2016年AMC 12B 真题：
Problem 1
What is the value of 
when 
?
Problem 2
The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of 
and 
is closest to which integer?
Problem 3
Let 
. What is the value of 
?
Problem 4
The ratio of the measures of two acute angles is 
, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
Problem 5
The War of 
started with a declaration of war on Thursday, June 
, . The peace treaty to end the war was signed 
days later, on December 
, 
. On what day of the week was the treaty signed?
Problem 6
All three vertices of 
lie on the parabola defined by 
, with 
at the origin and 
parallel to the 
-axis. The area of the triangle is 
. What is the length of 
?
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?
?
with 
by![\[\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor\]](https://latex.artofproblemsolving.com/9/5/c/95c21b1ce04c5f3c86f558991d39aa35a13bca21.png)
where 
denotes the greatest integer less than or equal to 
. What is the value of 
?
data values 
are all equal to 
). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?
coins in the second row, 
coins in the third row, and so on up to 
coins in the 
comic books in it, and she is adding to her collection at the rate of 
comic books in it, and he is adding to his collection at the rate of 
comic books per month. After how many months will LaShawn's collection have twice as many comic books as Kymbrea's?
satify the inequalities 
, 
, and 
. Which of the following numbers is necessarily positive?
. What is the value of 
?
kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 
kilometers per hour. In all it took her 
minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
![$[6,19,33,33,39,41,41,43,51,57]$](https://latex.artofproblemsolving.com/0/8/3/083926c0ff677e76283cde6dfcdb0c8e10c35098.png)
has median 
, first quartile 
, and third quartile 
. An outlier in a data set is a value that is more than 
times the interquartile range below the first quartile 
or more than 
, where the interquartile range is defined as 
. How many outliers does this data set have?
and 
as the endpoints of a diameter intersects the 
people who all know each other and 
through 
. She places the rods with lengths 
, 
, and 
on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was 
full of water. What is the ratio of the volume of the first container to the volume of the second container?
is not prime, then 
is prime." Which of the following values of 
onto the line segment 
so that the image of 
is 
and the image of 
is 
reflection in the 
counterclockwise rotation around the origin by 
translation by 
reflection in the 
clockwise rotation about the origin by 
. What is the sum of the digits of 
pieces of red candy, 
pieces of green candy, 
pieces of blue candy, or 
are 
are there in the plane such that the perimeter of 
is 
units and the area of 
square units?
inches is 
is 
of 
. What percent of 
?
yellow balls, 
blue balls, 
white balls, and 
black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 
?
and 
. What is the area of the triangle enclosed by these two lines and the line 
?
with a regular, infinite, recurring pattern of squares and line segments.![[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); [/asy]](https://latex.artofproblemsolving.com/a/2/f/a2f165e74d6ddc1b021c8af4c74374a1a78d2815.png)
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?
![[asy] import olympiad; unitsize(25); for (int i = 0; i < 3; ++i) { for (int j = 0; j < 9; ++j) { pair A = (j,i); } } for (int i = 0; i < 3; ++i) { for (int j = 0; j < 9; ++j) { if (j != 8) { draw((j,i)--(j+1,i), dashed); } if (i != 2) { draw((j,i)--(j,i+1), dashed); } } } draw((0,0)--(2,2),linewidth(2)); draw((2,0)--(2,2),linewidth(2)); draw((1,1)--(2,1),linewidth(2)); draw((3,0)--(3,2),linewidth(2)); draw((5,0)--(5,2),linewidth(2)); draw((4,1)--(3,2),linewidth(2)); draw((4,1)--(5,2),linewidth(2)); draw((6,0)--(8,0),linewidth(2)); draw((6,2)--(8,2),linewidth(2)); draw((6,0)--(6,2),linewidth(2)); [/asy]](https://latex.artofproblemsolving.com/6/6/4/6649eb6b62018218d4086bde7597f2da9b4b10e3.png)
miles per hour, during which her car gets 
per mile, and her only expense is gasoline at 
per gallon. What is her net rate of pay, in dollars per hour, after this expense?
and 
, inclusive) having only even digits are divisible by 
integers from 
to 
inclusive, can be arranged to form a 
![[asy] import olympiad; unitsize(25); filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7)); filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7)); filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7)); for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { pair A = (j,i); } } for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { if (j != 5) { draw((j,i)--(j+1,i)); } if (i != 4) { draw((j,i)--(j,i+1)); } } } [/asy]](https://latex.artofproblemsolving.com/0/9/8/098f8098c7103b004757d1977d2bd6d459831eb3.png)
![\[\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7}\]](https://latex.artofproblemsolving.com/3/5/9/359736c955004ea7d8e77812f4203f14ca07ba6e.png)
![\[\frac{100^2-7^2}{70^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)}\]](https://latex.artofproblemsolving.com/5/f/4/5f4f7b3d646f920fd9ef3f767aad765860768b41.png)
to 
, the ratio of 
, and the ratio of 
. What is the ratio of 
and 
, where 
and both 
of its games and team 
of its games. Also, team 
How many games has team 
the value of![\[\frac{(n+2)!-(n+1)!}{n!}\]](https://latex.artofproblemsolving.com/e/4/c/e4cf3e216ff29ec12bb85e389993280b1ca40180.png)
is always which of the following?
miles in 
minutes. His average speed during the first 
mph. What was his average speed, in mph, during the last 
. What is the distance between the 
contain at least one prime number?
cans of soda can be purchased from a vending machine for 
quarters. Which of the following expressions describes the number of cans of soda that can be purchased for 
dollars, where 
are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be 
? (No red balls are to be removed.)
each, 
each, and 
each. There are at least 
be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of 
for which the polynomials 
and 
have a root in common?
and 
, both the mean and the median of the set 
are equal to 
?
