2006 SMT/Algebra Problems

Problem 1

A finite sequence of positive integers $m_i$ for $i=1, 2, \cdots, 2006$ are defined so that $m_1=1$ and $m_i=10m_{i-1}+1$ for $i>1$ . How many of these integers are divisible by $37$ ?

Solution

Problem 2

Find the minimum value of $2x^2+2y^2+5z^2-2xy-4yz-4x-2z+15$ for real numbers $x, y, z$ .

Solution

Problem 3

A Gaussian prime is a Gaussian integer $z=a+bi$ (where $a$ and $b$ are integers) with no Guassian integer factors of smaller absolute value. Factor $-4+7i$ into Gaussian primes with positive real parts. $i$ is a symbol with the property that $i^2=-1$ .

Solution

Problem 4

Simplify: $\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)}$

Solution

Problem 5

Jerry is bored one day, so he makes an array of Cocoa pebbles. He makes $8$ equal rows with the pebbles remaining in a box. When Kramer drops by and eats one, Jerry yells at him until Kramer realizes he can make $9$ equal rows with the remaining pebbles. After Kramer eats another, he finds he can make $10$ equal rows with the remaining pebbles. Find the smallest number of pebbles that were in the box in the beginning.

Solution

Problem 6

Let $a, b, c$ be real numbers satisfying:

$ab-a=b+119$ $bc-b=c+59$ $ca-c=a+71$

Determine all possible values of $a+b+c$ .

Solution

Problem 7

Find all solutions to $aabb=n^4-6n^3$ , where $a$ and $b$ are nonzero digits, and $n$ is an integer. ( $a$ and $b$ are not necessarily distinct.)

Solution

Problem 8

Evaluate:

$\sum_{x=2}^{10}\frac{2}{x(x^2-1)}.$

Solution

Problem 9

Principal Skinner is thinking of two integers $m$ and $n$ and bets Superintendent Chalmers that he will not be able to determine these integers with a single piece of information. Chalmers asks Skinner the numerical value of $mn+13m+13n-m^2-n^2$ . From the value of this expression alone, he miraculously determines both $m$ and $n$ . What is the value of the above expression?