Difference between revisions of "AMC 12C 2020"
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| + | ==Problem 13== | ||
| + | The pentagon <math>ABCDE</math> rolls on a straight line as each side of the pentagon touches the ground at <math>1</math> stage in the entire cycle. What is the length of the path that vertex <math>C</math> travels throughout <math>1</math> whole cycle? | ||
==Problem 14== | ==Problem 14== | ||
Let <math>P(x)</math> be a polynomial with integral coefficients and <math>S(x) = \frac{P(x)}{x(1 - x)}</math> for all nonzero values of <math>x</math>. If <math>P(2) = P(3) = 8</math>, what is the numerical value of <math>P(100)</math>? | Let <math>P(x)</math> be a polynomial with integral coefficients and <math>S(x) = \frac{P(x)}{x(1 - x)}</math> for all nonzero values of <math>x</math>. If <math>P(2) = P(3) = 8</math>, what is the numerical value of <math>P(100)</math>? | ||
Revision as of 15:35, 20 April 2020
Contents
Problem 1
What is the sum of the solutions to the equation 
?
Problem 2
How many increasing subsets of 
contain no 
consecutive prime numbers?
Problem 3
A field is on the real 
plane in the shape of a circle, centered at 
with a a radius of 
. The area that is in the field but above the line 
is planted. What fraction of the field is planted?
Problem 4
What is the numerical value of 
?
Problem 5
cows can consume 
kilograms of grass in 
days. How many more cows are required such that it takes all of the cows to consume 
kilograms of grass in days?
Problem 6
candy canes and 
lollipops are to be distributed among children such that each child gets atleast 
candy. What is the probability that once the candies are distributed, no child has both types of candies?
Problem 7
Persons 
and 
can plough a field in days, persons and 
can plough the same field in 
days, and persons and can plough the same field in 
days. In how many days can all of them plough the field together?
Problem 8
The real value of 
that satisfies the equation 
can be written in the form ![\[a + \sqrt{b}\]](http://latex.artofproblemsolving.com/7/b/b/7bb956eb37defb8158983680cf3356613b01d582.png)
where 
and 
are integers. What is 
?
Problem 9
On a summer evening stargazing, the probability of seeing a shooting star in any given hour
on a sunny day is 
and the probability of seeing a shooting star on a rainy
day is 
. Both rainy and sunny days happen with equal chances. What is the
probability of seeing a shooting star in the second minutes of an hour stargazing on a
random night?
Problem 10
Let 
denote the number of trailing 
s in the numerical value of the expression 
, for example, 
since 
which has trailing zero. What is the sum
?
Problem 11
A line of hunters walk into a jungle where the distance between the first and last hunter is 
meters which maintains constant throughout their walk as the hunters walk at a constant speed of meters per second. A butterfly starts from the front of their line and flies to the back as they come forward and then turns and comes back as soon as it reaches the back of the line. When the butterfly is back at the front of the line, the hunter finds out that the butterfly has travelled a distance of 
meters. What was the speed of the butterfly?
Problem 12
How many positive base
integers are divisible by 
but the sum of their digits is not divisible by ?
Problem 13
The pentagon 
rolls on a straight line as each side of the pentagon touches the ground at stage in the entire cycle. What is the length of the path that vertex travels throughout whole cycle?
Problem 14
Let 
be a polynomial with integral coefficients and 
for all nonzero values of 
. If 
, what is the numerical value of 
?
