Difference between revisions of "2019 AIME II Problems"
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− | The 2019 | + | {{AIME Problems|year=2019|n=II}} |
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+ | ==Problem 1== | ||
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+ | Two different points, <math>C</math> and <math>D</math>, lie on the same side of line <math>AB</math> so that <math>\triangle ABC</math> and <math>\triangle BAD</math> are congruent with <math>AB=9,BC=AD=10</math>, and <math>CA=DB=17</math>. The intersection of these two triangular regions has area <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
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+ | [[2019 AIME II Problems/Problem 1 | Solution]] | ||
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+ | ==Problem 2== | ||
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+ | Lily pads <math>1,2,3,\ldots</math> lie in a row on a pond. A frog makes a sequence of jumps starting on pad <math>1</math>. From any pad <math>k</math> the frog jumps to either pad <math>k+1</math> or pad <math>k+2</math> chosen randomly with probability <math>\tfrac{1}{2}</math> and independently of other jumps. The probability that the frog visits pad <math>7</math> is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | ||
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+ | [[2019 AIME II Problems/Problem 2 | Solution]] | ||
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+ | ==Problem 3== | ||
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+ | Find the number of <math>7</math>-tuples of positive integers <math>(a,b,c,d,e,f,g)</math> that satisfy the following system of equations: | ||
+ | <cmath>abc=70</cmath> | ||
+ | <cmath>cde=71</cmath> | ||
+ | <cmath>efg=72.</cmath> | ||
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+ | [[2019 AIME II Problems/Problem 3 | Solution]] | ||
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+ | ==Problem 4== | ||
+ | A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
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+ | [[2019 AIME II Problems/Problem 4 | Solution]] | ||
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+ | ==Problem 5== | ||
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+ | [[2019 AIME II Problems/Problem 5 | Solution]] | ||
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+ | ==Problem 6== | ||
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+ | [[2019 AIME II Problems/Problem 6 | Solution]] | ||
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+ | ==Problem 7== | ||
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+ | [[2019 AIME II Problems/Problem 7 | Solution]] | ||
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+ | ==Problem 8== | ||
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+ | [[2019 AIME II Problems/Problem 8 | Solution]] | ||
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+ | ==Problem 9== | ||
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+ | [[2019 AIME II Problems/Problem 9 | Solution]] | ||
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+ | ==Problem 10== | ||
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+ | [[2019 AIME II Problems/Problem 10 | Solution]] | ||
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+ | ==Problem 11== | ||
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+ | [[2019 AIME II Problems/Problem 11 | Solution]] | ||
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+ | ==Problem 12== | ||
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+ | [[2019 AIME II Problems/Problem 12 | Solution]] | ||
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+ | ==Problem 13== | ||
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+ | [[2019 AIME II Problems/Problem 13 | Solution]] | ||
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+ | ==Problem 14== | ||
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+ | [[2019 AIME II Problems/Problem 14 | Solution]] | ||
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+ | ==Problem 15== | ||
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+ | [[2019 AIME II Problems/Problem 15 | Solution]] | ||
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+ | {{AIME box|year=2019|n=II|before=[[2019 AIME I]]|after=[[2020 AIME I]]}} | ||
+ | {{MAA Notice}} |
Revision as of 15:23, 22 March 2019
2019 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
[hide]Problem 1
Two different points, and , lie on the same side of line so that and are congruent with , and . The intersection of these two triangular regions has area , where and are relatively prime positive integers. Find .
Problem 2
Lily pads lie in a row on a pond. A frog makes a sequence of jumps starting on pad . From any pad the frog jumps to either pad or pad chosen randomly with probability and independently of other jumps. The probability that the frog visits pad is , where and are relatively prime positive integers. Find .
Problem 3
Find the number of -tuples of positive integers that satisfy the following system of equations:
Problem 4
A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is , where and are relatively prime positive integers. Find .
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
2019 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2019 AIME I |
Followed by 2020 AIME I | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.