Many maths problems begin with the question “Is it possible…?”. In these kinds of problems, what you need to do depends on what you think is true.
If you believe it is possible, then you must give an example that really satisfies the conditions in the problem.
If you believe it is not possible, then you must explain clearly why it cannot be done.
When trying to build an example, it often helps to ask yourself extra questions to narrow things down: “How could it be possible?”, or “What properties must a correct example have?”.
On the other hand, if you have been trying to build an example for a while and nothing works, perhaps the answer is that it is impossible. In that case, look for a property that any example would need to have — and then show why that property cannot actually happen. Let’s see some examples!
Jane wrote a number on the whiteboard. Then, she looked at it and she noticed it lacks her favourite digit: 5. So she wrote 5 at the end of it. She then realized the new number is larger than the original one by exactly 1661. What is the number written on the board?
Replace letters with digits to maximize the expression: \[NO + MORE + MATH\] (same letters stand for identical digits and different letters stand for different digits.)
Which triangle has the largest area? The dots form a regular grid.
What is the ratio between the red and blue area? All shapes are semicircles.
In a parallelogram \(ABCD\), point \(E\) belongs to the side \(CD\) and point \(F\) belongs to the side \(BC\). Show that the total red area is the same as the total blue area:
The figure below is a regular pentagram. What is larger, the black area or the blue area?
A circle was inscribed in a square, and another square was inscribed in the circle. Which area is larger, the blue or the orange one?
In a square, the midpoints of its sides were marked and some segments were drawn. There is another square formed in the centre. Find its area, if the side of the square has length \(10\).
In a parallelogram \(ABCD\), point \(E\) belongs to the side \(AB\), point \(F\) belongs to the side \(CD\) and point \(G\) belongs to the side \(AD\). What is more, the marked red segments \(AE\) and \(CF\) have equal lengths. Prove that the total grey area is equal to the total black area.