Q1401 Solve the recurrence relation

$$a(n)=6a(n-1)-9a(n-2)\ ,\quad a(0)=2\ ,\quad a(1)=21\ .$$

Comment. You can use the same method as in previous problems -- see, for example, the solution to problem~1393 -- but at one point you will find that things are a little different.

 

Q1402 Suppose that the three lines

$$y=ax+b\quad\hbox{and}\quad y=cx+d\quad\hbox{and}\quad y=ex+f$$

all have different gradients.  Find conditions on $a,b,c,d,e,f$ for the lines to intersect in a single point.

 

Q1403 Seven different real numbers are given.  Prove that there are two of them, say $x$ and $y$, for which

$$\frac{1+xy}{x-y}$$

is greater than $\sqrt3\,$.

 

Q1404 In the game of poker, a pack of cards (consisting of the usual $52$ cards) is shuffled and five cards are dealt to each player.  A hand is referred to as ``four of a kind'' if it contains four cards of the same value and one other card.  For example, $\spadesuit7,\,\heartsuit7,\,\diamondsuit7,\,\clubsuit7,\,\diamondsuit{\rm J}$ constitutes four of a kind.  Suppose that I deal two five--card hands from the same pack, one to myself and one to my opponent.