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

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== Problem 14 ==
 
== Problem 14 ==
  
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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>.  
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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>.  
  
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[[Northeastern WOOTers Mock AIME I Problems/Problem 14 | Solution]]  
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[[Northeastern WOOTers Mock AIME I Problems/Problem 14 | Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==

Revision as of 18:17, 10 March 2015

Problem 1

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

Solution

Problem 2

It is given that $181^2$ can be written as the difference of the cubes of two consecutive positive integers. Find the sum of these two integers.

Solution

Problem 3

Let $\triangle ABC$ be a triangle with $AC=15$, $BC=112$, and $AB=113$. Let $M$ be the midpoint of $BC$. Let $X$ and $Y$ be trisection points on $AB$. That is, $AX=XY=YB$. Let $D$ and $E$ be the points of intersection of $\overline{CX}$ and $\overline{CY}$ with the cevian $\overline{AM}$, respectively. Find the area of quadrilateral $XYED$.

Solution

Problem 4

Let the number of ordered tuples of positive odd integers $\left( x_1, x_2, \cdots, x_{42} \right)$ such that \[x_1 + x_2 + \cdots + x_{42} = 2014\] be $T$. Find the remainder when $T$ is divided by $1000$.

Solution

Problem 5

Let $x$, $y$, and $z$ be real numbers. Given that $x^2+2y^2+3z^2=1$, the maximum value of $\left( 3x+2y+z \right)^2$ can be represented $\frac{m}{n}$, where $m$ and $n$ are positive integers, where $m$ and $n$ are relatively prime. Find $m+n$.

Solution

Problem 6

Let $\mathcal{S}=\{1,...,10\}$. Two subsets, $A$ and $B$, of $S$ are chosen randomly with replacement, with $B$ chosen after $A$. The probability that $A$ is a subset of $B$ can be written as $\frac {p^a}{q^b}$, for some primes $p$ and $q$. Find $ab+p+q$.

Solution

Problem 7

Find the value of \[\sum_{n=0}^{100}\left\lfloor\frac{3n+4}{13}\right\rfloor-\left\lfloor\frac{n-28+\left\lfloor\frac{n-7}{13}\right\rfloor}{4}\right\rfloor.\]

Solution

Problem 8

Dai the Luzon bleeding-heart has numbered lillypads, $1$ through $100$. 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 $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Solution

Problem 9

Let $ABCDEF$ be a regular hexagon of unit side length. Line $\overline{BE}$ is extended to a point $P$ outside of the hexagon such that $FP=\sqrt{3}$. The line $\overline{AP}$ intersects the lines $\overline{BF}$ and $\overline{EF}$ at points $M$ and $N$, respectively. Let the area of quadrilateral $BENM$ be $S$. Then, the value of $S^2$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

[asy] size(200); defaultpen(linewidth(0.8)); string label[]={'A','B','C','D','E','F'}; pair hex[]; for(int i=0;i<=5;i=i+1) { hex[i]=dir(180-60*i); if(i!=4) label("$"+label[i]+"$",hex[i],dir((0,0)--hex[i])); else label("$E$",hex[4],dir(240)); } draw(hex[0]--hex[1]--hex[2]--hex[3]--hex[4]--hex[5]--cycle); pair X=2.25*hex[4]; draw(hex[1]--X); draw(X--hex[5],linetype("4 4")); [/asy]

Solution

Problem 10

If $a,b,c$ are complex numbers such that \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*} then find the value of $k^4$.

Solution

Problem 11

Cody and Toedy play a game. Cody guesses an integer between $1$ and $2014$ 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. What number should Cody guess to maximize his earnings?

Solution

Problem 12

Let $\triangle ABC$ be a triangle with $AB=39$, $BC=42$, and $CA=45$. A point $D$ is placed on the extension of $BC$ past $C$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle ABD$ and $\triangle ACD$ respectively. If $O_1O_2=29$, then the ratio $\frac{BD}{CD}$ can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m+n$.

Solution

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 $4 \times 4$ 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 $p$. Then, $p$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

[asy] size(100); defaultpen(linewidth(0.8)); draw(origin--(0,1)--(3,1)--(3,0)--cycle^^(1,1)--(1,-1)--(2,-1)--(2,1)); [/asy]

Solution

Problem 14

Consider three infinite sequences of real numbers: \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*} It is known that, for all integers $n$, the following statement holds: \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*}The elements of $Y$ are defined by the relation $y_n=2^{\frac{n}{2^n}}$. Let \[S =\sum_{n=1}^{\infty} \log_2 x_n \log_2 y_n \log_2 z_n.\]Then, $S$ can be represented as a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.

Solution

Problem 15

Find the sum of all integers $n\le96$ such that \[\phi(n)>n-\sqrt{n},\] where $\phi(n)$ denotes the number of integers less than or equal to $n$ that are relatively prime to $n$.

Solution

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