AMC 8 1990
1
What is the smallest sum of two 
-digit numbers that can be obtained by placing each of the six digits 
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]](//latex.artofproblemsolving.com/a/c/0/ac0ed1600dff3305fbf8b59fad0487b1019491dc.png)
-digit numbers that can be obtained by placing each of the six digits 
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]](http://latex.artofproblemsolving.com/a/c/0/ac0ed1600dff3305fbf8b59fad0487b1019491dc.png)
,
,![[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]](http://latex.artofproblemsolving.com/9/3/2/9326778edc85bbfb24c19c83fcc9ea5609620582.png)
4
Which of the following could not be the unit's digit [one's digit] of the square of a whole number?
?
7
When three different numbers from the set 
are multiplied, the largest possible product is
are multiplied, the largest possible product is
8
A dress originally priced at 80 dollars was put on sale for 
off. If 
tax was added to the sale price, then the total selling price (in dollars) of the dress was
off. If 
tax was added to the sale price, then the total selling price (in dollars) of the dress was
9
The grading scale shown is used at Jones Junior High. The fifteen scores in Mr. Freeman's class were:
![\[ \begin{tabular}[t]{lllllllll}89, & 72, & 54, & 97, & 77, & 92, & 85, & 74, & 75,\\ 63, & 84, & 78, & 71, & 80, & 90. & & &\\ \end{tabular} \]](//latex.artofproblemsolving.com/8/4/7/847f11805d6b56ec2fc25d100b58b9a576ec6bc9.png)
In Mr. Freeman's class, what percent of the students received a grade of C?
![\[ \boxed{\begin{tabular}[t]{cc}A: & 93-100\\ B: & 85-92\\ C: & 75-84\\ D: & 70-74\\ F: & 0-69\end{tabular}} \]](//latex.artofproblemsolving.com/7/3/d/73d5ce870d439929b73b11d4f30d5faa04c8f989.png)
![\[ \begin{tabular}[t]{lllllllll}89, & 72, & 54, & 97, & 77, & 92, & 85, & 74, & 75,\\ 63, & 84, & 78, & 71, & 80, & 90. & & &\\ \end{tabular} \]](http://latex.artofproblemsolving.com/8/4/7/847f11805d6b56ec2fc25d100b58b9a576ec6bc9.png)
In Mr. Freeman's class, what percent of the students received a grade of C?
![\[ \boxed{\begin{tabular}[t]{cc}A: & 93-100\\ B: & 85-92\\ C: & 75-84\\ D: & 70-74\\ F: & 0-69\end{tabular}} \]](http://latex.artofproblemsolving.com/7/3/d/73d5ce870d439929b73b11d4f30d5faa04c8f989.png)
10
On this monthly calendar, the date behind one of the letters is added to the date behind 
. If this sum equals the sum of the dates behind 
and 
, then the letter is
![[asy]
unitsize(12);
draw((1,1)--(23,1));
draw((0,5)--(23,5));
draw((0,9)--(23,9));
draw((0,13)--(23,13));
for(int a=0; a<6; ++a)
{
draw((4a+2,0)--(4a+2,14));
}
label("Tues.",(4,14),N); label("Wed.",(8,14),N); label("Thurs.",(12,14),N);
label("Fri.",(16,14),N); label("Sat.",(20,14),N);
label("C",(12,10.3),N); label("$\textbf{A}$",(16,10.3),N); label("Q",(12,6.3),N);
label("S",(4,2.3),N); label("$\textbf{B}$",(8,2.3),N); label("P",(12,2.3),N);
label("T",(16,2.3),N); label("R",(20,2.3),N);[/asy]](//latex.artofproblemsolving.com/f/b/9/fb9b9edd44ca25f5d555ce485baf09ebd4ba1af0.png)
. If this sum equals the sum of the dates behind 
and 
, then the letter is![[asy]
unitsize(12);
draw((1,1)--(23,1));
draw((0,5)--(23,5));
draw((0,9)--(23,9));
draw((0,13)--(23,13));
for(int a=0; a<6; ++a)
{
draw((4a+2,0)--(4a+2,14));
}
label("Tues.",(4,14),N); label("Wed.",(8,14),N); label("Thurs.",(12,14),N);
label("Fri.",(16,14),N); label("Sat.",(20,14),N);
label("C",(12,10.3),N); label("$\textbf{A}$",(16,10.3),N); label("Q",(12,6.3),N);
label("S",(4,2.3),N); label("$\textbf{B}$",(8,2.3),N); label("P",(12,2.3),N);
label("T",(16,2.3),N); label("R",(20,2.3),N);[/asy]](http://latex.artofproblemsolving.com/f/b/9/fb9b9edd44ca25f5d555ce485baf09ebd4ba1af0.png)
11
The numbers on the faces of this cube are consecutive whole numbers. The sums of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is
![[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3));
draw((3,3)--(5,5));
label("$15$",(1.5,1.2),N); label("$11$",(4,2.3),N); label("$14$",(2.5,3.7),N);[/asy]](//latex.artofproblemsolving.com/2/d/9/2d91f115b45d4b36eb70567ff50ef4bf311050b0.png)
![[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3));
draw((3,3)--(5,5));
label("$15$",(1.5,1.2),N); label("$11$",(4,2.3),N); label("$14$",(2.5,3.7),N);[/asy]](http://latex.artofproblemsolving.com/2/d/9/2d91f115b45d4b36eb70567ff50ef4bf311050b0.png)
