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Difference between revisions of "Mock AIME 2 Pre 2005 Problems"

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[[Mock AIME 2 Pre 2005 Problems/Problem 5|Solution]]
 
[[Mock AIME 2 Pre 2005 Problems/Problem 5|Solution]]
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== Problem 6 ==
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<math>ABC</math> is a scalene triangle. Points <math>D</math>, <math>E</math>, and <math>F</math> are selected on sides <math>BC</math>, <math>CA</math>, and <math>AB</math> respectively. The cevians <math>AD</math>, <math>BE</math>, and <math>CF</math> concur at point <math>P</math>. If <math>[AFP] = 126</math>, <math>[FBP] = 63</math>, and <math>[CEP] = 24</math>, determine the area of triangle <math>ABC</math>.
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[[Mock AIME 2 Pre 2005 Problems/Problem 6|Solution]]
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==Problem 7 ==
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Anders, Po-Ru, Reid, and Aaron are playing Bridge. After one hand, they notice that all of the cards of the two suits are split between Reid and Po-Ru's hands. Let <math>N</math> denote the number of ways <math>13</math> cards can be dealt to each player such that this is the case. Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>. (Bridge is a game played with the standard <math>52</math>-card deck.)
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[[Mock AIME 2 Pre 2005 Problems/Problem 7|Solution]]
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== Problem 8 ==
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Determine the remainder obtained when the expression <cmath>2004^{2003^{2002^{2001}}}</cmath> is divided by <math>1000</math>.
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[[Mock AIME 2 Pre 2005 Problems/Problem 8|Solution]]
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== Problem 9 ==
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Let <cmath>(1+x^3)\left(1+2x^{3^2}\right)\cdots \left(1+kx^{3^k}\right) \cdots \left(1+1997x^{3^{1997}}\right) = 1+a_1 x^{k_1} + a_2 x^{k_2} + \cdots + a_m x^{k_m}</cmath> where <math>a_i \ne 0</math> and <math>k_1 < k_2 < \cdots < k_m</math>. Determine the remainder obtained when <math>a_{1997}</math> is divided by <math>1000</math>.
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[[Mock AIME 2 Pre 2005 Problems/Problem 9|Solution]]
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== Problem 10 ==
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<math>ABCDE</math> is a cyclic pentagon with <math>BC = CD = DE</math>. The diagonals <math>AC</math> and <math>BE</math> intersect at <math>M</math>. <math>N</math> is the foot of the altitude from <math>M</math> to <math>AB</math>. We have <math>MA = 25</math>, <math>MD = 113</math>, and <math>MN = 15</math>. The area of triangle <math>ABE</math> can be expressed as <math>\tfrac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine the remainder obtained when <math>m+n</math> is divided by <math>1000</math>.
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[[Mock AIME 2 Pre 2005 Problems/Problem 10|Solution]]
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== Problem 11 ==
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<math>\alpha</math>, <math>\beta</math>, and <math>\gamma</math> are the roots of <math>x(x-200)(4x+1) = 1</math>. Let <cmath>\omega = \tan^{-1}(\alpha) + \tan^{-1}(\beta) + \tan^{-1} (\gamma).</cmath> The value of <math>\tan(\omega)</math> can be written as <math>\tfrac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine the value of <math>m+n</math>.
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[[Mock AIME 2 Pre 2005 Problems/Problem 11|Solution]]
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== Problem 12 ==
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<math>ABCD</math> is a cyclic quadrilateral with <math>AB = 8</math>, <math>BC = 4</math>, <math>CD = 1</math>, and <math>DA = 7</math>. Let <math>O</math> and <math>P</math> denote the circumcenter and intersection of <math>AC</math> and <math>BD</math> respectively. The value of <math>OP^2</math> can be expressed as <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine the remainder obtained when <math>m+n</math> is divided by <math>1000</math>.
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[[Mock AIME 2 Pre 2005 Problems/Problem 12|Solution]]
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== Problem 13 ==
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<math>P(x)</math> is a polynomial of minimal degree that satisfies <cmath>P(k) = \dfrac{1}{k(k+1)}</cmath> for <math>k = 1, 2, 3, \ldots, 10</math>. The value of <math>P(11)</math> can be written as <math>-\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine <math>m+n</math>.
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[[Mock AIME 2 Pre 2005 Problems/Problem 13|Solution]]
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== Problem 14 ==
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<math>3</math> Elm trees, <math>4</math> Dogwood trees, and <math>5</math> Oak trees are to be planted in a line in front of a library such that <cmath>\begin{align*} i&) \text{ No two Elm trees are next to each other.} \\ ii&) \text{ No Dogwood tree is adjacent to an Oak tree.} \\ iii&) \text{ All of the trees are planted.} \end{align*}</cmath> How many ways can the trees be situated in this manner?
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[[Mock AIME 2 Pre 2005 Problems/Problem 14|Solution]]
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== Problem 15 ==
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In triangle <math>ABC</math>, we have <math>BC = 13</math>, <math>CA = 37</math>, and <math>AB = 40</math>. Points <math>D</math>, <math>E</math>, and <math>F</math> are selected on <math>BC</math>, <math>CA</math>, and <math>AB</math> respectively such that <math>AD</math>, <math>BE</math>, and <math>CF</math> concur at the circumcenter of <math>ABC</math>. The value of <cmath>\dfrac{1}{AD} + \dfrac{1}{BE} + \dfrac{1}{CF}</cmath> can be expressed as <math>\tfrac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine <math>m+n</math>.
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[[Mock AIME 2 Pre 2005 Problems/Problem 15|Solution]]

Latest revision as of 12:52, 9 June 2020

Problem 1

Compute the largest integer $k$ such that $2004^k$ divides $2004!$.

