2020 FMC 12A Problems
Here are the problems that were on the 2020 FMC 12A.
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
Problem 1
The price of a bottle of soda in Fidgetville has increased from dollars to
dollars from
to
due to inflation. A standard pack of soda is defined as a collection of
identical bottles. Given that Amy spent
more dollars to buy a standard pack of soda in
than in
what is the cost per bottle, in dollars, of soda in
?
Problem 2
How many distinct real solutions are there to the equation
Problem 3
Call a repeating decimal standard if the repetend is exactly one digit long. For example, the repeating decimals and
are standard, but the repeating decimal
is not. Find the sum of all positive integers
such that
can be expressed as a standard repeating decimal that can not be expressed as a terminating decimal.
Problem 4
Adam, Brooks, Caleb, and Daniel are playing a game of Zheng Shang You, and they play for a total of four rounds. Assuming that for each round, each person has equal probability of winning, the probability that each person wins one round can be expressed as for relatively prime positive integers
and
. Find
Problem 5
The numbers are placed in the following
tokens that are arranged in a hexagonal order. The pairwise products of all opposite tokens are all equal. Some numbers are already placed on the tokens, what is the absolute difference between the possible numbers placed on the shaded token?
Problem 6
How many positive integers are there such that
can be expressed as
where
are digits in base
and
is nonzero?
Problem 7
Let and
be positive integers such that
Find the positive value of
.
Problem 8
It is given that an arithmetic sequence satisfies that the initial term
the final term
and that
Find the minimum possible value of
Problem 9
is a complex number such that
What is the positive value of
Problem 10
Alpha and Beta each choose numbers and
respectively with
without telling each other. Alpha wins if
Beta chooses first. To minimize Alpha's chance of winning, what is the sum of the two numbers Beta could choose?
Problem 11
Let denote the set of all positive integers
that satisfy
and
in base
is a four digit number that has an odd tens digit. Find the number of elements in
.
Problem 12
Consider a series of consecutive days. Mitsuha and Taki will switch bodies each day with a
probability, independently from other days. The expected number of instances when Mitsuha and Taki switch bodies on two consecutive days can be expressed as
where
and
are relatively prime positive integers. Find
. (For instance, the number of such instances for the series
is
where
denotes a day with switching bodies, and
a day without switching bodies.)
Problem 13
Jensen likes tossing his ball into baskets . When Jensen tosses his ball into basket
, he is forced to also toss his ball into basket
. In general the ball has
chance of missing its target for basket
. Jensen chooses a random basket from
to
, and the chance he makes both shots is equal to
for relatively prime positive integers
and
. The value of
is
Problem 14
In a right triangle , with
, let
be one of its internal angle bisector, with
on
. Let
be the foot of altitude from
onto
. Let
meet the line
at point
. Let
be a point on
, such that
. Then,
equals
Problem 15
How many -digit multiples of
can be written in the form
where
for
?
Problem 16
Let Find
.
Problem 17
In rectangle , diagonal
is drawn. Point
is selected uniformly at random on diagonal
. Suppose that the probability that the circumcenter of triangle
lies inside the rectangle is
. What is the ratio of the longer side of the rectangle to the shorter one?
Problem 18
Starting from define a conventional listing
of permutations of the first
positive integers to be an ordering of all aforementioned permutations such that the sets of conventional listings of permutations of the first
positive integers (which are ordered accordingly) not including
are ordered based on the placement of the
in the permutation. For instance, the conventional listing of permutations of the first
positive integers is
Let
denote the least positive integer
in a conventional listing of the first
positive integers such that the permutation
has leading number
Find the largest power of
that divides
Problem 19
Find the number of ordered quadruples of positive integers that satisfy
Problem 20
Henry's NT2 TA gave him this problem to work on: "Find the exponent of the largest power of that evenly divides
" where
is a fixed positive integer. However, Henry is bad at NT, so he just puts
as his answer for the aforementioned problem. Let
be the number of integer values
such that the positive difference between Henry's answer and the actual answer of the problem is exactly
Find the largest power of
that divides
.
Problem 21
Let the function denote the least positive integer value of
such that
is divisible by
.
Find the remainder when
is divided by
.
Problem 22
Call a sequence of rolls generated from a standard fair
-sided die fluctuating if the
sets of two consecutive rolls, placed in order, alternately fluctuate between having a sum greater than or equal to
and having a sum less than or equal to
. For example, the sequences
and
are fluctuating, but the sequence
is not. The probability that Jamie will produce a fluctuating sequence by rolling the said die
times can be expressed as
for relatively prime positive integers
. Find the number of positive factors that
contains.
Problem 23
Let the -set, for some fixed
, be the set {
}. For some fixed
, let the
-product, be the product of the greatest
rational values from its
-set. For example the
-product would be the product of the greatest
rational values from the set
. Let
the sum of all the
-products such that
is divisible by exactly one prime factor
such that
(note that
is not divisible by any other prime factors) and has at most
total factors.
Find the value of
modulo
Problem 24
A company has a system of levels of authority, and each level of authority has two people. One of the rules is that a person at the
th level of authority can fire another person at a level of authority that is no greater than
. Define a firing chain as a sequence of at least one firing event such that there exist a sequence of positive integers
such that member of the
th level of authority fires a member of the
th level of authority according to the firing rules for all
Note that if a member of the company is already fired, he cannot make any more actions for the company (this includes firing). Let
be the number of possible distinct firing chains starting from the
th level of authority, or otherwise, the founder and the co-founder of the company. Find the remainder when
is divided by
.
Problem 25
Carlos is bored during quarantine, so he writes a repeating decimal given by
Recall that
if
. If
is the period of
, let
be the number of positive factors of
. Find the remainder when
is divided by