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Difference between revisions of "2019 AIME II Problems"

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The 2019 Aime II took place on March 21, 2019.
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{{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:
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<cmath>abc=70</cmath>
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<cmath>cde=71</cmath>
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<cmath>efg=72.</cmath>
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[[2019 AIME II Problems/Problem 3 | Solution]]
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==Problem 4==
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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]]}}
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{{MAA Notice}}

Revision as of 15:23, 22 March 2019

2019 AIME II (Answer Key)
Printable version | AoPS Contest Collections • PDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB=9,BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 2

Lily pads $1,2,3,\ldots$ lie in a row on a pond. A frog makes a sequence of jumps starting on pad $1$. From any pad $k$ the frog jumps to either pad $k+1$ or pad $k+2$ chosen randomly with probability $\tfrac{1}{2}$ and independently of other jumps. The probability that the frog visits pad $7$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution

Problem 3

Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following system of equations: \[abc=70\] \[cde=71\] \[efg=72.\]

Solution

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 $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

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. AMC logo.png

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