Difference between revisions of "Mock AIME 1 Pre 2005 Problems"
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== 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 == | ||
− | + | 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 == | ||
− | + | <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 == | ||
− | + | 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 == | ||
− | + | 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>. | ||
− | + | == 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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− | |||
== Problem 7 == | == Problem 7 == | ||
− | + | 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>. | ||
− | + | Find the remainder when <math>N</math> is divided by <math>1000</math>. | |
− | |||
− | |||
− | |||
== Problem 8 == | == 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>. | |
+ | |||
== Problem 9 == | == 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>. | ||
− | |||
− | |||
− | |||
== Problem 10 == | == 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>. | |
== Problem 11 == | == Problem 11 == | ||
− | + | 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 == | ||
− | + | <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 == | ||
− | + | 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 == | ||
− | + | 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 == | ||
− | + | 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 20:33, 20 March 2008
Contents
[hide]Problem 1
Let denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when is divided by .
Problem 2
If , then the largest possible value of can be written as , where and are relatively prime positive integers. Determine .
Problem 3
and are collinear in that order such that and . If can be any point in space, what is the smallest possible value of ?
Problem 4
When is divided by , a remainder of is obtained. Determine the value of .
Problem 5
Let and be the two real values of for which The smaller of the two values can be expressed as , where and are integers. Compute .
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 denote the number of permutations of the -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 is divided by .
Problem 8
, a rectangle with and , is the base of pyramid , which has a height of . A plane parallel to is passed through , dividing into a frustum and a smaller pyramid . Let denote the center of the circumsphere of , and let denote the apex of . If the volume of is eight times that of , then the value of can be expressed as , where and are relatively prime positive integers. Compute the value of .
Problem 9
and are three non-zero integers such that and Compute .
Problem 10
is a regular heptagon inscribed in a unit circle centered at . is the line tangent to the circumcircle of at , and is a point on such that triangle is isosceles. Let denote the value of . Determine the value of .
Problem 11
Let denote the value of the sum Determine the remainder obtained when is divided by .
Problem 12
is a rectangular sheet of paper. and are points on and respectively such that . If is folded over , maps to on and maps to such that . If and , then the area of can be expressed as square units, where and are integers and is not divisible by the square of any prime. Compute .
Problem 13
A sequence obeys the recurrence for any integers . Additionally, and . Let
can be expressed as for two relatively prime positive integers and . Determine the value of .
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 has an inradius of and a circumradius of . If , then the area of triangle can be expressed as , where and are positive integers such that and are relatively prime and is not divisible by the square of any prime. Compute .