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A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

### Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

# AP Train

##### Stage: 5 Short Challenge Level:

A particular list of $N$ consecutive integers starts with $1111$ as follows:
$$1111, 1112, 1113, \dots, 1111 + N-1$$
The entire list is shifted $D$ places along the number line and the first number then excluded, leaving a list of $N-1$ larger consecutive numbers as follows:
$$1112+D, 1113+D, \dots, 1111 + N -1+ D$$

In each list the sum of the integers is the same.

What are the possibilities for $N$ and $D$?

Extension: Maybe you wish to try to create a similar problem to this one?
Did you know ... ?

Progressions of integers occur remarkably frequently in mathematics in applications from quantum mechanics to number theory and they have many beautiful properties. EvenĀ Carl Gauss, possibly the greatest mathematician of all time, fondly recalled his first encounter with sums of consecutive natural numbers, when he noticed that the sum of the first $100$ whole numbersĀ  equalled $50$ lots of $101$.