AMC 12C 2020 Problems
Contents
Problem 1
What is the the sum of the solutions of the equation ?
Problem 2
What is the numerical value of the sum
Problem 3
In a bag are marbles consisting of
blue marbles and
red marbles. If each marble is pulled out
at a time, what is the probability that the
marble pulled out red?
Problem 4
cows can consume
kilograms of grass in
days. How many more cows are required such that all the cows together can consume
kilograms of grass in
days?
Problem 5
Lambu the Lamb is tied to a post at the origin on the real
plane with a rope that measures
units.
wolves are tied with ropes of length
as well, both of them being at points
, and
. What is the area that the lamb can run around without being in the range of the wolves?
Problem 6
How many increasing(lower to higher numbered) subsets of contain no
consecutive prime numbers?
Problem 7
Let denote the sum of the factors of a positive integer
. What is the sum of the
least possible values of
such that
?
Problem 8
The real value of that satisfies the equation
can be written in the form
where
and
are integers. What is
?
Problem 9
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 10
In how many ways can candy canes and
lollipops be split between
children if each child must receive atleast
candy but no child receives both types?
Problem 11
Let be an isosceles trapezoid with
being parallel to
and
,
, and
. If
is the intersection of
and
, and
is the circumcenter of
, what is the length of
?
Problem 12
Rajbhog, Aditya, and Suman are racing a meter race. Aditya beats Rajbhog by
seconds and beats Suman by
meters. Given that Rajbhog beat Suman by
seconds, by how many meters would Aditya beat Rajbhog if they both were having a
meter race?
Problem 13
In how many ways can the first positive integers;
in red, blue, and green colors if no
numbers
, and
are the same color with
being even?
Problem 14
A function (Function along Complex Numbers) is defined by
, where
and
are positive real numbers. The function
has the property that there are complex numbers
such that
, and
form a triangle with an area of
. Given that
and
, there is a certain complex number
where
is minimized as much as possible satisfying this condition. What is
?
Problem 15
Let . What is the remainder when
is divided by
?
Problem 16
For some positive integer , let
satisfy the equation
.
What is the sum of the digits of
?
Problem 17
How many ordered pairs satisfy
where denotes the greatest common divisor and
denotes the least common multiple?