Mock AIME 4 2005-2006/Problems

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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? 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

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