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Difference between revisions of "Mock AIME 4 2005-2006/Problems"

 
 
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\begin{center} {\Large Mock Aime 2006} \qquad By: Alex Anderson (Altheman) \end{center}
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http://www.mathlinks.ro/Forum/latexrender/pictures/2ea0ce047615395691113a82d6c190b3.gif
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== Problem 1 ==
 
1. A 5-digit number is leet if and only if the sum of the first 2 digits, the sum of the last 2 digits and the middle digit are equal. How many 5-digit leet numbers exist?
 
1. A 5-digit number is leet if and only if the sum of the first 2 digits, the sum of the last 2 digits and the middle digit are equal. How many 5-digit leet numbers exist?
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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== Problem 2 ==
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2. Qin Shi Huang wants to count the number of warriors he has to invade China. He puts his warriors into lines with the most people such that they have even length. The people left over are the remainder. He makes 2 lines, with a remainder of 1, 3 lines with a remainder of 2, 4 lines with a remainder of 3, 5 lines with a remainder of 4, and 6 lines with a remainder of 5. Find the minimum number of warriors that he has.
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2. Qin Shi Huang wants to count the number of warriors he has to invade China. He puts his warriors into lines with the most people such that they have even length. The people left over are the remainder. He makes 2 lines, with a remainder of 1, 3 lines with a remainder of 2, 4 lines with a remainder of 3, 5 lines with a remainder of 4, and 6 lines with a remainder of 5. Find the minimum number of warriors that he has.  
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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== Problem 3 ==
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3. T_1 is a regular tetrahedron. Tetrahedron T_2 is formed by connecting the centers of the faces of T_1. Generally, a new tetrahedron T_{n+1} is formed by connecting the centers of the faces of T_n. V_n is the volume of tetrahedron T_n. \frac{V_{2006}}{V_1}=\frac{m}{n} where m and n are coprime positive integers, find the remainder when m+n is divided by 1000.
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3. <math>T_1</math> is a regular tetrahedron. Tetrahedron <math>T_2</math> is formed by connecting the centers of the faces of <math>T_1</math>. Generally, a new tetrahedron <math>T_{n+1}</math> is formed by connecting the centers of the faces of <math>T_n</math>. <math>V_n</math> is the volume of tetrahedron <math>T_n</math>. <math>\frac{V_{2006}}{V_1}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are coprime positive integers, find the remainder when <math>m+n</math> is divided by <math>1000</math>.
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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== Problem 4 ==
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4. Let P(x)=\sum_{i=1}^{20}(-x)^{20-i}(x+i)^i. Let K be the product of the roots. How many digits are does \lfloor K \rfloor have where \lfloor x \rfloor denotes the greatest integer less than or equal to x?
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4. Let <math>P(x)=\sum_{i=1}^{20}(-x)^{20-i}(x+i)^i</math>. Let <math>K</math> be the product of the roots. How many digits are does <math>\lfloor K \rfloor</math> have where <math>\lfloor x \rfloor</math> denotes the greatest integer less than or equal to <math>x</math>?
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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== Problem 5 ==
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5. A parabola P: y=x^2 is rotated 135 degrees clockwise about the origin to P'. This image is translated upward \frac{8+\sqrt{2}}{2} to P''. Point A: (0,0), B: (256,0), and C is in Quadrant I, on P''. If the area of \triangle ABC is at a maximum, it is a\sqrt{b}+c where a, b and c are integers and b is square free, find a+b+c.
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5. A parabola <math>P: y=x^2</math> is rotated <math>135</math> degrees clockwise about the origin to <math>P'</math>. This image is translated upward <math>\frac{8+\sqrt{2}}{2}</math> to <math>P''</math>. Point <math>A: (0,0)</math>, <math>B: (256,0)</math>, and <math>C</math> is in Quadrant I, on <math>P''</math>. If the area of <math>\triangle ABC</math> is at a maximum, it is <math>a\sqrt{b}+c</math> where <math>a</math>, <math>b</math> and <math>c</math> are integers and <math>b</math> is square free, find <math>a+b+c</math>.
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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== Problem 6 ==
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6. Define a sequence a_0=2006 and a_{n+1}=(n+1)^{a_n} for all positive integers n. Find the remainder when a_{2007} is divided by 1000.
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6. Define a sequence <math>a_0=2006</math> and <math>a_{n+1}=(n+1)^{a_n}</math> for all positive integers <math>n</math>. Find the remainder when <math>a_{2007}</math> is divided by <math>1000</math>.
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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== Problem 7 ==
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7. <math>f(x)</math> is a function that satisfies <math>3f(x)=2x+1-f(\frac{1}{1-x})</math> for all defined <math>x</math>. Suppose that the sum of the zeros of <math>f(x)=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are coprime positive integers, find <math>m^2+n^2</math>.
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7. f(x) is a function that satisfies 3f(x)=2x+1-f(\frac{1}{1-x}) for all defined x. Suppose that the sum of the zeros of f(x)=\frac{m}{n} where m and n are coprime positive integers, find m^2+n^2.
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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== Problem 8 ==
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8. <math>R</math> is a solution to <math>x+\frac{1}{x}=\frac{ \sin210^{\circ} }{\sin285^{\circ} }</math>. Suppose that <math>\frac{1}{R^{2006}}+R^{2006}=A</math> find <math>\lfloor A^{10} \rfloor</math> where <math>\lfloor x \rfloor</math> is the greatest integer less than or equal to <math>x</math>.
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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8. R is a solution to x+\frac{1}{x}=\frac{ \sin210^{\circ} }{\sin285^{\circ} }. Suppose that \frac{1}{R^{2006}}+R^{2006}=A find \lfloor A^{10} \rfloor where \lfloor x \rfloor is the greatest integer less than or equal to x.
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== Problem 9 ==
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9. Zeus, Athena, and Posideon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am, and stay for 3 hours. All three hours does not need to fall within 12 pm to 12 am. If any of the 2 gods see each other during 12 pm to 12 am, it will be a good day. The probability of it being a good day is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are coprime positive integers, find <math>m+n</math>.
