Difference between revisions of "AMC 12C 2020 Problems"
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==Problem 24== | ==Problem 24== | ||
− | Let <math>\lfloor x \rfloor</math> denote the greatest integer less than or equal to <math>x</math>. How many | + | Let <math>\lfloor x \rfloor</math> denote the greatest integer less than or equal to <math>x</math>. How many positive integers <math>x < 2020</math>, satisfy the equation |
<math>\frac{x^{4} + 2020}{108} = \lfloor \sqrt (x^{2} - x)\rfloor</math>? | <math>\frac{x^{4} + 2020}{108} = \lfloor \sqrt (x^{2} - x)\rfloor</math>? |
Revision as of 13:56, 26 April 2020
Contents
Problem 1
What is the the sum of the solutions of the equation ?
Problem 2
Rectangular Parallelepiped , the diagonal
and the length of the segment
. What is the volume of the triangular prism
?
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
The line has an equation
is rotated clockwise by
to obtain the line
. What is the distance between the
- intercepts of Lines
and
?
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
Let be the set of solutions to the equation
on the complex plane, where
.
points from
are chosen, such that a circle
passes through both points. What is the least possible area of
?
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
In rectangle ,
and
. Let the midpoint of
be
and let the midpoint of
be
. The centroids of Triangles
,
, and
are connected to from the minor triangle
. What is the length of largest altitude of
?
Problem 18
lays flat on the ground and has side lengths
, and
. Vertex
is then lifted up creating an elevation angle with the triangle and the ground of
. A wooden pole is dropped from
perpendicular to the ground, making an altitude of a
Dimensional figure. Ropes are connected from the foot of the pole,
, to form
other segments,
and
. What is the volume of
?
Problem 19
An urn left on a deserted island, consists of golden blocks,
silver blocks,
zinc blocks, and
wooden blocks.
pirates come to the island seeing the urn. Without noticing blocks are made of different materials, each of the pirates randomly grab an equal number of blocks from the urn, each at a time. The pirates then place the blocks back into the urn and then repeat the same process again. What is the probability that after the pirates repeat the same process
times, that no pirate who has more than
golden blocks has more than
silver blocks?
Problem 20
What is the maximum value of as
varies through all real numbers to the nearest integer?
Problem 24
Let denote the greatest integer less than or equal to
. How many positive integers
, satisfy the equation
?