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Difference between revisions of "2020 AIME II Problems"
m (I didn't put in 4, 7, 8, 9 because there were no problem on the solutions page.)
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==Problem 1==
 
==Problem 1==
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 +
Find the number of ordered pairs of positive integers <math>(m,n)</math> such that <math>{m^2n = 20 ^{20}}</math>.
  
 
[[2020 AIME II Problems/Problem 1 | Solution]]
 
[[2020 AIME II Problems/Problem 1 | Solution]]
  
 
==Problem 2==
 
==Problem 2==
 +
 +
Let <math>P</math> be a point chosen uniformly at random in the interior of the unit square with vertices at <math>(0,0), (1,0), (1,1)</math>, and <math>(0,1)</math>. The probability that the slope of the line determined by <math>P</math> and the point <math>\left(\frac58, \frac38 \right)</math> is greater than <math>\frac12</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
 
[[2020 AIME II Problems/Problem 2 | Solution]]
 
[[2020 AIME II Problems/Problem 2 | Solution]]
  
 
==Problem 3==
 
==Problem 3==
 +
 +
The value of <math>x</math> that satisfies <math>\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
 
[[2020 AIME II Problems/Problem 3 | Solution]]
 
[[2020 AIME II Problems/Problem 3 | Solution]]
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==Problem 5==
 
==Problem 5==
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 +
For each positive integer <math>n</math>, left <math>f(n)</math> be the sum of the digits in the base-four representation of <math>n</math> and let <math>g(n)</math> be the sum of the digits in the base-eight representation of <math>f(n)</math>. For example, <math>f(2020) = f(133210_{\text{four}}) = 10 = 12_{\text{eight}}</math>, and <math>g(2020) = \text{the digit sum of }12_{\text{eight}} = 3</math>. Let <math>N</math> be the least value of <math>n</math> such that the base-sixteen representation of <math>g(n)</math> cannot be expressed using only the digits <math>0</math> through <math>9</math>. Find the remainder when <math>N</math> is divided by <math>1000</math>.
  
 
[[2020 AIME II Problems/Problem 5 | Solution]]
 
[[2020 AIME II Problems/Problem 5 | Solution]]
  
 
==Problem 6==
 
==Problem 6==
 +
 +
Define a sequence recursively by <math>t_1 = 20</math>, <math>t_2 = 21</math>, and<cmath>t_n = \frac{5t_{n-1}+1}{25t_{n-2}}</cmath>for all <math>n \ge 3</math>. Then <math>t_{2020}</math> can be written as <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
  
 
[[2020 AIME II Problems/Problem 6 | Solution]]
 
[[2020 AIME II Problems/Problem 6 | Solution]]
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==Problem 10==
 
==Problem 10==
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 +
Find the sum of all positive integers <math>n</math> such that when <math>1^3+2^3+3^3+\cdots +n^3</math> is divided by <math>n+5</math>, the remainder is <math>17</math>.
  
 
[[2020 AIME II Problems/Problem 10 | Solution]]
 
[[2020 AIME II Problems/Problem 10 | Solution]]
  
 
==Problem 11==
 
==Problem 11==
 +
 +
Let <math>P(X) = x^2 - 3x - 7</math>, and let <math>Q(x)</math> and <math>R(x)</math> be two quadratic polynomials also with the coefficient of <math>x^2</math> equal to <math>1</math>. David computes each of the three sums <math>P + Q</math>, <math>P + R</math>, and <math>Q + R</math> and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If <math>Q(0) = 2</math>, then <math>R(0) = \frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
  
 
[[2020 AIME II Problems/Problem 11 | Solution]]
 
[[2020 AIME II Problems/Problem 11 | Solution]]
  
 
==Problem 12==
 
==Problem 12==
 +
 +
Let <math>m</math> and <math>n</math> be odd integers greater than <math>1.</math> An <math>m\times n</math> rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers <math>1</math> through <math>n</math>, those in the second row are numbered left to right with the integers <math>n + 1</math> through <math>2n</math>, and so on. Square <math>200</math> is in the top row, and square <math>2000</math> is in the bottom row. Find the number of ordered pairs <math>(m,n)</math> of odd integers greater than <math>1</math> with the property that, in the <math>m\times n</math> rectangle, the line through the centers of squares <math>200</math> and <math>2000</math> intersects the interior of square <math>1099</math>.
  
