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]] | ||
+ | |||
+ | == Problem 6 == | ||
+ | <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>. | ||
+ | |||
+ | [[Mock AIME 2 Pre 2005 Problems/Problem 6|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 <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.) | ||
+ | |||
+ | [[Mock AIME 2 Pre 2005 Problems/Problem 7|Solution]] | ||
+ | |||
+ | == Problem 8 == | ||
+ | Determine the remainder obtained when the expression <cmath>2004^{2003^{2002^{2001}}}</cmath> is divided by <math>1000</math>. | ||
+ | |||
+ | [[Mock AIME 2 Pre 2005 Problems/Problem 8|Solution]] | ||
+ | |||
+ | == Problem 9 == | ||
+ | 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>. | ||
+ | |||
+ | [[Mock AIME 2 Pre 2005 Problems/Problem 9|Solution]] | ||
+ | |||
+ | == Problem 10 == | ||
+ | <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>. | ||
+ | |||
+ | [[Mock AIME 2 Pre 2005 Problems/Problem 10|Solution]] | ||
+ | |||
+ | == Problem 11 == | ||
+ | <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>. | ||
+ | |||
+ | [[Mock AIME 2 Pre 2005 Problems/Problem 11|Solution]] | ||
+ | |||
+ | == Problem 12 == | ||
+ | <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>. | ||
+ | |||
+ | [[Mock AIME 2 Pre 2005 Problems/Problem 12|Solution]] | ||
+ | |||
+ | == Problem 13 == | ||
+ | <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>. | ||
+ | |||
+ | [[Mock AIME 2 Pre 2005 Problems/Problem 13|Solution]] | ||
+ | |||
+ | == Problem 14 == | ||
+ | <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? | ||
+ | |||
+ | [[Mock AIME 2 Pre 2005 Problems/Problem 14|Solution]] | ||
+ | |||
+ | == Problem 15 == | ||
+ | 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>. | ||
+ | |||
+ | [[Mock AIME 2 Pre 2005 Problems/Problem 15|Solution]] |
Latest revision as of 13:52, 9 June 2020
Contents
Problem 1
Compute the largest integer such that
divides
.
Problem 2
is a real number with the property that
. Let
. Determine the value of
.
Problem 3
In a box, there are green balls,
blue balls,
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
balls be removed. The probability that
balls are drawn can be expressed as
, where
and
are relatively prime positive integers. Compute
.
Problem 4
Let . Given that
has
digits, how many elements of
begin with the digit
?
Problem 5
Let be the set of integers
for which
, an infinite decimal that has the property that
for all positive integers
. Given that
is prime, how many positive integers are in
? (The
are digits.)
Problem 6
is a scalene triangle. Points
,
, and
are selected on sides
,
, and
respectively. The cevians
,
, and
concur at point
. If
,
, and
, determine the area of triangle
.
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 denote the number of ways
cards can be dealt to each player such that this is the case. Determine the remainder obtained when
is divided by
. (Bridge is a game played with the standard
-card deck.)
Problem 8
Determine the remainder obtained when the expression is divided by
.
Problem 9
Let where
and
. Determine the remainder obtained when
is divided by
.
Problem 10
is a cyclic pentagon with
. The diagonals
and
intersect at
.
is the foot of the altitude from
to
. We have
,
, and
. The area of triangle
can be expressed as
where
and
are relatively prime positive integers. Determine the remainder obtained when
is divided by
.
Problem 11
,
, and
are the roots of
. Let
The value of
can be written as
where
and
are relatively prime positive integers. Determine the value of
.
Problem 12
is a cyclic quadrilateral with
,
,
, and
. Let
and
denote the circumcenter and intersection of
and
respectively. The value of
can be expressed as
, where
and
are relatively prime positive integers. Determine the remainder obtained when
is divided by
.
Problem 13
is a polynomial of minimal degree that satisfies
for
. The value of
can be written as
, where
and
are relatively prime positive integers. Determine
.
Problem 14
Elm trees,
Dogwood trees, and
Oak trees are to be planted in a line in front of a library such that
How many ways can the trees be situated in this manner?
Problem 15
In triangle , we have
,
, and
. Points
,
, and
are selected on
,
, and
respectively such that
,
, and
concur at the circumcenter of
. The value of
can be expressed as
where
and
are relatively prime positive integers. Determine
.