Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

m (tex-ify)
m (fmtting)
Line 1: Line 1:
 
== Problem 1 ==
 
== 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.
+
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>.
 +
 
 
== Problem 2 ==
 
== Problem 2 ==
−
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.
+
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>.
 
== Problem 3 ==
 
== Problem 3 ==
−
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?
+
<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>?
 
== Problem 4 ==
 
== Problem 4 ==
−
4. When 1 + 7 + 7^2 + \cdots + 7^{2004} is divided by 1000, a remainder of N is obtained. Determine the value of N.
+
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>.
 
== Problem 5 ==
 
== Problem 5 ==
−
5. Let a and b be the two real values of x for which
+
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>.
  
−
\displaymode{ <br /> \sqrt[3]{x} + \sqrt[3]{20 - x} = 2 <br /> }
+
== 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.
  
−
The smaller of the two values can be expressed as p - \sqrt{q}, where p and q are integers. Compute p + q.
 
−
== Problem 6 ==
 
−
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.
 
 
== Problem 7 ==
 
== Problem 7 ==
−
7. Let N denote the number of permutations of the 15-character string AAAABBBBBCCCCCC such that
+
Let <math>N</math> denote the number of permutations of the <math>15</math>-character string <tt>AAAABBBBBCCCCCC</tt> such that
 +
 
 +
# None of the first four letter is an <tt>A</tt>.
 +
# None of the next five letters is a <tt>B</tt>.
 +
# None of the last six letters is a <tt>C</tt>.
  
−
(1) None of the first four letter is an A.
+
Find the remainder when <math>N</math> is divided by <math>1000</math>.
−
(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.
 
 
== Problem 8 ==
 
== Problem 8 ==
−
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.
+
<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>.
 +
 
 
== Problem 9 ==
 
== Problem 9 ==
−
9. p, q, and r are three non-zero integers such that p + q + r = 26 and
+
<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>.
  
−
\displaymode{ <br /> \frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{360}{pqr} = 1 <br /> }
 
−
 
−
Compute <math>pqr</math>.
 
 
== Problem 10 ==
 
== Problem 10 ==
−
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>.
+
<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>.
  
 
== Problem 11 ==
 
== Problem 11 ==
−
11. Let <math>S</math> denote the value of the sum
+
Let <math>S</math> denote the value of the sum
 
<cmath>\sum_{n=0}^{668} (-1)^{n} {2004 \choose 3n}</cmath>
 
<cmath>\sum_{n=0}^{668} (-1)^{n} {2004 \choose 3n}</cmath>
 
Determine the remainder obtained when <math>S</math> is divided by <math>1000</math>.
 
Determine the remainder obtained when <math>S</math> is divided by <math>1000</math>.
  
 
== Problem 12 ==
 
== Problem 12 ==
−
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>.
+
<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>.
 +
 
 
== Problem 13 ==
 
== Problem 13 ==
−
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
+
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
  
 
<cmath>S = \sum_{i=0}^{\infty} \frac{R_i}{2^i}</cmath>
 
<cmath>S = \sum_{i=0}^{\infty} \frac{R_i}{2^i}</cmath>
  
 
<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>.
 
<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>.
 +
 
== Problem 14 ==
 
== Problem 14 ==
−
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.)
+
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).
 +
 
 
== Problem 15 ==
 
== Problem 15 ==
−
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>.
+
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>.

Revision as of 21:33, 20 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 $1000$.

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 $m + n$.

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 $AP^2 + BP^2 + CP^2 + DP^2 + EP^2$?

Problem 4

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

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 $p + q$.

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.

Problem 7

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

  1. None of the first four letter 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 $1000$.

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 $m + n$.

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 $pqr$.

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 $p^2$.

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 $1000$.

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 $a + b + c$.

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 $m + n$.

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).

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 $a+b+c$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Mock_AIME_1_Pre_2005_Problems&oldid=24179"