12
There are twenty-four 4-digit numbers that use each of the four digits 2, 5, 7, and 4exactly once. Listed in numerical order from smallest to largest, the number in the 
position in the list is
position in the list is
13
One proposal for new postage rates for a letter was 
cents for the first ounce and 
cents for each additional ounce (or fraction of an ounce). The postage for a letter weighing 
ounces was
cents for the first ounce and 
cents for each additional ounce (or fraction of an ounce). The postage for a letter weighing 
ounces was
14
A bag contains only blue balls and green balls. There are 
blue balls. If the probability of drawing a blue ball at random from this bag is 
, then the number of green balls in the bag is
blue balls. If the probability of drawing a blue ball at random from this bag is 
, then the number of green balls in the bag is
.![[asy]
draw((0,2)--(2,2)--(2,1)--(3,1)--(3,0)--(1,0)--(1,1)--(0,1)--cycle,linewidth(1));
draw((1,2)--(1,1)--(2,1)--(2,0),dashed);[/asy]](http://latex.artofproblemsolving.com/b/0/5/b0582d0fb08519df7b9bd1a5f8bc977672bb504b.png)
17
A straight concrete sidewalk is to be feet wide, 
feet long, and inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards?
feet wide, 
feet long, and inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards?
18
Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have?
![[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3));
draw((3,3)--(5,5));
draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1));
draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1));[/asy]](//latex.artofproblemsolving.com/6/4/1/64195fa453f04b6517f67df0cf0f23a65f288446.png)
Assume that the planes cutting the prism do not intersect anywhere in or on the prism.
![[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3));
draw((3,3)--(5,5));
draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1));
draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1));[/asy]](http://latex.artofproblemsolving.com/6/4/1/64195fa453f04b6517f67df0cf0f23a65f288446.png)
Assume that the planes cutting the prism do not intersect anywhere in or on the prism.
19
There are 
seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone?
seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone?
20
The annual incomes of 
families range from 
dollars to 
dollars. In error, the largest income was entered on the computer as 
dollars. The difference between the mean of the incorrect data and the mean of the actual data is
families range from 
dollars to 
dollars. In error, the largest income was entered on the computer as 
dollars. The difference between the mean of the incorrect data and the mean of the actual data is
21
A list of 
numbers is formed by beginning with two given numbers. Each new number in the list is the product of the two previous numbers. Find the first number if the last three are shown:
![\[ \text{\underline{\hspace{3 mm}?\hspace{3 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{2 mm}16\hspace{2 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{2 mm}64\hspace{2 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{1 mm}1024\hspace{1 mm}}} \]](//latex.artofproblemsolving.com/8/a/8/8a866850417f06ab79826670f04b9d4df0a014d7.png)
numbers is formed by beginning with two given numbers. Each new number in the list is the product of the two previous numbers. Find the first number if the last three are shown:![\[ \text{\underline{\hspace{3 mm}?\hspace{3 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{2 mm}16\hspace{2 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{2 mm}64\hspace{2 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{1 mm}1024\hspace{1 mm}}} \]](http://latex.artofproblemsolving.com/8/a/8/8a866850417f06ab79826670f04b9d4df0a014d7.png)
22
Several students are seated at a large circular table. They pass around a bag containing 
pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then the number of students at the table could be
pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then the number of students at the table could be
23
The graph relates the distance traveled [in miles] to the time elapsed [in hours] on a trip taken by an experimental airplane. During which hour was the average speed of this airplane the largest?
