2021 Fall MIMC 10
The official solution will be released on November 17th, and feel free to add more solutions beneath the official solution after that date.
Contents
[hide]- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Additional Information
Problem 1
What is the sum of ?
Problem 2
Okestima is reading a page book. He reads a page every
minutes, and he pauses
minutes when he reaches the end of page 90 to take a break. He does not read at all during the break. After, he comes back with food and this slows down his reading speed. He reads one page in
minutes. If he starts to read at
, when does he finish the book?
Problem 3
Find the number of real solutions that satisfy the equation
.
Problem 4
Stiskwey wrote all the possible permutations of the letters (
is different from
). How many such permutations are there?
Problem 5
5. Given , Find
.
Problem 6
A worker cuts a piece of wire into two pieces. The two pieces, and
, enclose an equilateral triangle and a square with equal area, respectively. The ratio of the length of
to the length of
can be expressed as
in the simplest form. Find
.
Problem 7
Find the least integer such that
where
denotes
in base-
.
Problem 8
In the morning, Mr.Gavin always uses his alarm to wake him up. The alarm is special. It always rings in a cycle of ten rings. The first ring lasts second, and each ring after lasts twice the time than the previous ring. Given that Mr.Gavin has an equal probability of waking up at any time, what is the probability that Mr.Gavin wakes up and end the alarm during the tenth ring?
Problem 9
Find the largest number in the choices that divides .
Problem 10
If and
, find
.
Problem 11
How many factors of is a perfect cube or a perfect square?
Problem 12
Given that , what is
?
Problem 13
Given that Giant want to put green identical balls into
different boxes such that each box contains at least two balls, and that no box can contain
or more balls. Find the number of ways that Giant can accomplish this.
Problem 14
James randomly choose an ordered pair which both
and
are elements in the set
,
and
are not necessarily distinct, and all of the equations:
are divisible by
. Find the probability that James can do so.
Problem 15
Paul wrote all positive integers that's less than and wrote their base
representation. He randomly choose a number out the list. Paul insist that he want to choose a number that had only
and
as its digits, otherwise he will be depressed and relinquishes to do homework. How many numbers can he choose so that he can finish his homework?
Problem 16
Find the number of permutations of such that at exactly two
s are adjacent, and the
s are not adjacent.
Problem 17
The following expression can be expressed as
which both
and
are relatively prime positive integers. Find
.
Problem 18
What can be a description of the set of solutions for this: ?
Two overlapping circles with each area
.
Four not overlapping circles with each area
.
There are two overlapping circles on the right of the
-axis with each area
and the intersection area of two overlapping circles on the left of the
-axis with each area
.
Four overlapping circles with each area
.
There are two overlapping circles on the right of the
-axis with each area
and the intersection area of two overlapping circles on the left of the
-axis with each area
.
Problem 19
can be expressed as
in base
which
is a positive integer. Find the sum of the digits of
.
Problem 20
Given that . Given that the product of the even divisors is
, and the product of the odd divisors is
. Find
.
Problem 21
How many solutions are there for the equation . (Recall that
is the largest integer less than
, and
is the smallest integer larger than
.)
Problem 22
In the diagram, is a square with area
.
is a diagonal of square
. Square
has area
. Given that point
bisects line segment
, and
is a line segment. Extend
to meet diagonal
and mark the intersection point
. In addition,
is drawn so that
.
can be represented as
where
are not necessarily distinct integers. Given that
, and
does not have a perfect square factor. Find
.
Problem 23
On a coordinate plane, point denotes the origin which is the center of the diamond shape in the middle of the figure. Point
has coordinate
, and point
,
, and
are formed through
,
, and
rotation about the origin
, respectively. Quarter circle
(formed by the arc
and line segments
and
) has area
. Furthermore, another quarter circle
formed by arc
and line segments
,
is formed through a reflection of sector
across the line
. The small diamond centered at
is a square, and the area of the little square is
. Let
denote the area of the shaded region, and
denote the sum of the area of the regions
(formed by side
, arc
, and side
),
(formed by side
, arc
, and side
) and sectors
and
. Find
in the simplest radical form.
Problem 24
One semicircle is constructed with diameter and let the midpoint of
be
. Construct a point
on the side of segment
(closer to segment
than arc
) such that the distance from
to
is
, and that
is perpendicular to the diameter
. Three more such congruent semicircles are formed through multiple
rotations around the point
. Name the
endpoints of the diameters
,
,
,
,
,
in a circular direction from
to
. Another four congruent semicircles are constructed with diameters
, and that the distance from the diameters to the point
are less than the distance from the arcs to the point
. Connect
,
,
,
, and
. Find the ratio of the area of the pentagon
to the total area of the shape formed by arcs
,
,
,
,
,
,
,
.
Problem 25
Suppose that a researcher hosts an experiment. He tosses an equilateral triangle with area
onto a plane that has a strip every
horizontally. Find the expected number of intersections of the strips and the sides of the equilateral triangle.
Additional Information
1. The Committee on the Michael595 & Interstigation Math Contest (MIMC) reserves the right to re-examine students before deciding whether to grant official status to their scores. The MIMC also reserves the right to disqualify score from a test taker if it is determined that the required security procedures were not followed.
2. The publication, reproduction or communication of the problems or solutions of the MIMC 10 will result in disqualification. Dissemination via copier, telephone, e-mail, World Wide Web or media of any type during this period is a violation of the competition rules except the private discussion form.
Sincerely, the MIMC mock contest cannot come true without the contributions from the following testsolvers, problem writers and advisors: