Mock AIME 6 Pre 2005/Problems

$1$ . Find the remainder when $\displaystyle 11^{2005}$ is divided by $\displaystyle 1000$ .

$2$ . Suppose $\displaystyle \log_x{y}=3$ , $\displaystyle \log_y{z}=11$ , and $\displaystyle \log_z{w}=29$ . Given that $\displaystyle z=1024y$ and $\displaystyle wx=p^q$ , where $\displaystyle p$ is a prime, find $\displaystyle 3q$ .

$3$ . Suppose there are $\displaystyle 5$ red points and $\displaystyle 4$ blue points on a circle. Let $\displaystyle \frac{m}{n}$ be the probability that a convex polygon whose vertices are among the $\displaystyle 9$ points has at least one blue vertex, where $\displaystyle m$ and $\displaystyle n$ are relatively prime. Find $\displaystyle m+n$ .

$4$ . Let $\displaystyle O$ be the circumcircle of $\displaystyle \triangle ABC$ and let $\displaystyle D$ , $\displaystyle E$ , and $\displaystyle F$ be points on sides $\displaystyle BC$ , $\displaystyle AC$ , and $\displaystyle AB$ , respectively, such that $\displaystyle CD=8$ , $\displaystyle BD=6$ , $\displaystyle BF=3$ , $\displaystyle AF=2$ , and $\displaystyle AE=4$ , and $\displaystyle AD$ , $\displaystyle BE$ , and $\displaystyle CF$ are concurrent. Let $\displaystyle H$ be a point on minor arc $\displaystyle BC$ . Extend $\displaystyle AC$ to $\displaystyle G$ such that $\displaystyle GH$ is tangent to the circle at $\displaystyle H$ and $\displaystyle GC=12$ . Find $\displaystyle GH^2$ .

$5$ . Let $\displaystyle f(x)=\sqrt{x+2005\sqrt{x+2005\sqrt{\cdots}}}$ where $\displaystyle f(x)>0$ . Find the remainder when $\displaystyle f(0)f(2006)f(4014)f(6024)f(8036)$ is divided by $\displaystyle 1000$ .

$6$ . Let $\displaystyle z_0 = \cos{\frac{\pi}{2005}}+i\sin{\frac{\pi}{2005}}$ and $\displaystyle z_{n+1} = (z_n)^{n+1}$ for $\displaystyle n=0,1,2,\ldots$ . Find the smallest $\displaystyle n$ such that $\displaystyle z_n$ is an integer.

$7$ . Let $R$ be the region inside the graph $\displaystyle x^2+y^2-14x+4y-523=0$ and to the right of the line $\displaystyle x=-5$ . If the area of $\displaystyle R$ is in the form $\displaystyle a\pi+b\sqrt{c}$ where $\displaystyle c$ is squarefree, find $\displaystyle a+b+c$ .

$8$ . If $\displaystyle x\sqrt[3]{4}+\frac{\sqrt[3]{2}}{x}=3$ and $\displaystyle x$ is real, evaluate $\displaystyle 64x^9+\frac{8}{x^9}$ .

$9$ . You have $\displaystyle 5$ boxes and $\displaystyle 2005$ balls. $\displaystyle 286$ , $\displaystyle 645$ , and $\displaystyle 1074$ of these balls are blue, green, and red, respectively. Suppose the boxes are numbered $\displaystyle 1$ through $\displaystyle 5$ . You place $\displaystyle 1$ blue ball, $\displaystyle 3$ green balls, and $\displaystyle 3$ red balls in box $\displaystyle 1$ . Then $\displaystyle 2$ blue balls, $\displaystyle 5$ green balls, and $\displaystyle 7$ red balls in box $\displaystyle 2$ . Similarly, you put $\displaystyle n$ blue balls, $\displaystyle 2n+1$ green balls, and $\displaystyle 4n-1$ red balls in box $\displaystyle n$ for $\displaystyle n=3,4,5$ . Repeat the entire process (from boxes $\displaystyle 1$ to $\displaystyle 5$ ) until you run out of one color of balls. How many red balls are in boxes $\displaystyle 3$ , $\displaystyle 4$ , and $\displaystyle 5$ ? (NOTE: After placing the last ball of a certain color in a box, you still place the balls of the other colors in that box. You do not, however, place balls in the following box.)

$10$ . There are two ants on opposite corners of a cube. On each move, they can travel along an edge to an adjacent vertex. If the probability that they both return to their starting position after $\displaystyle 4$ moves is $\displaystyle \frac{m}{n}$ , where $\displaystyle m$ and $\displaystyle n$ are relatively prime, find $\displaystyle m+n$ . (NOTE: They do not stop if they collide.)

$11$ . Evaluate $\displaystyle \sum_{n=0}^{44}{\frac{\sin{(4n+1)}}{\cos{(2n)}\cos{(2n+1)}}} + \sum_{n=46}^{90}{\frac{\sin{(4n-1)}}{\cos{(2n-1)}\cos{(2n)}}}$ .

$12$ . A rigged coin has the property that when it is flipped $\displaystyle 2005$ times the probability of getting heads $\displaystyle 589$ times is equal to the probability of getting heads $\displaystyle 590$ times. If $\displaystyle \frac{m}{n}$ is the probability of getting a two heads in a row, where $\displaystyle m$ and $\displaystyle n$ are relatively prime, find $\displaystyle m+n$ .

$13$ . In $\triangle ABC$ , $\angle A=30^{\circ}$ , $AC=28\sqrt{3}$ , and $\displaystyle AB=42$ . Let $\displaystyle E$ be on side $\displaystyle AC$ and $\displaystyle D$ be on side $\displaystyle AB$ such that $\displaystyle DE$ is perpendicular to $\displaystyle AC$ and $\displaystyle DE=14$ . If $N = [\triangle ECB]-[\triangle EDB]$ , find $\lfloor N \rfloor$ . (NOTE: $[\triangle XYZ]$ denotes the area of $\triangle XYZ$ .)

$14$ . Let $\displaystyle f$ be a polynomial of degree $\displaystyle 2005$ with leading coefficient $\displaystyle 2006$ such that for $\displaystyle n=1,2,3,\ldots,2005$ , $\displaystyle f(n)=n$ . Find the number of $\displaystyle 0$ 's at the end of $\displaystyle f(2006)-2006$ .

$15$ . An AIME has $\displaystyle 15$ questions, $\displaystyle 5$ of each of three difficulties: easy, medium, and hard. Let $\displaystyle e(X)$ denote the number of easy questions up to question $\displaystyle X$ (including question $\displaystyle X$ ). Similarly define $\displaystyle m(X)$ and $\displaystyle h(X)$ . Let $\displaystyle N$ be the number of ways to arrange the questions in the AIME such that, for any $\displaystyle X$ , $\displaystyle e(X) \ge m(X) \ge h(X)$ and if a easy and hard problem are consecutive, the easy always comes first. Find the remainder when $\displaystyle N$ is divided by $\displaystyle 1000$ .

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Mock AIME 6 Pre 2005/Problems

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