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

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== Problem 1 ==
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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 1 Pre 2005 Problems/Problem 1|Solution]]
  
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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.
+
== 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>.
  
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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.
+
[[Mock AIME 1 Pre 2005 Problems/Problem 2|Solution]]
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== 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>?
  
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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?
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[[Mock AIME 1 Pre 2005 Problems/Problem 3|Solution]]
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== Problem 4 ==
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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>.
  
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4. When 1 + 7 + 7^2 + \cdots + 7^{2004} is divided by 1000, a remainder of N is obtained. Determine the value of N.
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[[Mock AIME 1 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
 +
<cmath>\sqrt[3]{x} + \sqrt[3]{20 - x} = 2</cmath>
 +
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>.
  
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5. Let a and b be the two real values of x for which
+
[[Mock AIME 1 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.
  
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\displaymode{ <br /> \sqrt[3]{x} + \sqrt[3]{20 - x} = 2 <br /> }
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[[Mock AIME 1 Pre 2005 Problems/Problem 6|Solution]]
 +
== Problem 7 ==
 +
Let <math>N</math> denote the number of permutations of the <math>15</math>-character string <math>AAAABBBBBCCCCCC</math> such that
  
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The smaller of the two values can be expressed as p - \sqrt{q}, where p and q are integers. Compute p + q.
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# None of the first four letters is an <math>A</math>.
 +
# None of the next five letters is a <math>B</math>.
 +
# None of the last six letters is a <math>C</math>.
  
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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.
+
Find the remainder when <math>N</math> is divided by <math>1000</math>.
  
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7. Let N denote the number of permutations of the 15-character string AAAABBBBBCCCCCC such that
+
[[Mock AIME 1 Pre 2005 Problems/Problem 7|Solution]]
  
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(1) None of the first four letter is an A.
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== Problem 8 ==
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(2) None of the next five letters is a B.
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<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>.
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(3) None of the last six letters is a C.
 
  
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Find the remainder when N is divided by 1000.
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[[Mock AIME 1 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
 +
<cmath> \frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{360}{pqr} = 1</cmath>
 +
Compute <math>pqr</math>.
  
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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.
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[[Mock AIME 1 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>.
  
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9. p, q, and r are three non-zero integers such that p + q + r = 26 and
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[[Mock AIME 1 Pre 2005 Problems/Problem 10|Solution]]
 +
== Problem 11 ==
 +
Let <math>S</math> denote the value of the sum
 +
<cmath>\sum_{n=0}^{668} (-1)^{n} {2004 \choose 3n}</cmath>
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Determine the remainder obtained when <math>S</math> is divided by <math>1000</math>.
  
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\displaymode{ <br /> \frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{360}{pqr} = 1 <br /> }
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[[Mock AIME 1 Pre 2005 Problems/Problem 11|Solution]]
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== Problem 12 ==
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<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>.
  
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Compute pqr.
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[[Mock AIME 1 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
  
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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.
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<cmath>S = \sum_{i=0}^{\infty} \frac{R_i}{2^i}</cmath>
  
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11. Let S denote the value of the sum
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<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>.
  
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\displaymode{ <br /> \sum_{n=0}^{668} (-1)^{n} {2004 \choose 3n} <br /> }
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[[Mock AIME 1 Pre 2005 Problems/Problem 13|Solution]]
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== Problem 14 ==
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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 Mr. Porter's 12 new keys on his keychain (rotations are considered the same and any two keys of the same type are identical).
  
−
Determine the remainder obtained when S is divided by 1000.
+
Note: The problem is meant to be interpreted so that if you cannot produce one arrangement from another by rotation, then the two arrangements are different, even if you can produce one from the other from a combination of rotation and reflection.
  
−
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.
+
[[Mock AIME 1 Pre 2005 Problems/Problem 14|Solution]]
  
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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
+
== 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>.
  
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\displaymode{ <br /> S = \sum_{i=0}^{\infty} \frac{R_i}{2^i} <br /> }
+
[[Mock AIME 1 Pre 2005 Problems/Problem 15|Solution]]
  
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S can be expressed as \frac{m}{n} for two relatively prime positive integers m and n. Determine the value of m + n.
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== See also ==
−
 
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*[[Mock AIME 1 Pre 2005]]
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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 (Where rotations are considered the same and any two keys of the same type are identical.)
 
−
 
 
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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.
 

Latest revision as of 21:35, 9 January 2016

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

  1. None of the first four letters is an $A$.
  2. None of the next five letters is a $B$.
  3. 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 Mr. Porter's 12 new keys on his keychain (rotations are considered the same and any two keys of the same type are identical).

Note: The problem is meant to be interpreted so that if you cannot produce one arrangement from another by rotation, then the two arrangements are different, even if you can produce one from the other from a combination of rotation and reflection.

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

See also

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