# Difference between revisions of "Northeastern WOOTers Mock AIME I Problems"

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== Problem 1 == | == Problem 1 == | ||

− | Let <math>u</math>, <math>v</math>, <math>x</math>, and <math>y</math> be digits, | + | Let <math>u</math>, <math>v</math>, <math>x</math>, and <math>y</math> be digits, not necessarily distinct and not necessarily non-zero. For how many quadruples <math>(u,v,x,y)</math> is it true that <cmath> N = \overline{uv.xy}+\overline{xy.uv} </cmath> is an integer? As an example, if <math>(u,v,x,y)=(0,1,2,3)</math>, then we have <math> N = 1.23 + 23.01 = 24.24 </math>, which is not an integer. |

− | [[Northeastern WOOTers Mock AIME I Problems/Problem 1 | Solution]] | + | [[Northeastern WOOTers Mock AIME I Problems/Problem 1 | Solution]] |

== Problem 2 == | == Problem 2 == | ||

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== Problem 10 == | == Problem 10 == | ||

− | If <math>a,b,c</math> are complex numbers such that <cmath> \begin {eqnarray*} \frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}&=&0,\\ \text{and} \qquad \frac{a\overline{b}+b\overline{c}+c\overline{a}-\overline{a}b-\overline{b}c-\overline{c}a}{\left(a-b\right)\left(\overline{a-b}\right)}&=&k, \end {eqnarray*} </cmath>then find the value of <math>k^4</math>. | + | If <math>a,b,c</math> are complex numbers such that |

+ | <cmath> \begin{eqnarray*} \frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}&=&0,\\ \text{and} \qquad \frac{a\overline{b}+b\overline{c}+c\overline{a}-\overline{a}b-\overline{b}c-\overline{c}a}{\left(a-b\right)\left(\overline{a-b}\right)}&=&k, \end{eqnarray*} </cmath> | ||

+ | then find the value of <math>k^4</math>. | ||

− | [[Northeastern WOOTers Mock AIME I Problems/Problem 10 | Solution]] | + | [[Northeastern WOOTers Mock AIME I Problems/Problem 10 | Solution]] |

== Problem 11 == | == Problem 11 == | ||

− | Cody and Toedy play a game. Cody guesses an integer between <math>1</math> and <math>2014</math> inclusive, and then Toedy guesses a different integer. A 2014-sided die is rolled. After rolling, whoever guessed closest to the number on the die wins. The winning player wins as much money as the number rolled. | + | Cody and Toedy play a game. Cody guesses an integer between <math>1</math> and <math>2014</math> inclusive, and then Toedy guesses a different integer. A 2014-sided die is rolled. After rolling, whoever guessed closest to the number on the die wins. The winning player wins as much money as the number rolled. Assuming both players play optimally, let <math>N</math> be the number should Cody guess to maximize his earnings. Find the remainder when <math>N</math> is divided by <math>1000</math>. |

[[Northeastern WOOTers Mock AIME I Problems/Problem 11 | Solution]] | [[Northeastern WOOTers Mock AIME I Problems/Problem 11 | Solution]] | ||

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== Problem 14 == | == Problem 14 == | ||

− | Consider three infinite sequences of real numbers: <cmath> \begin {eqnarray*} X &=& \left( x_1, x_2, \cdots \right), \\ Y &=& \left( y_1, y_2, \cdots \right), \\ Z &=& \left( z_1, z_2, \cdots \right). \end {eqnarray*} </cmath> It is known that, for all integers <math>n</math>, the following statement holds: <cmath> \left( \left( \log_2 x_n \right)^2 + \left( \log_2 y_n \right)^2 \right) \cdot \left( \left( \log_2 y_n \right)^2 + \left( \log_2 z_n \right)^2 \right) \\&= \left( \log_2 x_n \log_2 y_n + \log_2 y_n \log_2 z_n \right)^2. </cmath>The elements of <math>Y</math> are defined by the relation <math>y_n=2^{\frac{n}{2^n}}</math>. Let <cmath> S =\sum_{n=1}^{\infty} \log_2 x_n \log_2 y_n \log_2 z_n. </cmath>Then, <math>S</math> can be represented as a fraction <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m</math>. | + | Consider three infinite sequences of real numbers: <cmath> \begin{eqnarray*} X &=& \left( x_1, x_2, \cdots \right), \\ Y &=& \left( y_1, y_2, \cdots \right), \\ Z &=& \left( z_1, z_2, \cdots \right). \end{eqnarray*} </cmath> It is known that, for all integers <math>n</math>, the following statement holds: |

