Mock AIME 4 2005-2006/Problems

\begin{center} {\Large Mock Aime 2006} \qquad By: Alex Anderson (Altheman) \end{center}

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?

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.

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.

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?

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.

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.

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.

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.

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.

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.

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.

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.

13. \triangle VA_0A_1 is isosceles with base \overlineTemplate:A 1A 0. Construct A_2 on segment \overlineTemplate: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.

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.

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.