Problems

The marked pink segment (tangent to the inner circle) has length $1$. Find the area of the green annulus.

A triple of natural numbers $a,b,c$ such that $a^2+b^2=c^2$ is called a Pythagorean triple. There are some small Pythagorean triples that are well-known, like $3,4,5$ and $5,12,13$. Let us have a look at the latter one. We can notice an interesting thing: not only $5^2+{12}^2={13}^2$, but also $5^2=25=12+13$ and $13-12=1$. Use that as an inspiration to find an idea of how to generate some more Pythagorean triples. Check if they are correct by plugging them into the equation $a^2+b^2=c^2$.

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-angled triangle are all integers.

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

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 angles. 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 $3$.