m.} \qquad \text{(E)}\ \text{6:40 a.m.}$

Problem 6

$\frac{2}{1-\frac{2}{3}}=$

$\text{(A)}\ -3 \qquad \text{(B)}\ -\frac{4}{3} \qquad \text{(C)}\ \frac{2}{3} \qquad \text{(D)}\ 2 \qquad \text{(E)}\ 6$

Problem 7

How many whole numbers are between $\sqrt{8}$ and $\sqrt{80}$?

$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9$

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1990年AJHSME 真题及答案

1990年AJHSME 真题：

Problem 1

What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $4,5,6,7,8,9$ in one of the six boxes in this addition problem?

[asy] unitsize(12); draw((0,0)--(10,0)); draw((-1.5,1.5)--(-1.5,2.5)); draw((-1,2)--(-2,2)); draw((1,1)--(3,1)--(3,3)--(1,3)--cycle); draw((1,4)--(3,4)--(3,6)--(1,6)--cycle); draw((4,1)--(6,1)--(6,3)--(4,3)--cycle); draw((4,4)--(6,4)--(6,6)--(4,6)--cycle); draw((7,1)--(9,1)--(9,3)--(7,3)--cycle); draw((7,4)--(9,4)--(9,6)--(7,6)--cycle); [/asy]

$\text{(A)}\ 947 \qquad \text{(B)}\ 1037 \qquad \text{(C)}\ 1047 \qquad \text{(D)}\ 1056 \qquad \text{(E)}\ 1245$

Problem 2

Which digit of $.12345$, when changed to $9$, gives the largest number?

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

Problem 3

What fraction of the square is shaded?

[asy] draw((0,0)--(0,3)--(3,3)--(3,0)--cycle); draw((0,2)--(2,2)--(2,0)); draw((0,1)--(1,1)--(1,0)); draw((0,0)--(3,3)); fill((0,0)--(0,1)--(1,1)--cycle,grey); fill((1,0)--(1,1)--(2,2)--(2,0)--cycle,grey); fill((0,2)--(2,2)--(3,3)--(0,3)--cycle,grey); [/asy]

$\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{2}{5} \qquad \text{(C)}\ \frac{5}{12} \qquad \text{(D)}\ \frac{3}{7} \qquad \text{(E)}\ \frac{1}{2}$

Problem 4

Which of the following could not be the unit's digit [one's digit] of the square of a whole number?

$\text{(A)}\ 1 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$

Problem 5

Which of the following is closest to the product $(.48017)(.48017)(.48017)$?

$\text{(A)}\ 0.011 \qquad \text{(B)}\ 0.110 \qquad \text{(C)}\ 1.10 \qquad \text{(D)}\ 11.0 \qquad \text{(E)}\ 110$

Problem 6

Which of these five numbers is the largest?

$\text{(A)}\ 13579+\frac{1}{2468} \qquad \text{(B)}\ 13579-\frac{1}{2468} \qquad \text{(C)}\ 13579\times \frac{1}{2468}$

$\text{(D)}\ 13579\div \frac{1}{2468} \qquad \text{(E)}\ 13579.2468$

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1991年AJHSME 真题及答案

1991年AJHSME 真题：

Problem 1

$1,000,000,000,000-777,777,777,777=$

$\text{(A)}\ 222,222,222,222 \qquad \text{(B)}\ 222,222,222,223 \qquad \text{(C)}\ 233,333,333,333 \qquad \\ \text{(D)}\ 322,222,222,223 \qquad \text{(E)}\ 333,333,333,333$

Problem 2

$\frac{16+8}{4-2}=$

$\text{(A)}\ 4 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 20$

Problem 3

Two hundred thousand times two hundred thousand equals

$\text{(A)}\ \text{four hundred thousand} \qquad \text{(B)}\ \text{four million} \qquad \text{(C)}\ \text{forty thousand} \qquad \\ \text{(D)}\ \text{four hundred million} \qquad \text{(E)}\ \text{forty billion}$

Problem 4

If $991+993+995+997+999=5000-N$, then $N=$

$\text{(A)}\ 5 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 25$

Problem 5

A "domino" is made up of two small squares:[asy] unitsize(12); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); draw((1,1)--(2,1)--(2,0)--(1,0)); [/asy]Which of the "checkerboards" illustrated below CANNOT be covered exactly and completely by a whole number of non-overlapping dominoes?

