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Difference between revisions of "2019 AIME II Problems"

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==Problem 5==
 
==Problem 5==
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Four ambassadors and one advisor for each of then are to be seated at a round table with <math>12</math> chairs numbered in order <math>1</math> to <math>12</math>. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are <math>N</math> ways for the <math>8</math> people to be seated at the table under these conditions. Find the remainder when <math>N</math> is divided by <math>1000</math>.
+
Four ambassadors and one advisor for each of them are to be seated at a round table with <math>12</math> chairs numbered in order <math>1</math> to <math>12</math>. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are <math>N</math> ways for the <math>8</math> people to be seated at the table under these conditions. Find the remainder when <math>N</math> is divided by <math>1000</math>.
  
 
[[2019 AIME II Problems/Problem 5 | Solution]]
 
[[2019 AIME II Problems/Problem 5 | Solution]]
  
 
==Problem 6==
 
==Problem 6==
−
In a Martian civilization, all logarithms whose bases are not specified as assumed to be base <math>b</math>, for some fixed <math>b\ge2</math>. A Martian student writes down
+
In a Martian civilization, all logarithms whose bases are not specified are assumed to be base <math>b</math>, for some fixed <math>b\ge2</math>. A Martian student writes down
 
<cmath>3\log(\sqrt{x}\log x)=56</cmath>
 
<cmath>3\log(\sqrt{x}\log x)=56</cmath>
 
<cmath>\log_{\log x}(x)=54</cmath>
 
<cmath>\log_{\log x}(x)=54</cmath>
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==Problem 8==
 
==Problem 8==
−
The polynomial <math>f(z)=az^{2018}+bz^{2017}+cz^{2016}</math> has real coefficients not exceeding <math>2019</math> and <math>f\left(\tfrac{1+\sqrt3i}{2}\right)=2015+2019\sqrt3i</math>. Find the remainder when <math>f(1)</math> is divided by <math>1000</math>.
+
The polynomial <math>f(z)=az^{2018}+bz^{2017}+cz^{2016}</math> has real coefficients not exceeding <math>2019,</math> and <math>f\left(\tfrac{1+\sqrt3i}{2}\right)=2015+2019\sqrt3i</math>. Find the remainder when <math>f(1)</math> is divided by <math>1000</math>.
  
 
[[2019 AIME II Problems/Problem 8 | Solution]]
 
[[2019 AIME II Problems/Problem 8 | Solution]]
  
 
==Problem 9==
 
==Problem 9==
 +
Call a positive integer <math>n</math> <math>k</math>-<i>pretty</i> if <math>n</math> has exactly <math>k</math> positive divisors and <math>n</math> is divisible by <math>k</math>. For example, <math>18</math> is <math>6</math>-pretty. Let <math>S</math> be the sum of the positive integers less than <math>2019</math> that are <math>20</math>-pretty. Find <math>\tfrac{S}{20}</math>.
  
 
[[2019 AIME II Problems/Problem 9 | Solution]]
 
[[2019 AIME II Problems/Problem 9 | Solution]]
  
 
==Problem 10==
 
==Problem 10==
 +
There is a unique angle <math>\theta</math> between <math>0^\circ</math> and <math>90^\circ</math> such that for nonnegative integers <math>n,</math> the value of <math>\tan(2^n\theta)</math> is positive when <math>n</math> is a multiple of <math>3</math>, and negative otherwise. The degree measure of <math>\theta</math> is <math>\tfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
  
 
[[2019 AIME II Problems/Problem 10 | Solution]]
 
[[2019 AIME II Problems/Problem 10 | Solution]]
  
 
==Problem 11==
 
==Problem 11==
 +
Triangle <math>ABC</math> has side lengths <math>AB=7,BC=8,</math> and <math>CA=9.</math> Circle <math>\omega_1</math> passes through <math>B</math> and is tangent to line <math>AC</math> at <math>A.</math> Circle <math>\omega_2</math> passes through <math>C</math> and is tangent to line <math>AB</math> at <math>A.</math> Let <math>K</math> be the intersection of circles <math>\omega_1</math> and <math>\omega_2</math> not equal to <math>A.</math> Then <math>AK=\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math>
  
 
[[2019 AIME II Problems/Problem 11 | Solution]]
 
[[2019 AIME II Problems/Problem 11 | Solution]]
  
 
==Problem 12==
 
==Problem 12==
 +
For <math>n\ge1</math> call a finite sequence <math>(a_1,a_2,\ldots,a_n)</math> of positive integers <i>progressive</i> if <math>a_i<a_{i+1}</math> and <math>a_i</math> divides <math>a_{i+1}</math> for <math>1\le i\le n-1</math>. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to <math>360.</math>
  
 
[[2019 AIME II Problems/Problem 12 | Solution]]
 
[[2019 AIME II Problems/Problem 12 | Solution]]
  
