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Difference between revisions of "2022 AMC 12B Problems"

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==Problem 6 ==
 
==Problem 6 ==
  
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Consider the following <math>100</math> sets of <math>10</math> elements each:
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\begin{align*}
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&\{1,2,3,\cdots,10\}, \\
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&\{11,12,13,\cdots,20\},\\
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&\{21,22,23,\cdots,30\},\\
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&\vdots\\
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&\{991,992,993,\cdots,1000\}.
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\end{align*}How many of these sets contain exactly two multiples of <math>7</math>?
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<math>\textbf{(A)} 40\qquad\textbf{(B)} 42\qquad\textbf{(C)} 42\qquad\textbf{(D)} 49\qquad\textbf{(E)} 50</math>
  
 
[[2022 AMC 12B Problems/Problem 6|Solution]]
 
[[2022 AMC 12B Problems/Problem 6|Solution]]

Revision as of 15:36, 17 November 2022

2022 AMC 12B (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

Problem 1

Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y$. What is the value of\[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\] $\textbf{(A)}\ -2 \qquad \textbf{(B)}\ -1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

Solution

Problem 2

In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$?

[asy] import olympiad; size(180); real r = 3, s = 5, t = sqrt(r*r+s*s); defaultpen(linewidth(0.6) + fontsize(10)); pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0); draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D)); label("$A$",A,SW); label("$B$", B, NW); label("$C$",C,NE); label("$D$",D,SE); label("$P$",P,S); [/asy] $\textbf{(A) }3\sqrt 5 \qquad \textbf{(B) }10 \qquad \textbf{(C) }6\sqrt 5 \qquad \textbf{(D) }20\qquad \textbf{(E) }25$

Solution

Problem 3

How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers?

$\text{(A) } 0 \qquad \text{(B) }1 \qquad \text{(C) }2 \qquad \text{(D) }3 \qquad \text{(E) }4$

Solution

Problem 4

For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?

$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }14 \qquad \textbf{(E) }16$

Solution

Problem 5

The point $(-1, -2)$ is rotated $270^{\circ}$ counterclockwise about the point $(3, 1)$. What are the coordinates of its new position?

$\textbf{(A)}\ (-3, -4) \qquad \textbf{(B)}\ (0,5) \qquad \textbf{(C)}\ (2,-1) \qquad \textbf{(D)}\ (4,3) \qquad \textbf{(E)}\ (6,-3)$ Solution

Problem 6

Consider the following $100$ sets of $10$ elements each: \begin{align*} &\{1,2,3,\cdots,10\}, \\ &\{11,12,13,\cdots,20\},\\ &\{21,22,23,\cdots,30\},\\ &\vdots\\ &\{991,992,993,\cdots,1000\}. \end{align*}How many of these sets contain exactly two multiples of $7$?

$\textbf{(A)} 40\qquad\textbf{(B)} 42\qquad\textbf{(C)} 42\qquad\textbf{(D)} 49\qquad\textbf{(E)} 50$

Solution

Problem 7

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Problem 8

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Problem 9

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Problem 10

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Problem 11

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Problem 12

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Problem 13

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Problem 14

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Problem 15

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Problem 16

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Problem 17

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Problem 18

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Problem 19

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Problem 20

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Problem 21

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Problem 22

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Problem 23

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Problem 24

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Problem 25

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Solution

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