[asy] unitsize(12); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); fill((1,1)--(1,2)--(2,2)--(2,1)--cycle,black); fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,black); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,black); fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,black); fill((2,2)--(2,3)--(3,3)--(3,2)--cycle,black); draw((0,0)--(0,3)--(4,3)--(4,0)--cycle); draw((6,0)--(11,0)--(11,3)--(6,3)--cycle); fill((6,0)--(7,0)--(7,1)--(6,1)--cycle,black); fill((8,0)--(9,0)--(9,1)--(8,1)--cycle,black); fill((10,0)--(11,0)--(11,1)--(10,1)--cycle,black); fill((7,1)--(7,2)--(8,2)--(8,1)--cycle,black); fill((9,1)--(9,2)--(10,2)--(10,1)--cycle,black); fill((6,2)--(6,3)--(7,3)--(7,2)--cycle,black); fill((8,2)--(8,3)--(9,3)--(9,2)--cycle,black); fill((10,2)--(10,3)--(11,3)--(11,2)--cycle,black); draw((13,-1)--(13,3)--(17,3)--(17,-1)--cycle); fill((13,3)--(14,3)--(14,2)--(13,2)--cycle,black); fill((15,3)--(16,3)--(16,2)--(15,2)--cycle,black); fill((14,2)--(15,2)--(15,1)--(14,1)--cycle,black); fill((16,2)--(17,2)--(17,1)--(16,1)--cycle,black); fill((13,1)--(14,1)--(14,0)--(13,0)--cycle,black); fill((15,1)--(16,1)--(16,0)--(15,0)--cycle,black); fill((14,0)--(15,0)--(15,-1)--(14,-1)--cycle,black); fill((16,0)--(17,0)--(17,-1)--(16,-1)--cycle,black); draw((19,3)--(24,3)--(24,-1)--(19,-1)--cycle,black); fill((19,3)--(20,3)--(20,2)--(19,2)--cycle,black); fill((21,3)--(22,3)--(22,2)--(21,2)--cycle,black); fill((23,3)--(24,3)--(24,2)--(23,2)--cycle,black); fill((20,2)--(21,2)--(21,1)--(20,1)--cycle,black); fill((22,2)--(23,2)--(23,1)--(22,1)--cycle,black); fill((19,1)--(20,1)--(20,0)--(19,0)--cycle,black); fill((21,1)--(22,1)--(22,0)--(21,0)--cycle,black); fill((23,1)--(24,1)--(24,0)--(23,0)--cycle,black); fill((20,0)--(21,0)--(21,-1)--(20,-1)--cycle,black); fill((22,0)--(23,0)--(23,-1)--(22,-1)--cycle,black); draw((26,3)--(29,3)--(29,-3)--(26,-3)--cycle); fill((26,3)--(27,3)--(27,2)--(26,2)--cycle,black); fill((28,3)--(29,3)--(29,2)--(28,2)--cycle,black); fill((27,2)--(28,2)--(28,1)--(27,1)--cycle,black); fill((26,1)--(27,1)--(27,0)--(26,0)--cycle,black); fill((28,1)--(29,1)--(29,0)--(28,0)--cycle,black); fill((27,0)--(28,0)--(28,-1)--(27,-1)--cycle,black); fill((26,-1)--(27,-1)--(27,-2)--(26,-2)--cycle,black); fill((28,-1)--(29,-1)--(29,-2)--(28,-2)--cycle,black); fill((27,-2)--(28,-2)--(28,-3)--(27,-3)--cycle,black); [/asy]

$\text{(A)}\ 3\times 4 \qquad \text{(B)}\ 3\times 5 \qquad \text{(C)}\ 4\times 4 \qquad \text{(D)}\ 4\times 5 \qquad \text{(E)}\ 6\times 3$

Problem 6

Which number in the array below is both the largest in its column and the smallest in its row? (Columns go up and down, rows go right and left.)\[\begin{tabular}[t]{ccccc} 10 & 6 & 4 & 3 & 2 \\ 11 & 7 & 14 & 10 & 8 \\ 8 & 3 & 4 & 5 & 9 \\ 13 & 4 & 15 & 12 & 1 \\ 8 & 2 & 5 & 9 & 3 \end{tabular}\]

$\text{(A)}\ 1 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 15$

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1992年AJHSME 真题及答案

1992年AJHSME 真题：

Problem 1

$\dfrac{10-9+8-7+6-5+4-3+2-1}{1-2+3-4+5-6+7-8+9}=$

$\text{(A)}\ -1 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 10$

Problem 2

Which of the following is not equal to $\dfrac{5}{4}$?

$\text{(A)}\ \dfrac{10}{8} \qquad \text{(B)}\ 1\dfrac{1}{4} \qquad \text{(C)}\ 1\dfrac{3}{12} \qquad \text{(D)}\ 1\dfrac{1}{5} \qquad \text{(E)}\ 1\dfrac{10}{40}$

Problem 3

What is the largest difference that can be formed by subtracting two numbers chosen from the set $\{ -16,-4,0,2,4,12 \}$?

$\text{(A)}\ 10 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 16 \qquad \text{(D)}\ 28 \qquad \text{(E)}\ 48$

Problem 4

During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were singles. What percent of her hits were singles?

$\text{(A)}\ 28\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 75\% \qquad \text{(E)}\ 80\%$

Problem 5

A circle of diameter $1$ is removed from a $2\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?

