Find all triples $(a, b, c)$ of positive integers such that: $$a + b + c = 24$$$$a^2 + b^2 + c^2 = 210$$$$abc = 440$$

Suppose $a$ is a non-zero real number such that $a +\frac{1}{a}$ is a whole number. (a) Prove that $a^2 +\frac{1}{a^2}$ is also an integer. (b) Prove that $a^n+\frac{1}{a^n}$ is also an integer, for any integer value positive of $n$.

In the figure, $ABC$ and $CDE$ are right-angled and isosceles triangles. Segments $AD$ and $BC$ intersect at $P$, and segments $CD$ and $BE$ intersect at $Q$. (a) Show that segment$ PQ$ is parallel to segment $AE$. (b) If $BP = 4$ and $DQ = 9$, find the measure of segment $BD$.

The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. To each of the thirteen points marked are assigned a color: green or red. Prove that there will always be three points of the same color that are vertices of an equilateral triangle.

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$, and non-parallel sides $BC$ and $DA$. The angles $\angle BCD$ and $\angle CDA$ are acute. The lines $BC$ and $DA$ are cut at a point $E$. It is known that $AE = 2$, $AC = 6$, $CD =\sqrt{72}$ and area $( \vartriangle BCD)= 18$. (a) Find the height of the trapezoid $ABCD$. (b) Find the area of $\vartriangle ABC$.

Let $f$ be a function defined on $[0, 2022]$, such that $f(0) = f(2022) = 2022$, and $$|f(x) - f(y)| \le 2|x -y|,$$for all $x, y$ in $[0, 2022]$. Prove that for each $x, y$ in $[0, 2022]$, the distance between $f(x)$ and $f(y)$ does not exceed $2022$.

Let's call a natural number interesting if any of its two digits consecutive forms a number that is a multiple of $19$ or $21$. For example, The number $7638$ is interesting, because $76$ is a multiple of $19$, $63$ is multiple of $21$, and $38$ is a multiple of $19$. How many interesting numbers of $2022$ digits exist?

There are$ 1$ cm long bars with a number$ 1$, $2$ or $3$ written on each one from them. There is an unlimited supply of bars with each number. Two triangles formed by three bars are considered different if none of them can be built with the bars of the other triangle. (a) How many different triangles formed by three bars are possible? (b) An equilateral triangle of side length $3$ cm is formed using $18$ bars, , divided into $9$ equilateral triangles, different by pairs, $1$ cm long on each side. Find the largest sum possible from the numbers written on the $9$ bars of the border of the big triangle.

Let $\omega$ be a circle with center $O$ and diameter $AB$. A circle with center at $B$ intersects $\omega$ at C and $AB$ at $D$. The line $CD$ intersects $\omega$ at a point $E$ ($E\ne C$). The intersection of lines $OE$ and $BC$ is $F$. (a) Prove that triangle $OBF$ is isosceles. (b) If $D$ is the midpoint of $OB$, find the value of the ratio $\frac{FB}{BD}$.

Let's construct a family $\{K_n\}$ of geometric figures following the pattern shown in pictures: where each hexagon (like the starting one) is constructed by cutting the two corners tops of a square, in such a way that the two figures removed are identical isosceles triangles, and the three resulting upper sides have the same length. Continuing like this, a pattern is produced with which we can build the figures $K_n$, for integer $n \ge 0$ . Then, we denote by $P_n$ and $A_n$ the perimeter and area of the figure $K_n$, respectively. If the side of square to build $K_0$ measures $x$: (a) Calculate $P_0$ and $A_0$ (in terms of the length $x$). (b) Find an explicit formula for $P_n$, and for $A_n$, in terms of $x$ and of $n$. Simplify your answers. (c) If $P_{2022} = A_{2022}$, find the measure of the six sides of the figure $K_0$, in its simplest form.