![[asy]
unitsize(12);
for(int a=1; a<13; ++a)
{
draw((2a,-1)--(2a,1));
}
draw((-1,4)--(1,4)); draw((-1,8)--(1,8)); draw((-1,12)--(1,12)); draw((-1,16)--(1,16));
draw((0,0)--(0,17));
draw((-5,0)--(33,0));
label("$0$",(0,-1),S); label("$1$",(2,-1),S); label("$2$",(4,-1),S); label("$3$",(6,-1),S);
label("$4$",(8,-1),S); label("$5$",(10,-1),S); label("$6$",(12,-1),S); label("$7$",(14,-1),S);
label("$8$",(16,-1),S); label("$9$",(18,-1),S); label("$10$",(20,-1),S);
label("$11$",(22,-1),S); label("$12$",(24,-1),S);
label("Time in hours",(11,-2),S);
label("$500$",(-1,4),W); label("$1000$",(-1,8),W); label("$1500$",(-1,12),W);
label("$2000$",(-1,16),W);
label(rotate(90)*"Distance traveled in miles",(-4,10),W);
draw((0,0)--(2,3)--(4,7.2)--(6,8.5));
draw((6,8.5)--(16,12.5)--(18,14)--(24,15));[/asy]](//latex.artofproblemsolving.com/9/0/8/9086c681dd321e50ea70ab09bc47e053e96d3f61.png)
![[asy]
unitsize(12);
for(int a=1; a<13; ++a)
{
draw((2a,-1)--(2a,1));
}
draw((-1,4)--(1,4)); draw((-1,8)--(1,8)); draw((-1,12)--(1,12)); draw((-1,16)--(1,16));
draw((0,0)--(0,17));
draw((-5,0)--(33,0));
label("$0$",(0,-1),S); label("$1$",(2,-1),S); label("$2$",(4,-1),S); label("$3$",(6,-1),S);
label("$4$",(8,-1),S); label("$5$",(10,-1),S); label("$6$",(12,-1),S); label("$7$",(14,-1),S);
label("$8$",(16,-1),S); label("$9$",(18,-1),S); label("$10$",(20,-1),S);
label("$11$",(22,-1),S); label("$12$",(24,-1),S);
label("Time in hours",(11,-2),S);
label("$500$",(-1,4),W); label("$1000$",(-1,8),W); label("$1500$",(-1,12),W);
label("$2000$",(-1,16),W);
label(rotate(90)*"Distance traveled in miles",(-4,10),W);
draw((0,0)--(2,3)--(4,7.2)--(6,8.5));
draw((6,8.5)--(16,12.5)--(18,14)--(24,15));[/asy]](http://latex.artofproblemsolving.com/9/0/8/9086c681dd321e50ea70ab09bc47e053e96d3f61.png)
24
Three 
's and a 
will balance nine 
's. One will balance a and a .
![[asy]
unitsize(5.5);
fill((0,0)--(-4,-2)--(4,-2)--cycle,black);
draw((-12,2)--(-12,0)--(12,0)--(12,2));
draw(ellipse((-12,5),8,3)); draw(ellipse((12,5),8,3));
label("$\Delta \hspace{2 mm}\Delta \hspace{2 mm}\Delta \hspace{2 mm}\diamondsuit $",(-12,6.5),S);
label("$\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm} \bullet $",(12,5.2),N);
label("$\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet $",(12,5.2),S);
fill((44,0)--(40,-2)--(48,-2)--cycle,black);
draw((34,2)--(34,0)--(54,0)--(54,2));
draw(ellipse((34,5),6,3)); draw(ellipse((54,5),6,3));
label("$\Delta $",(34,6.5),S);
label("$\bullet \hspace{2 mm}\diamondsuit $",(54,6.5),S);[/asy]](//latex.artofproblemsolving.com/8/9/f/89fd9f6830d1892ea672f489bcc105b1f95ea37d.png)
How many's will balance the two 's in this balance?