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9. Zeus, Athena, and Posideon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am, and stay for 3 hours. All three hours does not need to fall within 12 pm to 12 am. If any of the 2 gods see each other during 12 pm to 12 am, it will be a good day. The probability of it being a good day is \frac{m}{n} where m and n are coprime positive integers, find m+n.
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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== Problem 10 ==
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10. Define <math>S= \tan^2{1^{\circ}}+\tan^2{3^{\circ}}+\tan^2{5^{\circ}}+...+\tan^2{87^{\circ}}+\tan^2{89^{\circ}}</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>.  
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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10. Define S= \tan^2{1^{\circ}}+\tan^2{3^{\circ}}+\tan^2{5^{\circ}}+...+\tan^2{87^{\circ}}+\tan^2{89^{\circ}}. Find the remainder when S is divided by 1000.
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== Problem 11 ==
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11. <math>\triangle ABC</math> is isosceles with <math>\angle C= 90^{\circ}</math>. A point <math>P</math> lies inside the triangle such that <math>AP=33</math>, <math>CP=28\sqrt{2}</math>, and <math>BP=65</math>. Let <math>A</math> be the area of <math>\triangle ABC</math>. Find the remainder when <math>2A</math> is divided by <math>1000</math>.
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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11. \triangle ABC is isosceles with \angle C= 90^{\circ}. A point P lies inside the triangle such that AP=33, CP=28\sqrt{2}, and BP=65. Let A be the area of \triangle ABC. Find the remainder when 2A is divided by 1000.
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== Problem 12 ==
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12. There exists a line <math>L</math> with points <math>D</math>,<math>E</math>,<math>F</math> with <math>E</math> in between <math>D</math> and <math>F</math>. Point <math>A</math>, not on the line is such that <math>\overline{AF}=6</math>, <math>\overline{AD}=\frac{36}{7}</math>, <math>\overline{AE}=\frac{12}{\sqrt{7}}</math> with <math>\angle AEF > 90</math>. Construct <math>E'</math> on ray <math>AE</math> such that <math>(\overline{AE})(\overline{AE'})=36</math> and <math>\overline{FE'}=3</math>. Point <math>G</math> is on ray <math>AD</math> such that <math>\overline{AG}=7</math>. If <math>2*(\overline{E'G})=a+\sqrt{b}</math> where <math>a</math> and <math>b</math> are integers, then find <math>a+b</math>.
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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== Problem 13 ==
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12. There exists a line L with points D,E,F with E in between D and F. Point A, not on the line is such that \overline{AF}=6, \overline{AD}=\frac{36}{7}, \overline{AE}=\frac{12}{\sqrt{7}} with \angle AEF > 90. Construct E' on ray AE such that (\overline{AE})(\overline{AE'})=36 and \overline{FE'}=3. Point G is on ray AD such that \overline{AG}=7. If 2*(\overline{E'G})=a+\sqrt{b} where a and b are integers, then find a+b.
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13. <math>\triangle VA_0A_1</math> is isosceles with base <math>\overline{{A_1A_0}}</math>. Construct <math>A_2</math> on segment <math>\overline{{A_0V}}</math> such that <math>\overline{A_0A_1}=\overline{A_1A_2}=b</math>. Construct <math>A_3</math> on <math>\overline{A_1V}</math> such that <math>b=\overline{A_2A_3}</math>. Contiue this pattern: construct <math>\overline{A_{2n}A_{2n+1}}=b</math> with <math>A_{2n+1}</math> on segment <math>\overline{VA_1}</math> and <math>\overline{A_{2n+1}A_{2n+2}}=b</math> with <math>A_{2n+2}</math> on segment <math>\overline{VA_0}</math>. The points <math>A_n</math> do not coincide and <math>\angle VA_1A_0=90-\frac{1}{2006}</math>. Suppose <math>A_k</math> is the last point you can construct on the perimeter of the triangle. Find the remainder when <math>k</math> is divided by <math>1000</math>.  
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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== Problem 14 ==
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14. <math>P</math> is the probability that if you flip a fair coin, <math>20</math> heads will occur before <math>19</math> tails. If <math>P=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find the remainder when <math>m+n</math> is divided by <math>1000</math>.
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13. \triangle VA_0A_1 is isosceles with base \overline{{A_1A_0}}. Construct A_2 on segment \overline{{A_0V}} such that \overline{A_0A_1}=\overline{A_1A_2}=b. Construct A_3 on \overline{A_1V} such that b=\overline{A_2A_3}. Contiue this pattern: construct \overline{A_{2n}A_{2n+1}}=b with A_{2n+1} on segment \overline{VA_1} and \overline{A_{2n+1}A_{2n+2}}=b with A_{2n+2} on segment \overline{VA_0}. The points A_n do not coincide and \angle VA_1A_0=90-\frac{1}{2006}. Suppose A_k is the last point you can construct on the perimeter of the triangle. Find the remainder when k is divided by 1000.
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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== Problem 15 ==
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15. A regular 61-gon with verticies <math>A_1</math>, <math>A_2</math>, <math>A_3</math>,...<math>A_{61}</math> is inscribed in a circle with a radius of <math>r</math>. Suppose <math>(\overline{A_1A_2})(\overline{A_1A_3})(\overline{A_1A_4})...(\overline{A_1A_{61}})=r</math>. If <math>r^{2006}=\frac{p}{q}</math> where <math>p</math> and <math>q</math> are coprime positive integers, find the remainder when <math>p+q</math> is divided by <math>1000</math>.
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14. P is the probability that if you flip a fair coin, 20 heads will occur before 19 tails. If P=\frac{m}{n} where m and n are relatively prime positive integers, find the remainder when m+n is divided by 1000.
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[[Mock AIME 5 2005-2006/Problem 14|Solution]]
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== See also ==
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* [[Mock AIME 4 2005-2006]]
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15. A regular 61-gon with verticies A_1, A_2, A_3,...A_{61} is inscribed in a circle with a radius of r. Suppose (\overline{A_1A_2})(\overline{A_1A_3})(\overline{A_1A_4})...(\overline{A_1A_{61}})=r. If r^{2006}=\frac{p}{q} where p and q are coprime positive integers, find the remainder when p+q is divided by 1000.
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* [[Mock AIME]]
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* [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=70988 Here is the page on AoPS]

