### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

# Unequal Statements

##### Stage: 4 Short Challenge Level:

I is true if and only if $-1 < x < 1$; II is true if $x> 1$ or if $x< -1$.
By considering the graph of $y = x - x^2$, which intersects the x - axis at (0,0) and (1,0) and has a maximum at ($\frac{1}{2}, \frac{1}{4}$), it may be seen that statement V is true is and only if $0 < x < 1$.
We see from the table below that a maximum of three statements may be true at any one time.
$x < -1$ then II true- 1
$x=-1$ then none true- 0
$-1< x< 0$ then I, III true- 2
$x=0$ then none true- 0
$0< x< 1$ then I, IV, V true- 3
$x=1$ then none true- 0
$x> 1$ then II true- 1

This problem is taken from the UKMT Mathematical Challenges.
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