Mock AIME 1 2007-2008 Problems

Problem 1

What is the coefficient of $x^3y^{13}$ in $\left(\frac 12x + y\right)^{17}$ ?

Problem 2

The expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $1:2:3$ that can be written in the form ${n\choose k} : {n\choose k+1} : {n \choose k+2}$ Find the sum of all possible values of $n+k$ .

Problem 3

A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent?

Problem 4

If $x$ is an odd number, then find the largest integer that always divides the expression $(10x+2)(10x+6)(5x+5)$

Problem 5

Let $S = (1+i)^{17} - (1-i)^{17}$ , where $i=\sqrt{-1}$ . Find $|S|$ .

Problem 6

[problem to be filled in]

Problem 7

Consider the following function $g(x)$ defined as $(x^{2^{2008}-1}-1)g(x) = (x+1)(x^2+1)(x^4+1)\cdots (x^{2^{2007}}+1) - 1$ Find $g(2)$ .

Problem 8

A sequence of ten $0$ s and/or $1$ s is randomly generated. If the probability that the sequence does not contain two consecutive $1$ s can be written in the form $\dfrac{m}{n}$ , where $m,n$ are relatively prime positive integers, find $m+n$ .

Problem 9

Let $n$ represent the smallest integer that satisfies the following conditions:

$\frac n2$ is a perfect square.

$\frac n3$ is a perfect cube.

$\frac n5$ is a perfect fifth.

How many divisors does $n$ have that are not multiples of 10?

Problem 10

An oreo shop sells $5$ different flavors of oreos and $3$ different flavors of milk. Alpha and Beta decide to purhcase some oreos. Since Alpha is picky, he will not order more than 1 of the same flavor. To be just as weird, Beta will only order oreos, but she will be willing to have repeats of flavors. How many ways could they have left the store with 3 products collectively? (A possible purchase is Alpha purchases 1 box of uh-oh oreos and 1 gallon of whole milk while Beta purchases 1 bag of strawberry milkshake oreos).

Problem 11

Problem 12

Let $d_1 = a^2 + 2^a + a \cdot 2^{(a+1)/2}$ and $d_2 = a^2 + 2^a - a \cdot 2^{(a+1)/2}$ . If $1 \le a \le 251$ , how many integral values of $a$ are there such that $d_1 \cdot d_2$ is a multiple of $5$ ?

Problem 13

Let $F(x)$ be a polynomial such that $F(6) = 15$ and $\frac{F(3x)}{F(x+3)} = 9-\frac{48x+54}{x^2+5x+6}$ for $x \in \mathbb{R}$ such that both sides are defined. Find $F(12)$ .

Problem 14

Points $A$ and $B$ lie on $\odot O$ , with radius $r$ , so that $\angle OAB$ is acute. Extend $AB$ to point $C$ so that $AB = BC$ . Let $D$ be the intersection of $\odot O$ and $OC$ such that $CD = \frac {1}{18}$ and $\cos(2\angle OAC) = \frac 38$ . If $r$ can be written as $\frac{a+b\sqrt{c}}{d}$ , where $a,b,$ and $d$ are relatively prime and $c$ is not divisible by the square of any prime, find $a+b+c+d$ .

Problem 15

The sum $\sum_{x=2}^{44} 2\sin{x}\sin{1}[1 + \sec (x-1) \sec (x+1)]$ can be written in the form $\sum_{n=1}^{4} (-1)^n \frac{\Phi(\theta_n)}{\Psi(\theta_n)}, where$ \Phi,\, \Psi$$ (Error compiling LaTeX. Unknown error_msg) are trigonometric functions and $\theta_1,\, \theta_2, \, \theta_3, \, \theta_4$ are degrees $\in [0,45]$ . Find $\theta_1 + \theta_2 + \theta_3 + \theta_4$ .

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