Latest revision as of 04:44, 26 February 2007

http://www.mathlinks.ro/Forum/latexrender/pictures/2ea0ce047615395691113a82d6c190b3.gif

Problem 1

1. A 5-digit number is leet if and only if the sum of the first 2 digits, the sum of the last 2 digits and the middle digit are equal. How many 5-digit leet numbers exist? Solution

Problem 2

2. Qin Shi Huang wants to count the number of warriors he has to invade China. He puts his warriors into lines with the most people such that they have even length. The people left over are the remainder. He makes 2 lines, with a remainder of 1, 3 lines with a remainder of 2, 4 lines with a remainder of 3, 5 lines with a remainder of 4, and 6 lines with a remainder of 5. Find the minimum number of warriors that he has. Solution

Problem 3

3. $T_1$ is a regular tetrahedron. Tetrahedron $T_2$ is formed by connecting the centers of the faces of $T_1$. Generally, a new tetrahedron $T_{n+1}$ is formed by connecting the centers of the faces of $T_n$. $V_n$ is the volume of tetrahedron $T_n$. $\frac{V_{2006}}{V_1}=\frac{m}{n}$ where $m$ and $n$ are coprime positive integers, find the remainder when $m+n$ is divided by $1000$. Solution

Problem 4

4. Let $P(x)=\sum_{i=1}^{20}(-x)^{20-i}(x+i)^i$. Let $K$ be the product of the roots. How many digits are does $\lfloor K \rfloor$ have where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$? Solution

