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

Revision as of 21:29, 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

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

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

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

5. Let a and b be the two real values of x for which

\displaymode{
\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

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

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

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

9. p, q, and r are three non-zero integers such that p + q + r = 26 and

\displaymode{
\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{360}{pqr} = 1
}

Compute $pqr$ .

Problem 10

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

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

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

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

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.)

Problem 15

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 $a+b+c$ .

@@ Line 1: / Line 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 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
@@ Line 14: / Line 13: @@
 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
@@ Line 24: / Line 23: @@
 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
 \displaymode{ <br /> \frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{360}{pqr} = 1 <br /> }
-Compute pqr.
+Compute <math>pqr</math>.
+== 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.
+. <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>.
-. Let S denote the value of the sum
-\displaymode{ <br /> \sum_{n=0}^{668} (-1)^{n} {2004 \choose 3n} <br /> }
-Determine the remainder obtained when S is divided by 1000.
-. 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.
-. 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 11 ==
+. Let <math>S</math> denote the value of the sum
+<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>.
-\displaymode{ <br /> S = \sum_{i=0}^{\infty} \frac{R_i}{2^i} <br /> }
+== 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>.
+== 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
-S can be expressed as \frac{m}{n} for two relatively prime positive integers m and n. Determine the value of m + n.
+<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>.
+== 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 (Where 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.
+. 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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