Mock AIME 4 Pre 2005/Problems

Problem 1

For how many integers $n>1$ is it possible to express $2005$ as the sum of $n$ distinct positive integers?

Problem 2

$a_1,a_2,\ldots$ is a sequence of real numbers where $a_n$ is the arithmetic mean of the previous $n-1$ terms for $n > 3$ and $a_{2004} = 7.$ $b_1, b_2, \ldots$ is a sequence of real numbers in which $b_n$ is the geometric mean of the previous $n - 1$ terms for $n > 3$ and $b_{2005} = 6.$ If $a_i = b_i$ for $i = 1, 2, 3$ and $a_1 = 3,$ then compute the value of $a_2^2+a_3^2$ .

Solution

Problem 3

Compute the largest integer $n$ such that $2005^{2^{100}}-2003^{2^{100}}$ is divisible by $2^n$ .

Solution

Problem 4

$ABCDEFG$ is a regular heptagon, and $P$ is a point in its interior such that $ABP$ is equilateral. There exists a unique pair $\{m, n\}$ of relatively prime positive integers such that $m\angle CPE=\left(\tfrac mn\right)^\circ$ . Compute the value of $m + n.$

Solution

Problem 5

Compute, to the nearest integer, the area of the region enclosed by the graph of $13x^2-20xy+52y^2-10x+52y=563.$

Solution

Problem 6

Determine the remainder when $1000!$ is divided by $2003$ .

Solution

Problem 7

$\mathcal P$ is a pyramid consisting of a square base and four slanted triangular faces such that all of its edges are equal in length. $\mathcal C$ is a cube of edge length $6$ . Six pyramids similar to $\mathcal P$ are constructed by taking points $P_i$ (all outside of $\mathcal C$ ) where $i = 1, 2, \ldots , 6$ and using the nearest face of $\mathcal C$ as the base of each pyramid exactly once. The volume of the octahedron formed by the $P_i$ (taking the convex hull) can be expressed as $m+n\sqrt p$ for some positive integers $m$ , $n$ , and $p$ , where $p$ is not divisible by the square of any prime. Determine the value of $m + n + p$ .

Solution

Problem 8

A single atom of Uranium-238 rests at the origin. Each second, the particle has a $1/4$ chance of moving one unit in the negative $x$ direction and a $1/2$ chance of moving in the positive $x$ direction. If the particle reaches $(-3, 0)$ , it ignites a fission that will consume the earth. If it reaches $(7, 0)$ , it is harmlessly diffused. The probability that, eventually, the particle is safely contained can be expressed as $\tfrac mn$ for some relatively prime positive integers $m$ and $n$ . Determine the remainder obtained when $m + n$ is divided by $1000$ .

Solution

Problem 9

The value of the sum $\sum_{n=1}^\infty\frac{(7n+32)\cdot3^n}{n\cdot(n+2)\cdot4^n}$ can be expressed in the form $\tfrac pq$ , for some relatively prime positive integers $p$ and $q$ . Compute the value of $p+q$ .

Solution

Problem 10

$100$ blocks are selected from a crate containing $33$ blocks of each of the following dimensions: $13 \times 17 \times 21, 13 \times 17 \times 37, 13 \times 21 \times 37,$ and $17 \times 21 \times 37.$ The chosen blocks are stacked on top of each other (one per cross section) forming a tower of height $h$ . Compute the number of possible values of $h$ .

Solution

Problem 11

$10$ lines and $10$ circles divide the plane into at most $n$ disjoint regions. Compute $n$ .

Solution

Problem 12

Determine the number of permutations of $1, 2, 3, 4, \ldots , 32$ such that if $m$ divides $n$ , the $m$ th number divides the $n$ th number.

Solution

Problem 13

$x$ , $y$ , and $z$ are distinct non-zero integers such that $-7 \le x, y, z \le 7.$ Compute the number of solutions $(x, y, z)$ to the equation $\frac 1x+\frac 1y+\frac 1z=\frac 1{x+y+z}.$

Solution

Problem 14

In triangle $ABC,$ $BC = 27, CA = 32,$ and $AB = 35.$ $P$ is the unique point such that the perimeters of triangles $BP C, CP A,$ and $AP B$ are equal. The value of $AP + BP + CP$ can be expressed as $\tfrac{p+q\sqrt r}{s},$ where $p, q, r,$ and $s$ are positive integers such that there is no prime divisor common to $p, q,$ and $s,$ and $r$ is not divisible by the square of any prime. Determine the value of $p + q + r + s.$

Solution

Problem 15

$ABCD$ is a convex quadrilateral in which $\overline{AB}\ ||\ \overline{CD}.$ Let $U$ denote the intersection of the extensions of $\overline{AD}$ and $\overline{BC}.$ $\omega_1$ is the circle tangent to line segment $\overline{BC}$ which also passes through $A$ and $D$ , and $\omega_2$ is the circle tangent to $\overline{AD}$ which passes through $B$ and $C.$ Call the points of tangency $M$ and $S.$ Let $O$ and $P$ be the points of intersection between $\omega_1$ and $\omega_2.$ Finally, $\overline{MS}$ intersects $\overline{OP}$ at $V$ . If $AB = 2,$ $BC = 2005,$ $CD = 4,$ and $DA = 2004,$ then the value of $UV^2$ is some integer $n.$ Determine the remainder obtained when $n$ is divided by $1000.$