Mock AIME 1 2006-2007/Problems

Problem 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 $\triangle ABC$).

Solution

Problem 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 $\star(m)$.

Solution

Problem 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 $1000$.

Solution

Problem 4

$\triangle ABC$ has all of its vertices 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 $\triangle ABC$.

Solution

Problem 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: $f(1)+f(2)+...+f(p^{2}-1)+f(p^{2})$.


Solution

Problem 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 $a+b+c$.

Solution

Problem 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 $\triangle ABC$).

Solution

Problem 8

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

Solution

Problem 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 $n$.

Solution

Problem 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 $m+n$.

Solution

Problem 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 $\mathcal{S}_{11}$.

Solution

Problem 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 $10^{2007}$.

Solution

Problem 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 $d_{7}$.

Solution

Problem 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 $m$.

Solution

Problem 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 $m+n$

Solution

See Also

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