Difference between revisions of "AMC 12C 2020 Problems"
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In rectangle <math>ABCD</math>, <math>\overline{AB} = 10</math> and <math>\overline{BC} = 6</math>. Let the midpoint of <math>\overline{AB}</math> be <math>M</math> and let the midpoint of <math>\overline{BC}</math> be <math>N</math>. The centroids of Triangles <math>\bigtriangleup ADM</math>, <math>\bigtriangleup CDN</math>, and <math>\bigtriangleup DMN</math> are connected to from the minor triangle <math>\bigtriangleup JKL</math>. What is the length of largest altitude of <math>\bigtriangleup JKL</math>? | In rectangle <math>ABCD</math>, <math>\overline{AB} = 10</math> and <math>\overline{BC} = 6</math>. Let the midpoint of <math>\overline{AB}</math> be <math>M</math> and let the midpoint of <math>\overline{BC}</math> be <math>N</math>. The centroids of Triangles <math>\bigtriangleup ADM</math>, <math>\bigtriangleup CDN</math>, and <math>\bigtriangleup DMN</math> are connected to from the minor triangle <math>\bigtriangleup JKL</math>. What is the length of largest altitude of <math>\bigtriangleup JKL</math>? | ||
− | ==Problem | + | ==Problem 18== |
<math>\bigtriangleup ABC</math> lays flat on the ground and has side lengths <math>\overline{AB} = 8, \overline{BC} = 15</math>, and <math>\overline{AC} = 17</math>. Vertex <math>A</math> is then lifted up creating an elevation angle with the triangle and the ground of <math>60^{\circ}</math>. A wooden pole is dropped from <math>A</math> perpendicular to the ground, making an altitude of a <math>3</math> Dimensional figure. Ropes are connected from the foot of the pole, <math>D</math>, to form <math>2</math> other segments, <math>\overline{BD}</math> and <math>\overline{CD}</math>. What is the volume of <math>ABCD</math>? | <math>\bigtriangleup ABC</math> lays flat on the ground and has side lengths <math>\overline{AB} = 8, \overline{BC} = 15</math>, and <math>\overline{AC} = 17</math>. Vertex <math>A</math> is then lifted up creating an elevation angle with the triangle and the ground of <math>60^{\circ}</math>. A wooden pole is dropped from <math>A</math> perpendicular to the ground, making an altitude of a <math>3</math> Dimensional figure. Ropes are connected from the foot of the pole, <math>D</math>, to form <math>2</math> other segments, <math>\overline{BD}</math> and <math>\overline{CD}</math>. What is the volume of <math>ABCD</math>? |
Revision as of 23:13, 18 May 2020
Contents
Problem 1
glass bowls hang on
sides of a balance each having a weight of
pounds,
bowl having
lemons and the other bowl having
. If lemons weigh
pounds each, how many lemons should be added to the lighter bowl to balance the scale?
Problem 2
has side lengths of
,
, and
, and
. What is the smallest possible measure of
?
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
A spaceship flies in space at a speed of miles/hour and the spaceship is paid
dollars for each
miles traveled. It’s only expense is fuel in which it pays
dollars per gallon, while going at a rate of
hours per gallon. Traveling
miles, how much money would the spaceship have gained?
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
An ant is lost inside a square with an unknown side length. The ant is
units away from
,
units away from
, and
units away from
. By how,any units is the ant away from
?
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
Let be a cubic polynomial with integral coefficients and roots
,
, and
. What is the least possible sum of the coefficients of
?
Problem 20
What is the maximum value of as
varies through all real numbers to the nearest integer?
Problem 21
Let denote the greatest integer less than or equal to
. How many positive integers
, satisfy the equation
?