What is the first order predicate calculus statement equivalent to the following?
Every teacher is liked by some student

What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs $$(a, b)$$ and $$(c, d)$$ in the chosen set such that $$a \equiv c$$ mod $$3$$ and $$b \equiv d$$ mode $$5$$

Consider the following system of equations in three real variables $$x1, x2$$ and $$x3$$ :
$$2x1 - x2 + 3x3 = 1$$
$$3x1 + 2x2 + 5x3 = 2$$
$$ - x1 + 4x2 + x3 = 3$$
This system of equations has

Let $$G\left( x \right) = 1/\left( {1 - x} \right)2 = \sum\limits_{i = 0}^\infty {g\left( i \right)\,{x^1}} \,\,\,,$$
where $$\left| x \right| < 1$$ What is $$g(i)$$?