Problems

Two semicircles and one circle were drawn on the sides of a right triangle. The circle whose centre is in the midpoint of the hypothenuse actually goes through the right angle corner – this is a general fact, but you don’t need to prove it here. If the two shorter sides of the triangle are $3$ and $4$, what is the total area of the red region?

The lengths of three sides of a right triangle are all integer numbers.

a) Show that one of them is divisible by $3$.

b*) Show that one of them is divisible by $5$.

A segment $AB$ is a base of an isosceles triangle $ABC$. A line perpendicular to the segment $AC$ was drawn through point $A$ – this line crosses an extension of the segment $BC$ at point $D$. There is also a point $E$ somewhere, such that angles $\mathrm{\angle }ECB$ and $\mathrm{\angle }EBA$ are both right. Point $F$ is on the extension of the segment $AB$, such that $B$ is between $A$ and $F$. We also know that $BF=AD$. Show that $ED=EF$.

What is a remainder in division by $3$ of the sum $1+2+\cdots +2018$?

What is a remainder in division by $3$ of the number $8^{2019}$?

Show that a number $3333333333332$ is not a perfect square (without using a calculator).

What is a remainder in division by $3$ of the number $5^{21}+{17}^6\times 7^{2019}$?

Show that the sum of any three consecutive integers is divisible by $3$.

In a country far far away, there are only two types of coins: 1 crown and 3 crowns coins. Molly had a bag with only 3 crown coins in it. She used some of these coins to buy herself hat and she got one 1 crown coin back. The next day, all of her friends were jealous of her hat, so she decided to buy identical hats for them. She again only had 3 crown coins in her purse, and she used them to pay for 7 hats. Show that she got a single 1 crown coin back.

Show that numbers $12n+1$ and $12n+7$ are relatively prime.