Difference between revisions of "AMC 12C 2020"
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==Problem 14== | ==Problem 14== | ||
− | Let <math>P(x)</math> be a polynomial with integral coefficients and <math>S(x) = \frac{P^{2}(x)}{x( | + | Let <math>P(x)</math> be a polynomial with integral coefficients and <math>S(x) = \frac{P^{2}(x)}{x(2 - x)}</math> for all nonzero values of <math>x</math>. If <math>P(2) = P(3) = 8</math>, what is the sum of the digits in the numerical value of <math>P(100)</math>? |
==Problem 15== | ==Problem 15== |
Revision as of 17:30, 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
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 sum of the digits in the numerical value of
?
Problem 15
Let be
. (All the way till the number consisting of
zeroes starting with a
. What is the remainder when N is divided by
?
Problem 16
red balls and
blue balls are to be placed in a grid of
squares in which each ball must be placed in
square in which each square contains at most
ball. In how many ways can the gals be places such that each row contains exactly
of each a red ball and a blue ball and each column also contains exactly
of each a red ball and a blue ball?