Problems
Could the difference of two integers multiplied by their product be equal to the number 1999?
a) There are 21 coins on a table with the tails side facing upwards. In one operation, you are allowed to turn over any 20 coins. Is it possible to achieve the arrangement were all coins are facing with the heads side upwards in a few operations?
b) The same question, if there are 20 coins, but you are allowed to turn over 19.
Two friends went simultaneously from A to B. The first went by bicycle, the second – by car at a speed five times faster than the first. Halfway along the route, the car was in an accident, and the rest of the way the motorist walked on foot at a speed half of the speed of the cyclist. Which of them arrived at B first?
Andrew drives his car at a speed of 60 km/h. He wants to travel every kilometre 1 minute faster. By how much should he increase his speed?
A tourist walked 3.5 hours, and for every period of time, in one hour, he walked exactly 5 km. Does this mean that his average speed is 5 km/h?
Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.
One of the four angles formed when two straight lines intersect is \(41^{\circ}\). What are the other three angles equal to?
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!
Cut a square into two equal:
1. Triangles.
2. Pentagons
3. Hexagons.
