Cool problem of the day! (4 Viewers)

[deswa1]

Interesting one for a 4U student that I saw in one of our school's past papers:

<a href="http://www.codecogs.com/eqnedit.php?latex=\textup{Given }|z-2|=2\textup{ and }0<\textup{argz}<\frac{\pi}{2}, \textup{ find the value of k (real) for which:} \\ \textup{arg}(z-2)=\textup{karg}(z^2-2z)" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\textup{Given }|z-2|=2\textup{ and }0<\textup{argz}<\frac{\pi}{2}, \textup{ find the value of k (real) for which:} \\ \textup{arg}(z-2)=\textup{karg}(z^2-2z)" title="\textup{Given }|z-2|=2\textup{ and }0<\textup{argz}<\frac{\pi}{2}, \textup{ find the value of k (real) for which:} \\ \textup{arg}(z-2)=\textup{karg}(z^2-2z)" /></a>

 

[qwerty44]

Is it sqrt(3)?

That's what i got, but I don't have answer lol

 

[bleakarcher]

Interesting one for a 4U student that I saw in one of our school's past papers:

<a href="http://www.codecogs.com/eqnedit.php?latex=\textup{Given }|z-2|=2\textup{ and }0<\textup{argz}<\frac{\pi}{2}, \textup{ find the value of k (real) for which:} \\ \textup{arg}(z-2)=\textup{karg}(z^2-2z)" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\textup{Given }|z-2|=2\textup{ and }0<\textup{argz}<\frac{\pi}{2}, \textup{ find the value of k (real) for which:} \\ \textup{arg}(z-2)=\textup{karg}(z^2-2z)" title="\textup{Given }|z-2|=2\textup{ and }0<\textup{argz}<\frac{\pi}{2}, \textup{ find the value of k (real) for which:} \\ \textup{arg}(z-2)=\textup{karg}(z^2-2z)" /></a>

I remember doing this a while ago now.

k=2/3

 

[qwerty44]

Here is another good one:

The lengths of the sides of the octagon are 1, 2, 3, 4, 5, 6, 7 and 8 units in some
order. Find the maximum area of the hexagon (square units).

 

[qwerty44]

Is it sqrt(3)?

Yeh sqrt(3) confirmed.

 

[AlbertEinstein]

Post more questions please I wanna try solving them.

 

[qwerty44]

Post more questions please I wanna try solving them.

Here is another good one:

The lengths of the sides of the octagon are 1, 2, 3, 4, 5, 6, 7 and 8 units in some
order. Find the maximum area of the hexagon (square units).

.

 

[largarithmic]

Here is another good one:

The lengths of the sides of the octagon are 1, 2, 3, 4, 5, 6, 7 and 8 units in some
order. Find the maximum area of the hexagon (square units).

This reminds me of a really neat (but actually pretty hard) problem:

Given a polygon, we define a "flip-stick" to be the following process: take a 'cut' of the polygon (i.e. a line segment that cuts the polygon into exactly two new polygons), then take one of the polygons you produced, reflect it and stick it back on: as long as the flipped thing doesnt overlap with the rest of the polygon, in which case the move is illegal. So essentially you cut of a bit, flip it and stick it back on if its allowed. Does there exist a sequence of "flip-sticks" that can transform a square into an equilateral triangle?

 

[qwerty44]

Here is another good one:

The lengths of the sides of the octagon are 1, 2, 3, 4, 5, 6, 7 and 8 units in some
order. Find the maximum area of the hexagon (square units).

I got 30 units^2.

Anyone else care to try?

 

[qwerty44]

The least common multiple of positive integers a, b, c and d is equal to a + b + c + d.
Prove that abcd is divisible by at least one of 3 and 5.