![[asy]
unitsize(5.5);
fill((0,0)--(-4,-2)--(4,-2)--cycle,black);
draw((-12,4)--(-12,2)--(12,-2)--(12,0));
draw(ellipse((-12,7),6.5,3)); draw(ellipse((12,3),6.5,3));
label("$?$",(-12,8.5),S);
label("$\diamondsuit \hspace{2 mm}\diamondsuit $",(12,4.5),S);[/asy]](//latex.artofproblemsolving.com/d/c/7/dc7bbf671c8e780ffadf0483c7c2d4890d5707e9.png)
's and a 
will balance nine 
's. One will balance a and a .![[asy]
unitsize(5.5);
fill((0,0)--(-4,-2)--(4,-2)--cycle,black);
draw((-12,2)--(-12,0)--(12,0)--(12,2));
draw(ellipse((-12,5),8,3)); draw(ellipse((12,5),8,3));
label("$\Delta \hspace{2 mm}\Delta \hspace{2 mm}\Delta \hspace{2 mm}\diamondsuit $",(-12,6.5),S);
label("$\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm} \bullet $",(12,5.2),N);
label("$\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet $",(12,5.2),S);
fill((44,0)--(40,-2)--(48,-2)--cycle,black);
draw((34,2)--(34,0)--(54,0)--(54,2));
draw(ellipse((34,5),6,3)); draw(ellipse((54,5),6,3));
label("$\Delta $",(34,6.5),S);
label("$\bullet \hspace{2 mm}\diamondsuit $",(54,6.5),S);[/asy]](http://latex.artofproblemsolving.com/8/9/f/89fd9f6830d1892ea672f489bcc105b1f95ea37d.png)
How many
's will balance the two 's in this balance?![[asy]
unitsize(5.5);
fill((0,0)--(-4,-2)--(4,-2)--cycle,black);
draw((-12,4)--(-12,2)--(12,-2)--(12,0));
draw(ellipse((-12,7),6.5,3)); draw(ellipse((12,3),6.5,3));
label("$?$",(-12,8.5),S);
label("$\diamondsuit \hspace{2 mm}\diamondsuit $",(12,4.5),S);[/asy]](http://latex.artofproblemsolving.com/d/c/7/dc7bbf671c8e780ffadf0483c7c2d4890d5707e9.png)
25
How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different.
![[asy]
fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,gray); fill((1,2)--(2,2)--(2,3)--(1,3)--cycle,gray);
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,linewidth(1));
draw((2,0)--(2,3),linewidth(1)); draw((0,1)--(3,1),linewidth(1));
draw((1,0)--(1,3),linewidth(1)); draw((0,2)--(3,2),linewidth(1));
fill((6,0)--(8,0)--(8,1)--(6,1)--cycle,gray);
draw((6,0)--(9,0)--(9,3)--(6,3)--cycle,linewidth(1));
draw((8,0)--(8,3),linewidth(1)); draw((6,1)--(9,1),linewidth(1));
draw((7,0)--(7,3),linewidth(1)); draw((6,2)--(9,2),linewidth(1));
fill((14,1)--(15,1)--(15,3)--(14,3)--cycle,gray);
draw((12,0)--(15,0)--(15,3)--(12,3)--cycle,linewidth(1));
draw((14,0)--(14,3),linewidth(1)); draw((12,1)--(15,1),linewidth(1));
draw((13,0)--(13,3),linewidth(1)); draw((12,2)--(15,2),linewidth(1));
fill((18,1)--(19,1)--(19,3)--(18,3)--cycle,gray);
draw((18,0)--(21,0)--(21,3)--(18,3)--cycle,linewidth(1));
draw((20,0)--(20,3),linewidth(1)); draw((18,1)--(21,1),linewidth(1));
draw((19,0)--(19,3),linewidth(1)); draw((18,2)--(21,2),linewidth(1));[/asy]](//latex.artofproblemsolving.com/2/9/8/29838ea5993bbb7d16a3fcf99cd9802cb31a371f.png)
![[asy]
fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,gray); fill((1,2)--(2,2)--(2,3)--(1,3)--cycle,gray);
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,linewidth(1));
draw((2,0)--(2,3),linewidth(1)); draw((0,1)--(3,1),linewidth(1));
draw((1,0)--(1,3),linewidth(1)); draw((0,2)--(3,2),linewidth(1));
fill((6,0)--(8,0)--(8,1)--(6,1)--cycle,gray);
draw((6,0)--(9,0)--(9,3)--(6,3)--cycle,linewidth(1));
draw((8,0)--(8,3),linewidth(1)); draw((6,1)--(9,1),linewidth(1));
draw((7,0)--(7,3),linewidth(1)); draw((6,2)--(9,2),linewidth(1));
fill((14,1)--(15,1)--(15,3)--(14,3)--cycle,gray);
draw((12,0)--(15,0)--(15,3)--(12,3)--cycle,linewidth(1));
draw((14,0)--(14,3),linewidth(1)); draw((12,1)--(15,1),linewidth(1));
draw((13,0)--(13,3),linewidth(1)); draw((12,2)--(15,2),linewidth(1));
fill((18,1)--(19,1)--(19,3)--(18,3)--cycle,gray);
draw((18,0)--(21,0)--(21,3)--(18,3)--cycle,linewidth(1));
draw((20,0)--(20,3),linewidth(1)); draw((18,1)--(21,1),linewidth(1));
draw((19,0)--(19,3),linewidth(1)); draw((18,2)--(21,2),linewidth(1));[/asy]](http://latex.artofproblemsolving.com/2/9/8/29838ea5993bbb7d16a3fcf99cd9802cb31a371f.png)
These problems are copyright 
