2020 Mock Combo AMC 10
Here are the problems from the 2020 Mock Combo AMC 10 II, a mock contest created by the AoPS user fidgetboss_4000.
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
Fred, his girlfriend Sara, along with eight other classmates, are randomly seated along a row of ten chairs. What is the probability that Fred sits next to Sara?
Problem 2
In Mega Tetris, the player receives tetrominoes falling in a 32 block high, 16 block wide pit in which the player has to clear 16 block wide lines in order to score points. Which of these choices shown below is a possible number of monominoes in the grid at any given point in the game? (Note that a tetromino is a polyomino with four unit squares, and a monomino is a single unit square.)
Problem 3
Let be the answer to this problem. Compute
Problem 4
Let be the set of the first
positive integers, and let
be a random subset of
integers in
. Let
be the number of positive integers
such that the expected value of the sum of the integers in
is an integer. Find the sum of the digits of
.
Problem 5
There exists an integer such that
holds. Find the sum of the digits of
.
Problem 6
A committee includes men and
women, and a subset of the committee is chosen at random. The probability that more women than men are chosen can be expressed as
where
are positive integers and
is odd. Find
.
Problem 7
How many permutations of the string are there such that no two identical letters are adjacent to each other?
Problem 8
Let . Henry the ant wants to around the square with vertices
,
,
and
while abiding to the following rules:
He must land (not necessarily stay) on the borders (this includes corners) of the square.
He must walk in a clockwise direction at all times.
The length of every step he makes must be an integer.
The number of ways that he can transverse around the square while abiding to all three aforementioned rules can be expressed as
for positive integers
where
.
Find
.
Problem 9
Let be the number of distinct paths from the origin
to a point on the line
such that each step is from
to either
or
. Find the remainder when
is divided by
.
Problem 10
Call two unit squares neighboring if they share a side. Pablo will start on the center unit square on a -by-
grid of squares, and each second, given that he is on Square
, he will move to a square neighboring
, each of the possible neighboring squares with equal probability. Find the expected value of the number of seconds he will take to move back to the center square.
Problem 11
A recursive sequence is defined as
Let
be the sum of all
such that
and
.
Find the last two digits of
.
Problem 12
Misha starts on Point on a regular hexagon, and is trying to reach Point
, which is diametrically opposite of Point
. Given that one move consists of a move to an adjacent vertex, how many ways are there for him to reach Point
in exactly
moves? (This means that he won’t reach Point
anytime before he makes his
th move.)
Problem 13
Lyndon is initially on Square , which is located as the topmost leftmost unit square on a
-by-
grid of unit squares. He wants to travel to Square
, which is located as the bottommost rightmost unit square on the grid. Given that Squares
and
are both black and that Lyndon can only move one unit square right, one unit square down, or one unit square diagonally right and down and can only travel on black squares, find the number of ways that the
-by-
grid can be tiled with white unit squares and black unit squares such that it is possible for Lyndon to travel from
to
while abiding by all the aforementioned rules.
Problem 14
Find the remainder obtained upon dividing
by
.
Problem 15
Charlie and Sylvia are betting on a pair of dice. Charlie bets that a roll with sum will come first, while Sylvia bets that the sum will be
two times consecutively.The probability Charlie wins can be expressed as as
for relatively prime positive integers
. Find
.
Problem 16
Consider a cube with side length and with center
in the 3-dimensional coordinate plane. Integers
are randomly and independently drawn such that
Then, a new cube with side length
is formed with center
. The expected value of the volume intersected by the two cubes can be expressed as
where
and
are relatively prime positive integers. Find the remainder obtained upon dividing
by
.
Problem 17
How many sequences of positive integers satisfy
and
for all
?
Problem 18
Call a quadruple of positive integers
-plausible if there exists a permutation
of the first
positive integers such that
for some
such that there exists no other set of four consecutive integers in the permutation with sum greater than
. Let
be the sum of the distinct values of
over
-plausible quadruples
. Find the value of
.
Problem 19
Consider the set that includes all polynomials
of degree
such that
are integers and
. For all
polynomial
is generated by taking a random element of set
. Then, a new polynomial
is created, where
are integer coefficients for all
. Find the largest power of two that evenly divides the expected value of
.
Problem 20
On Square with side length
, Point
is drawn at a random location on Diagonal
. What is the sum of the digits of the floor of the expected value of
?
Problem 21
A subset of the set is chosen randomly, and the product
of the numbers in that subset is taken (if the aforementioned subset is empty,
is defined as
.) The value of the expected value of
can be expressed as
where
and
are positive integers and
is odd. Find the value of
.
Problem 22
In Level 2020 of the game , the player has to defeat a boss named The Unbeatable Dragon Boss in order to win, which initially spawns at Generation
. After defeating a Generation
boss, there is a
probability that the game will spawn one Generation
boss and a
probability that it will spawn two Generation
bosses. The level is considered completed after the player defeats all Generation
bosses. The expected value of the number of dragons that spawn during the level, given that the player completes the level, can be written as
, where
are positive integers and
is odd. Find the remainder when
is divided by
. (Note that no bosses with Generation number greater than 2020 will spawn.)
Problem 23
How many sequences satisfy that
for all
and that
is divisible by
if and only if
?
Problem 24
Let denote the number of ordered
-tuples
(where
) that satisfy
and
for all
. Find the remainder when
is divided by
.
Problem 25
Define the of a permutation
of the first
positive integers to be
Let
be the average
of all permutations of the first
positive integers, and let
be the maximum possible
of a permutation of the first
positive integers.
Find the remainder when
is divided by
.