Difference between revisions of "2022 AMC 12B Problems"
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==Problem 10 == | ==Problem 10 == | ||
− | + | Regular hexagon <math>ABCDEF</math> has side length <math>2</math>. Let <math>G</math> be the midpoint of <math>\overline{AB}</math>, and let <math>H</math> be the midpoint of <math>\overline{DE}</math>. What is the perimeter of <math>GCHF</math>? | |
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+ | <math> \textbf{(A)}\ 4\sqrt3 \qquad | ||
+ | \textbf{(B)}\ 8 \qquad | ||
+ | \textbf{(C)}\ 4\sqrt5 \qquad | ||
+ | \textbf{(D)}\ 4\sqrt7 \qquad | ||
+ | \textbf{(E)}\ 12</math> | ||
[[2022 AMC 12B Problems/Problem 10|Solution]] | [[2022 AMC 12B Problems/Problem 10|Solution]] | ||
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==Problem 11 == | ==Problem 11 == |
Revision as of 15:41, 17 November 2022
2022 AMC 12B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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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 |
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
Define to be for all real numbers and . What is the value of
Problem 2
In rhombus , point lies on segment such that , , and . What is the area of ?
[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,SW); label("", B, NW); label("",C,NE); label("",D,SE); label("",P,S); [/asy]
Problem 3
How many of the first ten numbers of the sequence , , , ... are prime numbers?
Problem 4
For how many values of the constant will the polynomial have two distinct integer roots?
Problem 5
The point is rotated counterclockwise about the point . What are the coordinates of its new position?
Problem 6
Consider the following sets of elements each: How many of these sets contain exactly two multiples of ?
Problem 7
Camila writes down five positive integers. The unique mode of these integers is greater than their median, and the median is greater than their arithmetic mean. What is the least possible value for the mode?
Problem 8
What is the graph of in the coordinate plane?
Problem 9
The sequence is a strictly increasing arithmetic sequence of positive integers such that What is the minimum possible value of ?
Problem 10
Regular hexagon has side length . Let be the midpoint of , and let be the midpoint of . What is the perimeter of ?
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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