Problem 5

5. A parabola $P: y=x^2$ is rotated $135$ degrees clockwise about the origin to $P'$. This image is translated upward $\frac{8+\sqrt{2}}{2}$ to $P''$. Point $A: (0,0)$, $B: (256,0)$, and $C$ is in Quadrant I, on $P''$. If the area of $\triangle ABC$ is at a maximum, it is $a\sqrt{b}+c$ where $a$, $b$ and $c$ are integers and $b$ is square free, find $a+b+c$. Solution

Problem 6

6. Define a sequence $a_0=2006$ and $a_{n+1}=(n+1)^{a_n}$ for all positive integers $n$. Find the remainder when $a_{2007}$ is divided by $1000$. Solution

Problem 7

7. $f(x)$ is a function that satisfies $3f(x)=2x+1-f(\frac{1}{1-x})$ for all defined $x$. Suppose that the sum of the zeros of $f(x)=\frac{m}{n}$ where $m$ and $n$ are coprime positive integers, find $m^2+n^2$. Solution

Problem 8

8. $R$ is a solution to $x+\frac{1}{x}=\frac{ \sin210^{\circ} }{\sin285^{\circ} }$. Suppose that $\frac{1}{R^{2006}}+R^{2006}=A$ find $\lfloor A^{10} \rfloor$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. Solution

Problem 9

9. Zeus, Athena, and Posideon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am, and stay for 3 hours. All three hours does not need to fall within 12 pm to 12 am. If any of the 2 gods see each other during 12 pm to 12 am, it will be a good day. The probability of it being a good day is $\frac{m}{n}$ where $m$ and $n$ are coprime positive integers, find $m+n$. Solution

Problem 10

10. Define $S= \tan^2{1^{\circ}}+\tan^2{3^{\circ}}+\tan^2{5^{\circ}}+...+\tan^2{87^{\circ}}+\tan^2{89^{\circ}}$. Find the remainder when $S$ is divided by $1000$. Solution

Problem 11

11. $\triangle ABC$ is isosceles with $\angle C= 90^{\circ}$. A point $P$ lies inside the triangle such that $AP=33$, $CP=28\sqrt{2}$, and $BP=65$. Let $A$ be the area of $\triangle ABC$. Find the remainder when $2A$ is divided by $1000$. Solution

Problem 12

12. There exists a line $L$ with points $D$,$E$,$F$ with $E$ in between $D$ and $F$. Point $A$, not on the line is such that $\overline{AF}=6$, $\overline{AD}=\frac{36}{7}$, $\overline{AE}=\frac{12}{\sqrt{7}}$ with $\angle AEF > 90$. Construct $E'$ on ray $AE$ such that $(\overline{AE})(\overline{AE'})=36$ and $\overline{FE'}=3$. Point $G$ is on ray $AD$ such that $\overline{AG}=7$. If $2*(\overline{E'G})=a+\sqrt{b}$ where $a$ and $b$ are integers, then find $a+b$. Solution

Problem 13

13. $\triangle VA_0A_1$ is isosceles with base $\overline{{A_1A_0}}$. Construct $A_2$ on segment $\overline{{A_0V}}$ such that $\overline{A_0A_1}=\overline{A_1A_2}=b$. Construct $A_3$ on $\overline{A_1V}$ such that $b=\overline{A_2A_3}$. Contiue this pattern: construct $\overline{A_{2n}A_{2n+1}}=b$ with $A_{2n+1}$ on segment $\overline{VA_1}$ and $\overline{A_{2n+1}A_{2n+2}}=b$ with $A_{2n+2}$ on segment $\overline{VA_0}$. The points $A_n$ do not coincide and $\angle VA_1A_0=90-\frac{1}{2006}$. Suppose $A_k$ is the last point you can construct on the perimeter of the triangle. Find the remainder when $k$ is divided by $1000$. Solution

Problem 14

14. $P$ is the probability that if you flip a fair coin, $20$ heads will occur before $19$ tails. If $P=\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find the remainder when $m+n$ is divided by $1000$. Solution

Problem 15

15. A regular 61-gon with verticies $A_1$, $A_2$, $A_3$,...$A_{61}$ is inscribed in a circle with a radius of $r$. Suppose $(\overline{A_1A_2})(\overline{A_1A_3})(\overline{A_1A_4})...(\overline{A_1A_{61}})=r$. If $r^{2006}=\frac{p}{q}$ where $p$ and $q$ are coprime positive integers, find the remainder when $p+q$ is divided by $1000$. Solution

See also

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