# First Epsilons

This is our first collection of stage 5 epsilons. An epsilon problem is quick to read and understand. They might not be so easy to solve!

Epsilons will be good for filling spare mathematical moments. Why not carry a few around with you?

### Purr-fection

##### Age 16 to 18 Challenge Level:

What is the smallest perfect square that ends with the four digits 9009?

### Real(ly) Numbers

##### Age 16 to 18 Challenge Level:

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

### Cut Cube

##### Age 16 to 18 Challenge Level:

Find the shape and symmetries of the two pieces of this cut cube.

### Shades of Fermat's Last Theorem

##### Age 16 to 18 Challenge Level:

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

### Be Reasonable

##### Age 16 to 18 Challenge Level:

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

### Shape and Territory

##### Age 16 to 18 Challenge Level:

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?