# Difference between revisions of "2020 Mock Combo AMC 10"

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Let <math>A = (0, 0)</math>. Henry the ant wants to around the square with vertices <math>A</math>, <math>(7, 0)</math>, <math>(7, 7)</math> and <math>(0, 7)</math> while abiding to the following rules: | Let <math>A = (0, 0)</math>. Henry the ant wants to around the square with vertices <math>A</math>, <math>(7, 0)</math>, <math>(7, 7)</math> and <math>(0, 7)</math> while abiding to the following rules: | ||

+ | |||

<math>\bullet</math> He must land (not necessarily stay) on the borders (this includes corners) of the square. | <math>\bullet</math> He must land (not necessarily stay) on the borders (this includes corners) of the square. | ||

+ | |||

<math>\bullet</math> He must walk in a clockwise direction at all times. | <math>\bullet</math> He must walk in a clockwise direction at all times. | ||

+ | |||

<math>\bullet</math> The length of every step he makes must be an integer. | <math>\bullet</math> 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 | The number of ways that he can transverse around the square while abiding to all three aforementioned rules can be expressed as | ||

<cmath>2^s + 2^t + 2^u + 2^v + 2^w + 2^x + 2^y + 2^z,</cmath>for positive integers <math>s, t, u, v, w, x, y, z</math> where <math>s > t > u > v > w > x > y > z</math>. | <cmath>2^s + 2^t + 2^u + 2^v + 2^w + 2^x + 2^y + 2^z,</cmath>for positive integers <math>s, t, u, v, w, x, y, z</math> where <math>s > t > u > v > w > x > y > z</math>. | ||

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Misha starts on Point <math>A</math> on a regular hexagon, and is trying to reach Point <math>B</math>, which is diametrically opposite of Point <math>A</math>. Given that one move consists of a move to an adjacent vertex, how many ways are there for him to reach Point <math>B</math> in exactly <math>11</math> moves? (This means that he won’t reach Point <math>B</math> anytime before he makes his <math>11</math>th move.) | Misha starts on Point <math>A</math> on a regular hexagon, and is trying to reach Point <math>B</math>, which is diametrically opposite of Point <math>A</math>. Given that one move consists of a move to an adjacent vertex, how many ways are there for him to reach Point <math>B</math> in exactly <math>11</math> moves? (This means that he won’t reach Point <math>B</math> anytime before he makes his <math>11</math>th move.) | ||

− | + | <asy> size(3.5cm); pair A[]; for (int i=0; i<7; ++i) { A[i] = rotate(30+60*i)*(1,0); } filldraw(A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--cycle,gray,black); for (int i=0; i<6; ++i) { dot(A[i]); } </asy> | |

<math>\textbf{(A)}\ 54 \qquad\textbf{(B)}\ 81 \qquad\textbf{(C)}\ 108 \qquad\textbf{(D)}\ 162 \qquad\textbf{(E)}\ 243 </math> | <math>\textbf{(A)}\ 54 \qquad\textbf{(B)}\ 81 \qquad\textbf{(C)}\ 108 \qquad\textbf{(D)}\ 162 \qquad\textbf{(E)}\ 243 </math> | ||

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==Problem 19== | ==Problem 19== | ||

− | Consider the set <math>S</math> that includes all polynomials <math>ax^3 + bx^2 + cx + d</math> of degree <math>3</math> such that <math>a, b, c, d</math> are integers and <math>1 \leq a, b, c, d \leq 15</math>. For all <math>1 \leq i \leq 15,</math> polynomial <math>H_i(x)</math> is generated by taking a random element of set <math>S</math>. Then, a new polynomial<cmath>P(x) = \prod_{i=1}^{15} H_i(x) = \sum_{i=0}^{45} a_ix^i</cmath>is created, where <math>a_i</math> are integer coefficients for all <math>0 \leq i \leq 45</math>. Find the largest power of two that evenly divides the expected value of <math>\sum_{i=0}^{44} a_i</math>. | + | Consider the set <math>S</math> that includes all polynomials <math>ax^3 + bx^2 + cx + d</math> of degree <math>3</math> such that <math>a, b, c, d</math> are integers and <math>1 \leq a, b, c, d \leq 15</math>. For all <math>1 \leq i \leq 15,</math> polynomial <math>H_i(x)</math> is generated by taking a random element of set <math>S</math>. Then, a new polynomial |

+ | <cmath>P(x) = \prod_{i=1}^{15} H_i(x) = \sum_{i=0}^{45} a_ix^i</cmath> | ||

+ | is created, where <math>a_i</math> are integer coefficients for all <math>0 \leq i \leq 45</math>. Find the largest power of two that evenly divides the expected value of <math>\sum_{i=0}^{44} a_i</math>. | ||

<math>\textbf{(A) }5\qquad\textbf{(B) }15\qquad\textbf{(C) }30\qquad\textbf{(D) }45\qquad\textbf{(E) }75</math> | <math>\textbf{(A) }5\qquad\textbf{(B) }15\qquad\textbf{(C) }30\qquad\textbf{(D) }45\qquad\textbf{(E) }75</math> |

## Revision as of 11:53, 7 April 2020

Here are the problems from the 2020 Mock Combo AMC 10 II, a mock contest created by the AoPS user fidgetboss_4000.

## Contents

- 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 aswhere 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 thatThen, 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 .