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Difference between revisions of "2020 AIME II Problems"

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==Problem 11==
 
==Problem 11==
  
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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>.
 
  
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Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks!
 
[[2020 AIME II Problems/Problem 11 | Solution]]
 
[[2020 AIME II Problems/Problem 11 | Solution]]
  

Revision as of 12:58, 14 December 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

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Solution

Problem 2

Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks!

Solution

Problem 3

Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Solution

Problem 4

Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks!

Solution

Problem 5

Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks!

Solution

Problem 6

Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks!

Solution

Problem 7

Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks!

Solution

Problem 8

Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks!

Solution

Problem 9

Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks!

Solution

Problem 10

Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Solution

Problem 11

Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! Scorpio Sucks! 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)
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2020 AIME I Problems 		Followed by
2021 AIME I Problems
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