Difference between revisions of "Mock AIME 1 Pre 2005 Problems"

Revision as of 12:20, 21 March 2008

Problem 1

Let $S$ denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when $S$ is divided by $1000$ .

Solution

Problem 2

If $x^2 + y^2 - 30x - 40y + 24^2 = 0$ , then the largest possible value of $\frac{y}{x}$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Determine $m + n$ .

Solution

Problem 3

$A, B, C, D,$ and $E$ are collinear in that order such that $AB = BC = 1, CD = 2,$ and $DE = 9$ . If $P$ can be any point in space, what is the smallest possible value of $AP^2 + BP^2 + CP^2 + DP^2 + EP^2$ ?

Solution

Problem 4

When $1 + 7 + 7^2 + \cdots + 7^{2004}$ is divided by $1000$ , a remainder of $N$ is obtained. Determine the value of $N$ .

Solution

Problem 5

Let $a$ and $b$ be the two real values of $x$ for which $\sqrt[3]{x} + \sqrt[3]{20 - x} = 2$ The smaller of the two values can be expressed as $p - \sqrt{q}$ , where $p$ and $q$ are integers. Compute $p + q$ .

Solution

Problem 6

A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner.

Solution

Problem 7

Let $N$ denote the number of permutations of the $15$ -character string AAAABBBBBCCCCCC such that

None of the first four letter is an A.
None of the next five letters is a B.
None of the last six letters is a C.

Find the remainder when $N$ is divided by $1000$ .

Solution

Problem 8

$ABCD$ , a rectangle with $AB = 12$ and $BC = 16$ , is the base of pyramid $P$ , which has a height of $24$ . A plane parallel to $ABCD$ is passed through $P$ , dividing $P$ into a frustum $F$ and a smaller pyramid $P'$ . Let $X$ denote the center of the circumsphere of $F$ , and let $T$ denote the apex of $P$ . If the volume of $P$ is eight times that of $P'$ , then the value of $XT$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute the value of $m + n$ .

Solution

Problem 9

$p, q,$ and $r$ are three non-zero integers such that $p + q + r = 26$ and $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{360}{pqr} = 1$ Compute $pqr$ .

Solution

Problem 10

$ABCDEFG$ is a regular heptagon inscribed in a unit circle centered at $O$ . $l$ is the line tangent to the circumcircle of $ABCDEFG$ at $A$ , and $P$ is a point on $l$ such that triangle $AOP$ is isosceles. Let $p$ denote the value of $AP \cdot BP \cdot CP \cdot DP \cdot EP \cdot FP \cdot GP$ . Determine the value of $p^2$ .

Solution

Problem 11

Let $S$ denote the value of the sum $\sum_{n=0}^{668} (-1)^{n} {2004 \choose 3n}$ Determine the remainder obtained when $S$ is divided by $1000$ .

Solution

Problem 12

$ABCD$ is a rectangular sheet of paper. $E$ and $F$ are points on $AB$ and $CD$ respectively such that $BE < CF$ . If $BCFE$ is folded over $EF$ , $C$ maps to $C'$ on $AD$ and $B$ maps to $B'$ such that $\angle{AB'C'} \cong \angle{B'EA}$ . If $AB' = 5$ and $BE = 23$ , then the area of $ABCD$ can be expressed as $a + b\sqrt{c}$ square units, where $a, b,$ and $c$ are integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$ .

Solution

Problem 13

A sequence $\{R_n\}_{n \ge 0}$ obeys the recurrence $7R_n = 64 - 2R_{n-1} + 9R_{n-2}$ for any integers $n \ge 2$ . Additionally, $R_0 = 10$ and $R_1 = -2$ . Let

$S = \sum_{i=0}^{\infty} \frac{R_i}{2^i}$

$S$ can be expressed as $\frac{m}{n}$ for two relatively prime positive integers $m$ and $n$ . Determine the value of $m + n$ .

Solution

Problem 14

Wally's Key Company makes and sells two types of keys. Mr. Porter buys a total of 12 keys from Wally's. Determine the number of possible arrangements of My. Porter's 12 new keys on his keychain (rotations are considered the same and any two keys of the same type are identical).

Solution

Problem 15

Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$ . If $2\cos{B} = \cos{A} + \cos{C}$ , then the area of triangle $ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$ , where $a, b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Compute $a+b+c$ .