[asy] fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray); draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1)); fill(circle((1,5/4),1/2),white); draw(circle((1,5/4),1/2),linewidth(1)); [/asy]

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

Problem 6

Suppose that[asy] unitsize(18); draw((0,0)--(2,0)--(1,sqrt(3))--cycle); label("$a$",(1,sqrt(3)-0.2),S); label("$b$",(sqrt(3)/10,0.1),ENE); label("$c$",(2-sqrt(3)/10,0.1),WNW); [/asy]means $a+b-c$. For example,[asy] unitsize(18); draw((0,0)--(2,0)--(1,sqrt(3))--cycle); label("$5$",(1,sqrt(3)-0.2),S); label("$4$",(sqrt(3)/10,0.1),ENE); label("$6$",(2-sqrt(3)/10,0.1),WNW); [/asy]is $5+4-6 = 3$. Then the sum[asy] unitsize(18); draw((0,0)--(2,0)--(1,sqrt(3))--cycle); label("$1$",(1,sqrt(3)-0.2),S); label("$3$",(sqrt(3)/10,0.1),ENE); label("$4$",(2-sqrt(3)/10,0.1),WNW); draw((3,0)--(5,0)--(4,sqrt(3))--cycle); label("$2$",(4,sqrt(3)-0.2),S); label("$5$",(3+sqrt(3)/10,0.1),ENE); label("$6$",(5-sqrt(3)/10,0.1),WNW); label("$+$",(2.5,-0.1),N); [/asy]is

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

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1993年AJHSME 真题及答案

1993年AJHSME 真题：

Problem 1

Which pair of numbers does NOT have a product equal to $36$?

$\text{(A)}\ \{ -4,-9\} \qquad \text{(B)}\ \{ -3,-12\} \qquad \text{(C)}\ \left\{ \dfrac{1}{2},-72\right\} \qquad \text{(D)}\ \{ 1,36\} \qquad \text{(E)}\ \left\{\dfrac{3}{2},24\right\}$

Problem 2

When the fraction $\dfrac{49}{84}$ is expressed in simplest form, then the sum of the numerator and the denominator will be

$\text{(A)}\ 11 \qquad \text{(B)}\ 17 \qquad \text{(C)}\ 19 \qquad \text{(D)}\ 33 \qquad \text{(E)}\ 133$

Problem 3

Which of the following numbers has the largest prime factor?

$\text{(A)}\ 39 \qquad \text{(B)}\ 51 \qquad \text{(C)}\ 77 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 121$

Problem 4

$1000\times 1993 \times 0.1993 \times 10 =$

$\text{(A)}\ 1.993\times 10^3 \qquad \text{(B)}\ 1993.1993 \qquad \text{(C)}\ (199.3)^2 \qquad \text{(D)}\ 1,993,001.993 \qquad \text{(E)}\ (1993)^2$

Problem 5

Which one of the following bar graphs could represent the data from the circle graph?

[asy] unitsize(36); draw(circle((0,0),1),gray); fill((0,0)--arc((0,0),(0,-1),(1,0))--cycle,gray); fill((0,0)--arc((0,0),(1,0),(0,1))--cycle,black); [/asy]

[asy] unitsize(4); fill((1,0)--(1,15)--(5,15)--(5,0)--cycle,gray); fill((6,0)--(6,15)--(10,15)--(10,0)--cycle,black); draw((11,0)--(11,20)--(15,20)--(15,0)); fill((26,0)--(26,15)--(30,15)--(30,0)--cycle,gray); fill((31,0)--(31,15)--(35,15)--(35,0)--cycle,black); draw((36,0)--(36,15)--(40,15)--(40,0)); fill((51,0)--(51,10)--(55,10)--(55,0)--cycle,gray); fill((56,0)--(56,10)--(60,10)--(60,0)--cycle,black); draw((61,0)--(61,20)--(65,20)--(65,0)); fill((76,0)--(76,10)--(80,10)--(80,0)--cycle,gray); fill((81,0)--(81,15)--(85,15)--(85,0)--cycle,black); draw((86,0)--(86,20)--(90,20)--(90,0)); fill((101,0)--(101,15)--(105,15)--(105,0)--cycle,gray); fill((106,0)--(106,10)--(110,10)--(110,0)--cycle,black); draw((111,0)--(111,20)--(115,20)--(115,0)); for(int a = 0; a < 5; ++a) { draw((25*a,21)--(25*a,0)--(25*a+16,0)); } label("(A)",(8,21),N); label("(B)",(33,21),N); label("(C)",(58,21),N); label("(D)",(83,21),N); label("(E)",(108,21),N); [/asy]

Problem 6

A can of soup can feed $3$ adults or $5$ children. If there are $5$ cans of soup and $15$ children are fed, then how many adults would the remaining soup feed?

$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 25$

Problem 7

$3^3+3^3+3^3 =$

$\text{(A)}\ 3^4 \qquad \text{(B)}\ 9^3 \qquad \text{(C)}\ 3^9 \qquad \text{(D)}\ 27^3 \qquad \text{(E)}\ 3^{27}$

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