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Difference between revisions of "2020 Mock Combo AMC 10"

(Created page with "Here are the problems from the 2020 Mock Combo AMC 10 II, a mock contest created by the AoPS user fidgetboss_4000. ==Problem 1== Fred, his girlfriend Sara, along with eight o...")
 
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Line 59: Line 59:
  
 
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>.
Line 100: Line 104:
 
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]
+
<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>
Line 157: Line 161:
 
==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 10:53, 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.

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?

$\textbf{(A)}\ \frac{9}{55} \qquad\textbf{(B)}\ \frac{2}{11} \qquad\textbf{(C)}\ \frac{1}{5} \qquad\textbf{(D)}\ \frac{2}{9} \qquad\textbf{(E)}\ \frac{2}{5}$

Solution

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.)

$\textbf{(A)}\ 477 \qquad\textbf{(B)}\ 480 \qquad\textbf{(C)}\ 483 \qquad\textbf{(D)}\ 486 \qquad\textbf{(E)}\ 492$

Solution

Problem 3

Let $A$ be the answer to this problem. Compute\[A^4 - 4A^3 + 6A^2 - 3A + 1.\] $\textbf{(A)}\ -1 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{the answer is a complex number}$

Solution

Problem 4

Let $S$ be the set of the first $2n$ positive integers, and let $R$ be a random subset of $n$ integers in $S$. Let $N$ be the number of positive integers $1 \leq n \leq 2019$ such that the expected value of the sum of the integers in $R$ is an integer. Find the sum of the digits of $N$.

$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

Solution

Problem 5

There exists an integer $n$ such that \[\dbinom{n}{4} : \dbinom{n}{5} : \dbinom{n}{6} = 1 : 2 : 3\]holds. Find the sum of the digits of $n$.

$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 8$

Solution

Problem 6

A committee includes $12$ men and $12$ women, and a subset of the committee is chosen at random. The probability that more women than men are chosen can be expressed as\[\frac{m}{2^n},\]where $m, n$ are positive integers and $m$ is odd. Find $n$.

$\textbf{(A)}\ 21 \qquad\textbf{(B)}\ 22 \qquad\textbf{(C)}\ 23 \qquad\textbf{(D)}\ 24 \qquad\textbf{(E)}\ 25$

Solution

Problem 7

How many permutations of the string $AABBCC$ are there such that no two identical letters are adjacent to each other?

$\textbf{(A)}\ 28 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 32 \qquad\textbf{(D)}\ 34 \qquad\textbf{(E)}\ 36$

Solution

Problem 8

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

$\bullet$ He must land (not necessarily stay) on the borders (this includes corners) of the square.

$\bullet$ He must walk in a clockwise direction at all times.

$\bullet$ 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 \[2^s + 2^t + 2^u + 2^v + 2^w + 2^x + 2^y + 2^z,\]for positive integers $s, t, u, v, w, x, y, z$ where $s > t > u > v > w > x > y > z$. Find $2(z + u) + s + t + v + w + x + y$.

$\textbf{(A)}\ 131 \qquad\textbf{(B)}\ 132 \qquad\textbf{(C)}\ 133 \qquad\textbf{(D)}\ 134 \qquad\textbf{(E)}\ 135$

Solution

Problem 9

Let $N$ be the number of distinct paths from the origin $(0, 0)$ to a point on the line $y=2020-2x$ such that each step is from $(x, y)$ to either $(x+1, y)$ or $(x, y+1)$. Find the remainder when $N$ is divided by $11$.

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 10$

Solution

Problem 10

Call two unit squares neighboring if they share a side. Pablo will start on the center unit square on a $3$-by-$3$ grid of squares, and each second, given that he is on Square $d$, he will move to a square neighboring $d$, 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.

$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 12$

Solution

Problem 11

A recursive sequence $r_n$ is defined as \[r_1 = 1\]\[r_n = nr_{n-1} + n\]Let $S$ be the sum of all $n$ such that $1 \leq n \leq 1000$ and $40 | r_n$. Find the last two digits of $S$.

$\textbf{(A)}\ 00 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 40 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 80$

Solution

Problem 12

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

$\textbf{(A)}\ 54 \qquad\textbf{(B)}\ 81 \qquad\textbf{(C)}\ 108 \qquad\textbf{(D)}\ 162 \qquad\textbf{(E)}\ 243$

Solution

Problem 13

Lyndon is initially on Square $A$, which is located as the topmost leftmost unit square on a $3$-by-$3$ grid of unit squares. He wants to travel to Square $B$, which is located as the bottommost rightmost unit square on the grid. Given that Squares $A$ and $B$ 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 $3$-by-$3$ grid can be tiled with white unit squares and black unit squares such that it is possible for Lyndon to travel from $A$ to $B$ while abiding by all the aforementioned rules.

$\textbf{(A)}\ 72 \qquad\textbf{(B)}\ 80 \qquad\textbf{(C)}\ 86 \qquad\textbf{(D)}\ 92 \qquad\textbf{(E)}\ 96$

Solution

Problem 14

Find the remainder obtained upon dividing \[1 + 11 + 11^2 + 11^3 + … + 11^{2019}\]by $1000$.

$\textbf{(A)}\ 120 \qquad\textbf{(B)}\ 320 \qquad\textbf{(C)}\ 520 \qquad\textbf{(D)}\ 720 \qquad\textbf{(E)}\ 920$

Solution

Problem 15

Charlie and Sylvia are betting on a pair of dice. Charlie bets that a roll with sum $12$ will come first, while Sylvia bets that the sum will be $7$ two times consecutively.The probability Charlie wins can be expressed as as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p+q$.

