problem

Favourite

### Tens

When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?

problem
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Tens

Favourite

When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?

problem
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Converging Product

Favourite

In the limit you get the sum of an infinite geometric series. What
about an infinite product (1+x)(1+x^2)(1+x^4)... ?

problem
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Gosh Cosh

Explore the hyperbolic functions sinh and cosh using what you know about the exponential function.

problem
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Particularly general

By proving these particular identities, prove the existence of general cases.

problem
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Binary Squares

If a number N is expressed in binary by using only 'ones,' what can
you say about its square (in binary)?

problem
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Golden Powers

You add 1 to the golden ratio to get its square. How do you find higher powers?

problem
###
One Basket or Group Photo

Libby Jared helped to set up NRICH and this is one of her favourite
problems. It's a problem suitable for a wide age range and best
tackled practically.

problem
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Walkabout

A walk is made up of diagonal steps from left to right, starting at
the origin and ending on the x-axis. How many paths are there for 4
steps, for 6 steps, for 8 steps?

problem
###
Counting Binary Ops

How many ways can the terms in an ordered list be combined by
repeating a single binary operation. Show that for 4 terms there
are 5 cases and find the number of cases for 5 terms and 6 terms.

problem
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Dirisibly Yours

Find and explain a short and neat proof that 5^(2n+1) + 11^(2n+1) +
17^(2n+1) is divisible by 33 for every non negative integer n.