Problems

Seven nines written out in a series: 9 9 9 9 9 9 9. Put some “\(+\)” or “\(-\)” between some of them, so that the resultant expression equals 1989.

The tower clock chimes three times in 12 seconds. How long will six chimes last?

Fred always tells the truth, but George always lies. What question could you ask them so that they would give the same answer?

Woodchucks are sawing a log. They made 10 cuts. How many pieces were made?

A bagel is cut into sectors. Ten cuts were made. How many pieces did this make?

1. A bagel is cut into sectors. Ten cuts were made. How many pieces did this make?

2. Woodchucks are sawing a log. They made 10 cuts. How many pieces were made? How can we explain why the answers in the previous two questions are different?

Five oaks are planted along two linear park alleys in such a way that there are three oaks along each alley, see picture. Where should we plant the sixth oak so that it will be possible to lay two more linear alleys, along each of which there would also be three oak trees growing?

The angle at the top of a crane is \(20^{\circ}\). How will the magnitude of this angle change when looking at the crane with binoculars which triple the size of everything?

Prove that in any group of 7 natural numbers – not necessarily consecutive – it is possible to choose three numbers such that their sum is divisible by 3.

10 friends sent one another greetings cards; each sent 5 cards. Prove that there will be two friends who sent cards to one another.