 
==Problem 13==
 
==Problem 13==
 +
Regular octagon <math>A_1A_2A_3A_4A_5A_6A_7A_8</math> is inscribed in a circle of area <math>1.</math> Point <math>P</math> lies inside the circle so that the region bounded by <math>\overline{PA_1},\overline{PA_2},</math> and the minor arc <math>\widehat{A_1A_2}</math> of the circle has area <math>\tfrac{1}{7},</math> while the region bounded by <math>\overline{PA_3},\overline{PA_4},</math> and the minor arc <math>\widehat{A_3A_4}</math> of the circle has area <math>\tfrac{1}{9}.</math> There is a positive integer <math>n</math> such that the area of the region bounded by <math>\overline{PA_6},\overline{PA_7},</math> and the minor arc <math>\widehat{A_6A_7}</math> of the circle is equal to <math>\tfrac{1}{8}-\tfrac{\sqrt2}{n}.</math> Find <math>n.</math>
  
 
[[2019 AIME II Problems/Problem 13 | Solution]]
 
[[2019 AIME II Problems/Problem 13 | Solution]]
  
 
==Problem 14==
 
==Problem 14==
 +
Find the sum of all positive integers <math>n</math> such that, given an unlimited supply of stamps of denominations <math>5,n,</math> and <math>n+1</math> cents, <math>91</math> cents is the greatest postage that cannot be formed.
  
 
[[2019 AIME II Problems/Problem 14 | Solution]]
 
[[2019 AIME II Problems/Problem 14 | Solution]]
  
 
==Problem 15==
 
==Problem 15==
 +
In acute triangle <math>ABC,</math> points <math>P</math> and <math>Q</math> are the feet of the perpendiculars from <math>C</math> to <math>\overline{AB}</math> and from <math>B</math> to <math>\overline{AC}</math>, respectively. Line <math>PQ</math> intersects the circumcircle of <math>\triangle ABC</math> in two distinct points, <math>X</math> and <math>Y</math>. Suppose <math>XP=10</math>, <math>PQ=25</math>, and <math>QY=15</math>. The value of <math>AB\cdot AC</math> can be written in the form <math>m\sqrt n</math> where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>.
  
 
[[2019 AIME II Problems/Problem 15 | Solution]]
 
[[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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{{AIME box|year=2019|n=II|before=[[2019 AIME I Problems]]|after=[[2020 AIME I Problems]]}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 13:45, 21 August 2023

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 $\tfrac{p}{q}$, 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

Four ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order $1$ to $12$. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the $8$ people to be seated at the table under these conditions. Find the remainder when $N$ is divided by $1000$.

Solution

Problem 6

In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b\ge2$. A Martian student writes down \[3\log(\sqrt{x}\log x)=56\] \[\log_{\log x}(x)=54\] and finds that this system of equations has a single real number solution $x>1$. Find $b$.

Solution

Problem 7

Triangle $ABC$ has side lengths $AB=120,BC=220$, and $AC=180$. Lines $\ell_A,\ell_B$, and $\ell_C$ are drawn parallel to $\overline{BC},\overline{AC}$, and $\overline{AB}$, respectively, such that the intersections of $\ell_A,\ell_B$, and $\ell_C$ with the interior of $\triangle ABC$ are segments of lengths $55,45$, and $15$, respectively. Find the perimeter of the triangle whose sides lie on lines $\ell_A,\ell_B$, and $\ell_C$.

Solution

Problem 8

The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019,$ and $f\left(\tfrac{1+\sqrt3i}{2}\right)=2015+2019\sqrt3i$. Find the remainder when $f(1)$ is divided by $1000$.

Solution

Problem 9

Call a positive integer $n$ $k$-pretty if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$-pretty. Let $S$ be the sum of the positive integers less than $2019$ that are $20$-pretty. Find $\tfrac{S}{20}$.

Solution

Problem 10

There is a unique angle $\theta$ between $0^\circ$ and $90^\circ$ such that for nonnegative integers $n,$ the value of $\tan(2^n\theta)$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\theta$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution

Problem 11

Triangle $ABC$ has side lengths $AB=7,BC=8,$ and $CA=9.$ Circle $\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\omega_1$ and $\omega_2$ not equal to $A.$ Then $AK=\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 12

For $n\ge1$ call a finite sequence $(a_1,a_2,\ldots,a_n)$ of positive integers progressive if $a_i<a_{i+1}$ and $a_i$ divides $a_{i+1}$ for $1\le i\le n-1$. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360.$

Solution

Problem 13

Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1.$ Point $P$ lies inside the circle so that the region bounded by $\overline{PA_1},\overline{PA_2},$ and the minor arc $\widehat{A_1A_2}$ of the circle has area $\tfrac{1}{7},$ while the region bounded by $\overline{PA_3},\overline{PA_4},$ and the minor arc $\widehat{A_3A_4}$ of the circle has area $\tfrac{1}{9}.$ There is a positive integer $n$ such that the area of the region bounded by $\overline{PA_6},\overline{PA_7},$ and the minor arc $\widehat{A_6A_7}$ of the circle is equal to $\tfrac{1}{8}-\tfrac{\sqrt2}{n}.$ Find $n.$

Solution

Problem 14

Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed.

Solution

Problem 15

In acute triangle $ABC,$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\overline{AB}$ and from $B$ to $\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\triangle ABC$ in two distinct points, $X$ and $Y$. Suppose $XP=10$, $PQ=25$, and $QY=15$. The value of $AB\cdot AC$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

Solution

2019 AIME II (Problems • Answer Key • Resources)
Preceded by
2019 AIME I Problems 		Followed by
2020 AIME I Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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