$\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 19 \qquad\textbf{(E)}\ 20$

Solution

Problem 16

Consider a cube with side length $2020$ and with center $A = (0, 0, 0)$ in the 3-dimensional coordinate plane. Integers $x, y, z$ are randomly and independently drawn such that\[x, y, z \in \{-2020, -2019, -2018, …, 2018, 2019, 2020\}.\]Then, a new cube with side length $2020$ is formed with center $B = (x, y, z)$. The expected value of the volume intersected by the two cubes can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find the remainder obtained upon dividing $m + n$ by $1000$.

$\textbf{(A) }000\qquad\textbf{(B) }088\qquad\textbf{(C) }507\qquad\textbf{(D) }681\qquad\textbf{(E) }921$

Solution

Problem 17

How many sequences of positive integers $(a_1, a_2, a_3, a_4, a_5)$ satisfy $1 \leq a_1, a_2, a_3, a_4, a_5 \leq 32$ and $a_{k+1} \geq 2a_k$ for all $1 \leq k \leq 4$?

$\textbf{(A)}\ 201 \qquad\textbf{(B)}\ 202 \qquad\textbf{(C)}\ 203 \qquad\textbf{(D)}\ 204 \qquad\textbf{(E)}\ 205$

Solution

Problem 18

Call a quadruple $(w, x, y, z)$ of positive integers $n$-plausible if there exists a permutation $p_1p_2…p_n$ of the first $n$ positive integers such that $w, x, y, z = p_{k-3}, p_{k-2}, p_{k-1}, p_k$ for some $4 \leq k \leq n$ such that there exists no other set of four consecutive integers in the permutation with sum greater than $w+x+y+z$. Let $N$ be the sum of the distinct values of $w+x+y+z$ over $8$-plausible quadruples $(w, x, y, z)$. Find the value of $N$.

$\textbf{(A)}\ 141 \qquad\textbf{(B)}\ 161 \qquad\textbf{(C)}\ 180 \qquad\textbf{(D)}\ 198 \qquad\textbf{(E)}\ 215$

Solution

Problem 19

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

$\textbf{(A) }5\qquad\textbf{(B) }15\qquad\textbf{(C) }30\qquad\textbf{(D) }45\qquad\textbf{(E) }75$

Solution

Problem 20

On Square $ABCD$ with side length $16$, Point $P$ is drawn at a random location on Diagonal $AC$. What is the sum of the digits of the floor of the expected value of $BP \times DP$?

$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 10$

Solution

Problem 21

A subset of the set $\{1, 2, 3, …, 2020\}$ is chosen randomly, and the product $P$ of the numbers in that subset is taken (if the aforementioned subset is empty, $P$ is defined as $1$.) The value of the expected value of $P$ can be expressed as $\frac{m}{2^n}$ where $m$ and $n$ are positive integers and $m$ is odd. Find the value of $n$.

$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 10$

Solution

Problem 22

In Level 2020 of the game $\text{Dreadful Dragons}$, the player has to defeat a boss named The Unbeatable Dragon Boss in order to win, which initially spawns at Generation $0$. After defeating a Generation $k$ boss, there is a $\frac{1}{2}$ probability that the game will spawn one Generation $k+1$ boss and a $\frac{1}{2}$ probability that it will spawn two Generation $k+1$ bosses. The level is considered completed after the player defeats all Generation $2020$ 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 $\frac{m}{2^n}$, where $m, n$ are positive integers and $m$ is odd. Find the remainder when $m+n$ is divided by $1000$. (Note that no bosses with Generation number greater than 2020 will spawn.)

$\textbf{(A)}\ 071 \qquad\textbf{(B)}\ 072 \qquad\textbf{(C)}\ 325 \qquad\textbf{(D)}\ 845 \qquad\textbf{(E)}\ 846$

Solution

Problem 23

How many sequences $a_1, a_2, …, a_{15}$ satisfy that $a_i \in \{2, 3\}$ for all $1 \leq i\leq 15$ and that \[\sum_{j=1}^{k} a_j\]is divisible by $5$ if and only if $k=15$?

$\textbf{(A)}\ 144 \qquad\textbf{(B)}\ 178 \qquad\textbf{(C)}\ 233 \qquad\textbf{(D)}\ 288 \qquad\textbf{(E)}\ 466$

Solution

Problem 24

Let $N$ denote the number of ordered $n$-tuples $(a_1, a_2, …, a_{n})$ (where $n \geq 1$) that satisfy \[\sum_{j=1}^{n} a_j = 2004\]and $a_i \in \{1, 2, …, 1001\}$ for all $1 \leq i\leq n$. Find the remainder when $N$ is divided by $1000$.

$\textbf{(A)}\ 001 \qquad\textbf{(B)}\ 489 \qquad\textbf{(C)}\ 753 \qquad\textbf{(D)}\ 969 \qquad\textbf{(E)}\ 993$

Solution

Problem 25

Define the $\text{juicyness}$ of a permutation $p_1p_2p_3…p_n$ of the first $n$ positive integers to be \[\sum_{j=1}^{n-1} |p_{j+1} - p_{j}|\]Let $m_k$ be the average $\text{juicyness}$ of all permutations of the first $k$ positive integers, and let $v_k$ be the maximum possible $\text{juicyness}$ of a permutation of the first $k$ positive integers. Find the remainder when $m_{2020} + v_{2020}$ is divided by $1000$.

$\textbf{(A)}\ 332 \qquad\textbf{(B)}\ 374 \qquad\textbf{(C)}\ 679 \qquad\textbf{(D)}\ 721 \qquad\textbf{(E)}\ 932$

Solution

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