Mock AIME 1 2006-2007/Problems

1. $\triangle ABC$ has positive integer side lengths of $x$ , $y$ , and $17$ . The angle bisector of $\angle BAC$ hits $BC$ at $D$ . If $\angle C=90^\circ$ , and the maximum value of $\frac{[ABD]}{[ACD]}=\frac{m}{n}$ where $m$ and $n$ are relatively prime positive intgers, find $m+n$ . (Note that $[ABC]$ denotes the area of $\triangle ABC$ ).

2. Let $\star (x)$ be the sum of the digits of a positive integer $x$ . $\mathcal{S}$ is the set of positive integers such that for all elements $n$ in $\mathcal{S}$ , we have that $\star (n)=12$ and $0\le n< 10^{7}$ . If $m$ is the number of elements in $\mathcal{S}$ , compute $\star(m)$ .

3. Let $\triangle ABC$ have $BC=\sqrt{7}$ , $CA=1$ , and $AB=3$ . If $\angle A=\frac{\pi}{n}$ where $n$ is an integer, find the remainder when $n^{2007}$ is divided by $1000$ .

4. $\triangle ABC$ has all of it's verticies on the parabola $y=x^{2}$ . The slopes of $AB$ and $BC$ are $10$ and $-9$ , respectively. If the x-coordinate of the triangle's centroid is $1$ , find the area of $\triangle ABC$ .

5. Let $p$ be a prime and $f(n)$ satisfy $0\le f(n) <p$ for all integers $n$ . $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$ . If for fixed $n$ , there exists an integer $0\le y < p$ such that:

$ny-p\left\lfloor \frac{ny}{p}\right\rfloor=1$

then $f(n)=y$ . If there is no such $y$ , then $f(n)=0$ . If $p=11$ , find the sum: $f(1)+f(2)+...+f(p^{2}-1)+f(p^{2})$ .

6. Let $P_{1}: y=x^{2}+\frac{101}{100}$ and $P_{2}: x=y^{2}+\frac{45}{4}$ be two parabolas in the cartesian plane. Let $\mathcal{L}$ be the common tangent of $P_{1}$ and $P_{2}$ that has a rational slope. If $\mathcal{L}$ is written in the form $ax+by=c$ for positive integers $a,b,c$ where $\gcd(a,b,c)=1$ . Find $a+b+c$ .

7. Let $\triangle ABC$ have $AC=6$ and $BC=3$ . Point $E$ is such that $CE=1$ and $AE=5$ . Construct point $F$ on segment $BC$ such that $\angle AEB=\angle AFB$ . $EF$ and $AB$ are extended to meet at $D$ . If $\frac{[AEF]}{[CFD]}=\frac{m}{n}$ where $m$ and $n$ are positive integers, find $m+n$ (note: $[ABC]$ denotes the area of $\triangle ABC$ ).

8. Let $ABCDE$ be a convex pentagon with $AB\sqrt{2}=BC=CD=DE$ , $\angle ABC=150^\circ$ , $\angle BCD=75^\circ$ , and $\angle CDE=165^\circ$ . If $\angle ABE=\frac{m}{n}^\circ$ where $m$ and $n$ are relatively prime positive integers, find $m+n$ .

9. Let $a_{n}$ be a geometric sequence for $n\in\mathbb{Z}$ with $a_{0}=1024$ and $a_{10}=1$ . Let $S$ denote the infinite sum: $a_{10}+a_{11}+a_{12}+...$ . If the sum of all distinct values of $S$ is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, then compute the sum of the positive prime factors of $n$ .

10. In $\triangle ABC$ , $AB$ , $BC$ , and $CA$ have lengths $3$ , $4$ , and $5$ , respectively. Let the incircle, circle $I$ , of $\triangle ABC$ touch $AB$ , $BC$ , and $CA$ at $C'$ , $A'$ , and $B'$ , respectively. Construct three circles, $A''$ , $B''$ , and $C''$ , externally tangent to the other two and circles $A''$ , $B''$ , and $C''$ are internally tangent to the circle $I$ at $A'$ , $B'$ , and $C'$ , respectively. Let circles $A''$ , $B''$ , $C''$ , and $I$ have radii $a$ , $b$ , $c$ , and $r$ , respectively. If $\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}$ where $m$ and $n$ are positive integers, find $m+n$ .

11. Let $\mathcal{S}_{n}$ be the set of strings with only 0's or 1's with length $n$ such that any 3 adjacent place numbers sum to at least 1. For example, $00100$ works, but $10001$ does not. Find the number of elements in $\mathcal{S}_{11}$ .

12. Let $k$ be a positive integer with a first digit four such that after removing the first digit, you get another positive integer, $m$ , that satisfies $14m+1=k$ . Find the number of possible values of $m$ between $0$ and $10^{2007}$ .

13. Let $a_{n}$ , $b_{n}$ , and $c_{n}$ be geometric sequences with different common ratios and let $a_{n}+b_{n}+c_{n}=d_{n}$ for all integers $n$ . If $d_{1}=1$ , $d_{2}=2$ , $d_{3}=3$ , $d_{4}=-7$ , $d_{5}=13$ , and $d_{6}=-16$ , find $d_{7}$ .

14. Three points $A$ , $B$ , and $T$ are fixed such that $T$ lies on segment $AB$ , closer to point $A$ . Let $AT=m$ and $BT=n$ where $m$ and $n$ are positive integers. Construct circle $O$ with a variable radius that is tangent to $AB$ at $T$ . Let $P$ be the point such that circle $O$ is the incircle of $\triangle APB$ . Construct $M$ as the midpoint of $AB$ . Let $f(m,n)$ denote the maximum value $\tan^{2}\angle AMP$ for fixed $m$ and $n$ where $n>m$ . If $f(m,49)$ is an integer, find the sum of all possible values of $m$ .

15. Let $S$ be the set of integers $0,1,2,...,10^{11}-1$ . An element $x\in S$ (in) is chosen at random. Let $\star (x)$ denote the sum of the digits of $x$ . The probability that $\star (x)$ is divisible by 11 is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute the last 3 digits of $m+n$

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Mock AIME 1 2006-2007/Problems