Solution

Problem 2

$x$ is a real number with the property that $x+\tfrac1x = 3$. Let $S_m = x^m + \tfrac{1}{x^m}$. Determine the value of $S_7$.

Solution

Problem 3

In a box, there are $4$ green balls, $4$ blue balls, $2$ red balls, a brown ball, a white ball, and a black ball. These balls are randomly drawn out of the box one at a time (without replacement) until two of the same color have been removed. This process requires that at most $7$ balls be removed. The probability that $7$ balls are drawn can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Solution

Problem 4

Let $S = \{5^k | k \in \textbf{Z}, 0 \le k \le 2004 \}$. Given that $5^{2004} = 5443 \cdots 0625$ has $1401$ digits, how many elements of $S$ begin with the digit $1$?

Solution

Problem 5

Let $S$ be the set of integers $n > 1$ for which $\tfrac1n = 0.d_1d_2d_3d_4\ldots$, an infinite decimal that has the property that $d_i = d_{i+12}$ for all positive integers $i$. Given that $9901$ is prime, how many positive integers are in $S$? (The $d_i$ are digits.)

Solution

Problem 6

$ABC$ is a scalene triangle. Points $D$, $E$, and $F$ are selected on sides $BC$, $CA$, and $AB$ respectively. The cevians $AD$, $BE$, and $CF$ concur at point $P$. If $[AFP] = 126$, $[FBP] = 63$, and $[CEP] = 24$, determine the area of triangle $ABC$.

Solution

Problem 7

Anders, Po-Ru, Reid, and Aaron are playing Bridge. After one hand, they notice that all of the cards of the two suits are split between Reid and Po-Ru's hands. Let $N$ denote the number of ways $13$ cards can be dealt to each player such that this is the case. Determine the remainder obtained when $N$ is divided by $1000$. (Bridge is a game played with the standard $52$-card deck.)

Solution

Problem 8

Determine the remainder obtained when the expression \[2004^{2003^{2002^{2001}}}\] is divided by $1000$.

Solution

Problem 9

Let \[(1+x^3)\left(1+2x^{3^2}\right)\cdots \left(1+kx^{3^k}\right) \cdots \left(1+1997x^{3^{1997}}\right) = 1+a_1 x^{k_1} + a_2 x^{k_2} + \cdots + a_m x^{k_m}\] where $a_i \ne 0$ and $k_1 < k_2 < \cdots < k_m$. Determine the remainder obtained when $a_{1997}$ is divided by $1000$.

Solution

Problem 10

$ABCDE$ is a cyclic pentagon with $BC = CD = DE$. The diagonals $AC$ and $BE$ intersect at $M$. $N$ is the foot of the altitude from $M$ to $AB$. We have $MA = 25$, $MD = 113$, and $MN = 15$. The area of triangle $ABE$ can be expressed as $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Determine the remainder obtained when $m+n$ is divided by $1000$.

Solution

Problem 11

$\alpha$, $\beta$, and $\gamma$ are the roots of $x(x-200)(4x+1) = 1$. Let \[\omega = \tan^{-1}(\alpha) + \tan^{-1}(\beta) + \tan^{-1} (\gamma).\] The value of $\tan(\omega)$ can be written as $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Determine the value of $m+n$.

Solution

Problem 12

$ABCD$ is a cyclic quadrilateral with $AB = 8$, $BC = 4$, $CD = 1$, and $DA = 7$. Let $O$ and $P$ denote the circumcenter and intersection of $AC$ and $BD$ respectively. The value of $OP^2$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine the remainder obtained when $m+n$ is divided by $1000$.

Solution

Problem 13

$P(x)$ is a polynomial of minimal degree that satisfies \[P(k) = \dfrac{1}{k(k+1)}\] for $k = 1, 2, 3, \ldots, 10$. The value of $P(11)$ can be written as $-\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $m+n$.

Solution

Problem 14

$3$ Elm trees, $4$ Dogwood trees, and $5$ Oak trees are to be planted in a line in front of a library such that \begin{align*} i&) \text{ No two Elm trees are next to each other.} \\ ii&) \text{ No Dogwood tree is adjacent to an Oak tree.} \\ iii&) \text{ All of the trees are planted.} \end{align*} How many ways can the trees be situated in this manner?

Solution

Problem 15

In triangle $ABC$, we have $BC = 13$, $CA = 37$, and $AB = 40$. Points $D$, $E$, and $F$ are selected on $BC$, $CA$, and $AB$ respectively such that $AD$, $BE$, and $CF$ concur at the circumcenter of $ABC$. The value of \[\dfrac{1}{AD} + \dfrac{1}{BE} + \dfrac{1}{CF}\] can be expressed as $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Determine $m+n$.

Solution

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