Solution

@@ Line 1: / Line 1: @@
 == Problem 1 ==
 Let <math>S</math> denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when <math>S</math> is divided by <math>1000</math>.
+[[Mock AIME I Pre 2005 Problems/Problem 1|Solution]]
 == Problem 2 ==
 If <math>x^2 + y^2 - 30x - 40y + 24^2 = 0</math>, then the largest possible value of <math>\frac{y}{x}</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine <math>m + n</math>.
+[[Mock AIME I Pre 2005 Problems/Problem 2|Solution]]
 == Problem 3 ==
 <math>A, B, C, D,</math> and <math>E</math> are collinear in that order such that <math>AB = BC = 1, CD = 2,</math> and <math>DE = 9</math>. If <math>P</math> can be any point in space, what is the smallest possible value of <math>AP^2 + BP^2 + CP^2 + DP^2 + EP^2</math>?
+[[Mock AIME I Pre 2005 Problems/Problem 3|Solution]]
 == Problem 4 ==
 When <math>1 + 7 + 7^2 + \cdots + 7^{2004}</math> is divided by <math>1000</math>, a remainder of <math>N</math> is obtained. Determine the value of <math>N</math>.
+[[Mock AIME I Pre 2005 Problems/Problem 4|Solution]]
 == Problem 5 ==
 Let <math>a</math> and <math>b</math> be the two real values of <math>x</math> for which
@@ Line 13: / Line 21: @@
 The smaller of the two values can be expressed as <math>p - \sqrt{q}</math>, where <math>p</math> and <math>q</math> are integers. Compute <math>p + q</math>.
+[[Mock AIME I Pre 2005 Problems/Problem 5|Solution]]
 == Problem 6 ==
 A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner.
+[[Mock AIME I Pre 2005 Problems/Problem 6|Solution]]
 == Problem 7 ==
 Let <math>N</math> denote the number of permutations of the <math>15</math>-character string <tt>AAAABBBBBCCCCCC</tt> such that
@@ Line 25: / Line 35: @@
 Find the remainder when <math>N</math> is divided by <math>1000</math>.
+[[Mock AIME I Pre 2005 Problems/Problem 7|Solution]]
 == Problem 8 ==
 <math>ABCD</math>, a rectangle with <math>AB = 12</math> and <math>BC = 16</math>, is the base of pyramid <math>P</math>, which has a height of <math>24</math>. A plane parallel to <math>ABCD</math> is passed through <math>P</math>, dividing <math>P</math> into a frustum <math>F</math> and a smaller pyramid <math>P'</math>. Let <math>X</math> denote the center of the circumsphere of <math>F</math>, and let <math>T</math> denote the apex of <math>P</math>. If the volume of <math>P</math> is eight times that of <math>P'</math>, then the value of <math>XT</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute the value of <math>m + n</math>.
+[[Mock AIME I Pre 2005 Problems/Problem 8|Solution]]
 == Problem 9 ==
 <math>p, q,</math> and <math>r</math> are three non-zero integers such that <math>p + q + r = 26</math> and
@@ Line 33: / Line 45: @@
 Compute <math>pqr</math>.
+[[Mock AIME I Pre 2005 Problems/Problem 9|Solution]]
 == Problem 10 ==
 <math>ABCDEFG</math> is a regular heptagon inscribed in a unit circle centered at <math>O</math>. <math>l</math> is the line tangent to the circumcircle of <math>ABCDEFG</math> at <math>A</math>, and <math>P</math> is a point on <math>l</math> such that triangle <math>AOP</math> is isosceles. Let <math>p</math> denote the value of <math>AP \cdot BP \cdot CP \cdot DP \cdot EP \cdot FP \cdot GP</math>. Determine the value of <math>p^2</math>.
+[[Mock AIME I Pre 2005 Problems/Problem 10|Solution]]
 == Problem 11 ==
 Let <math>S</math> denote the value of the sum
@@ Line 41: / Line 55: @@
 Determine the remainder obtained when <math>S</math> is divided by <math>1000</math>.
+[[Mock AIME I Pre 2005 Problems/Problem 11|Solution]]
 == Problem 12 ==
 <math>ABCD</math> is a rectangular sheet of paper. <math>E</math> and <math>F</math> are points on <math>AB</math> and <math>CD</math> respectively such that <math>BE < CF</math>. If <math>BCFE</math> is folded over <math>EF</math>, <math>C</math> maps to <math>C'</math> on <math>AD</math> and <math>B</math> maps to <math>B'</math> such that <math>\angle{AB'C'} \cong \angle{B'EA}</math>. If <math>AB' = 5</math> and <math>BE = 23</math>, then the area of <math>ABCD</math> can be expressed as <math>a + b\sqrt{c}</math> square units, where <math>a, b,</math> and <math>c</math> are integers and <math>c</math> is not divisible by the square of any prime. Compute <math>a + b + c</math>.
+[[Mock AIME I Pre 2005 Problems/Problem 12|Solution]]
 == Problem 13 ==
 A sequence <math>\{R_n\}_{n \ge 0}</math> obeys the recurrence <math>7R_n = 64 - 2R_{n-1} + 9R_{n-2}</math> for any integers <math>n \ge 2</math>. Additionally, <math>R_0 = 10</math> and <math>R_1 = -2</math>. Let
@@ Line 51: / Line 67: @@
 <math>S</math> can be expressed as <math>\frac{m}{n}</math> for two relatively prime positive integers <math>m</math> and <math>n</math>. Determine the value of <math>m + n</math>.
+[[Mock AIME I Pre 2005 Problems/Problem 13|Solution]]
 == Problem 14 ==
 Wally's Key Company makes and sells two types of keys. Mr. Porter buys a total of 12 keys from Wally's. Determine the number of possible arrangements of My. Porter's 12 new keys on his keychain (rotations are considered the same and any two keys of the same type are identical).
+[[Mock AIME I Pre 2005 Problems/Problem 14|Solution]]
 == Problem 15 ==
 Triangle <math>ABC</math> has an inradius of <math>5</math> and a circumradius of <math>16</math>. If <math>2\cos{B} = \cos{A} + \cos{C}</math>, then the area of triangle <math>ABC</math> can be expressed as <math>\frac{a\sqrt{b}}{c}</math>, where <math>a, b,</math> and <math>c</math> are positive integers such that <math>a</math> and <math>c</math> are relatively prime and <math>b</math> is not divisible by the square of any prime. Compute <math>a+b+c</math>.
+[[Mock AIME I Pre 2005 Problems/Problem 15|Solution]]

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Difference between revisions of "Mock AIME 1 Pre 2005 Problems"

Revision as of 12:20, 21 March 2008

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15