Reasoning, convincing and proving

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

Can you express every recurring decimal as a fraction?

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage stamps? Prove that all other values can be made up.

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?