+ | <cmath>\begin{align*} \left( \left( \log_2 x_n \right)^2 + \left( \log_2 y_n \right)^2 \right) \cdot \left( \left( \log_2 y_n \right)^2 + \left( \log_2 z_n \right)^2 \right) \\ | ||

+ | &= \left( \log_2 x_n \log_2 y_n + \log_2 y_n \log_2 z_n \right)^2.\end{align*} </cmath>The elements of <math>Y</math> are defined by the relation <math>y_n=2^{\frac{n}{2^n}}</math>. Let <cmath> S =\sum_{n=1}^{\infty} \log_2 x_n \log_2 y_n \log_2 z_n. </cmath>Then, <math>S</math> can be represented as a fraction <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m</math>. | ||

− | [[Northeastern WOOTers Mock AIME I Problems/Problem 14 | Solution]] | + | [[Northeastern WOOTers Mock AIME I Problems/Problem 14 | Solution]] |

== Problem 15 == | == Problem 15 == |

## Latest revision as of 09:10, 30 May 2020

## Contents

## Problem 1

Let , , , and be digits, not necessarily distinct and not necessarily non-zero. For how many quadruples is it true that is an integer? As an example, if , then we have , which is not an integer.

## Problem 2

It is given that can be written as the difference of the cubes of two consecutive positive integers. Find the sum of these two integers.

## Problem 3

Let be a triangle with , , and . Let be the midpoint of . Let and be trisection points on . That is, . Let and be the points of intersection of and with the cevian , respectively. Find the area of quadrilateral .

## Problem 4

Let the number of ordered tuples of positive odd integers such that be . Find the remainder when is divided by .

## Problem 5

Let , , and be real numbers. Given that , the maximum value of can be represented , where and are positive integers, where and are relatively prime. Find .

## Problem 6

Let . Two subsets, and , of are chosen randomly with replacement, with chosen after . The probability that is a subset of can be written as , for some primes and . Find .

## Problem 7

Find the value of

## Problem 8

Dai the Luzon bleeding-heart has numbered lillypads, through . Then, Ryan the alligator eats those lillypads with the intention of eating Dai. Dai starts on a random lillypad and flies around between lillypads randomly every minute. Ryan also eats a random lillypad every minute. If the expected number of minutes left for Dai to live is , where and are relatively prime positive integers, find .

## Problem 9

Let be a regular hexagon of unit side length. Line is extended to a point outside of the hexagon such that . The line intersects the lines and at points and , respectively. Let the area of quadrilateral be . Then, the value of can be expressed in the form , where and are relatively prime positive integers. Find .

## Problem 10

If are complex numbers such that then find the value of .

## Problem 11

Cody and Toedy play a game. Cody guesses an integer between and inclusive, and then Toedy guesses a different integer. A 2014-sided die is rolled. After rolling, whoever guessed closest to the number on the die wins. The winning player wins as much money as the number rolled. Assuming both players play optimally, let be the number should Cody guess to maximize his earnings. Find the remainder when is divided by .

## Problem 12

Let be a triangle with , , and . A point is placed on the extension of past . Let and be the circumcenters of and respectively. If , then the ratio can be written in the form for relatively prime positive integers and . Find .

## Problem 13

Define a *T-Polyomino* to be a set of 4 cells in a grid that form a T, as shown below. Dai wants to place T-Polyominos onto a grid such that there is no overlap. He continues to place T-Polyominos randomly until he can no longer do so. Let the probability that he will cover the entire board be . Then, can be expressed as , where and are relatively prime positive integers. Find .

## Problem 14

Consider three infinite sequences of real numbers: It is known that, for all integers , the following statement holds: The elements of are defined by the relation . Let Then, can be represented as a fraction , where and are relatively prime positive integers. Find .

## Problem 15

Find the sum of all integers such that where denotes the number of integers less than or equal to that are relatively prime to .