2021 MECC Mock AMC 10
Contents
[hide]Problem 1
Compute
Problem 2
Define a binary operation . Find the number of possible ordered pair of positive integers
such that
.
Problem 3
can be expressed as
. Find
.
Problem 4
Compute the number of ways to arrange 2 distinguishable apples and five indistinguishable books.
Problem 5
Galieo, Neton, Timiel, Fidgety and Jay are participants of a game in soccer. Their coach, Mr.Tom, will allocate them into two INDISTINGUISHABLE groups for practice purpose(People in the teams are interchangable). Given that the coach will not put Galieo and Timiel into the same team because they just had a fight. Find the number of ways the coach can put them into two such groups.
Problem 6
Let be a sequence of positive integers with
and
and
for all integers
such that
. Find
.
Problem 7
Find the sum of all the solutions of , where
can be any number. The roots may be repeated.
Problem 8
Define the number of real numbers
such that
is a perfect square. Find
.
Problem 9
A unit cube ABCDEFGH is shown below. is reflected across the plane that contains line
and line
. Then, it is reflected again across the plane that contains line
and
. Call the new point
. Find
.
Problem 10
Find the number of nonempty subsets of such that the product of all the numbers in the subset is NOT divisible by
.
Problem 11
In square with side length
, point
and
are on side
and
respectively, such that
is perpendicular to
and
. Find the area enclosed by the quadrilateral
.
Problem 12
Given that ,
, and
, find
.
Problem 13
Let be a
term sequence of positive integers such that
,
,
,
,
. Find the number of such sequences
such that all of
.
Problem 14
The answer of this problem can be expressed as
which
are not necessarily distinct positive integers, and all of
are not divisible by any square number. Find
.
Problem 15
Find the number of positive real numbers that are less than or equal to
such that
is a four digit terminating decimal
which
.
Problem 16
Find the remainder when expressed in base
is divided by 1000.
Problem 17
There exists a polynomial which
and
are both integers. How many of the following statements are true about all quadratics
?
1. For every possible , there are at least
of them such that
but two quadratic that
if the such
has all integer roots.
2. For all roots of any quadratic in
, there exists infinite number of quadratic
such that
if and only if
has all real solutions and all terms of
are real numbers.
3. For any quadratics in , there exists at least one quadratics such that they shares exactly one of the roots of
and all of the roots are positive integers.
4. Statement
5. Statement
6. Statement
Problem 18
Given that , Find the area of region enclosed by the intersection point of
,
, and the new point formed through rotations of
and
about the origin.
Problem 19
In a circle with a radius of , four arcs are drawn inside the circle. Smaller circles are inscribed inside the eye-shape diagrams. Find the ratio between the area of the smaller circles to the area of the star-diagram.
Problem 20
In a square with length , two overlapping quarter circle centered at two of the vertices of the square is drawn. Find the ratio of the shaded region and the area of the entire square.
Problem 21
In square with side length
, equilateral triangles are drawn on each side of the square. Additionally,
arcs are drawn inside the square. A circle with center
is externally tangent to all the four arcs in the square. Let the midpoint of
be
. The secant
cuts circle
. What is the probability that a randomly chosen point on line segment
will be a point on the portion that cuts through the circle?