 
[[2020 AIME II Problems/Problem 12 | Solution]]
 
[[2020 AIME II Problems/Problem 12 | Solution]]
  
 
==Problem 13==
 
==Problem 13==
 +
 +
Convex pentagon <math>ABCDE</math> has side lengths <math>AB=5</math>, <math>BC=CD=DE=6</math>, and <math>EA=7</math>. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of <math>ABCDE</math>.
  
 
[[2020 AIME II Problems/Problem 13 | Solution]]
 
[[2020 AIME II Problems/Problem 13 | Solution]]
  
 
==Problem 14==
 
==Problem 14==
 +
 +
For real number <math>x</math> let <math>\lfloor x\rfloor</math> be the greatest integer less than or equal to <math>x</math>, and define <math>\{x\} = x - \lfloor x \rfloor</math> to be the fractional part of <math>x</math>. For example, <math>\{3\} = 0</math> and <math>\{4.56\} = 0.56</math>. Define <math>f(x)=x\{x\}</math>, and let <math>N</math> be the number of real-valued solutions to the equation <math>f(f(f(x)))=17</math> for <math>0\leq x\leq 2020</math>. Find the remainder when <math>N</math> is divided by <math>1000</math>.
  
 
[[2020 AIME II Problems/Problem 14 | Solution]]
 
[[2020 AIME II Problems/Problem 14 | Solution]]
  
 
==Problem 15==
 
==Problem 15==
 +
 +
Let <math>\triangle ABC</math> be an acute scalene triangle with circumcircle <math>\omega</math>. The tangents to <math>\omega</math> at <math>B</math> and <math>C</math> intersect at <math>T</math>. Let <math>X</math> and <math>Y</math> be the projections of <math>T</math> onto lines <math>AB</math> and <math>AC</math>, respectively. Suppose <math>BT = CT = 16</math>, <math>BC = 22</math>, and <math>TX^2 + TY^2 + XY^2 = 1143</math>. Find <math>XY^2</math>.
  
 
[[2020 AIME II Problems/Problem 15 | Solution]]
 
[[2020 AIME II Problems/Problem 15 | Solution]]

Revision as of 02:09, 8 June 2020

2020 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

Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$.

Solution

Problem 2

Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\left(\frac58, \frac38 \right)$ is greater than $\frac12$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 3

The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 4

Solution

Problem 5

For each positive integer $n$, left $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\text{four}}) = 10 = 12_{\text{eight}}$, and $g(2020) = \text{the digit sum of }12_{\text{eight}} = 3$. Let $N$ be the least value of $n$ such that the base-sixteen representation of $g(n)$ cannot be expressed using only the digits $0$ through $9$. Find the remainder when $N$ is divided by $1000$.

Solution

Problem 6

Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\[t_n = \frac{5t_{n-1}+1}{25t_{n-2}}\]for all $n \ge 3$. Then $t_{2020}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Find the sum of all positive integers $n$ such that when $1^3+2^3+3^3+\cdots +n^3$ is divided by $n+5$, the remainder is $17$.

Solution

Problem 11

Let $P(X) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution

Problem 12

Let $m$ and $n$ be odd integers greater than $1.$ An $m\times n$ rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers $1$ through $n$, those in the second row are numbered left to right with the integers $n + 1$ through $2n$, and so on. Square $200$ is in the top row, and square $2000$ is in the bottom row. Find the number of ordered pairs $(m,n)$ of odd integers greater than $1$ with the property that, in the $m\times n$ rectangle, the line through the centers of squares $200$ and $2000$ intersects the interior of square $1099$.

Solution

Problem 13

Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$.

Solution

Problem 14

For real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$, and define $\{x\} = x - \lfloor x \rfloor$ to be the fractional part of $x$. For example, $\{3\} = 0$ and $\{4.56\} = 0.56$. Define $f(x)=x\{x\}$, and let $N$ be the number of real-valued solutions to the equation $f(f(f(x)))=17$ for $0\leq x\leq 2020$. Find the remainder when $N$ is divided by $1000$.

Solution

Problem 15

Let $\triangle ABC$ be an acute scalene triangle with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT = CT = 16$, $BC = 22$, and $TX^2 + TY^2 + XY^2 = 1143$. Find $XY^2$.

Solution

2020 AIME II (Problems • Answer Key • Resources)
Preceded by
2020 AIME I Problems 		Followed by
2021 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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