Problems

Cut an arbitrary triangle into parts that can be used to build a triangle that is symmetrical to the original triangle with respect to some straight line (the pieces cannot be inverted, they can only be rotated on the plane).

Find all solutions of the puzzle $HE\times HE=SHE$. Different letters denote different digits, while the same letters correspond to the same digits.

Alex writes natural numbers in a row: $123456789101112...$ Counting from the beginning, in what places do the digits $555$ first appear? For example, $101$ first appears in the 10th, 11th and 12th places.

Frodo can meet either Sam, or Pippin, or Merry in the fog. One day everyone came out to meet Frodo, but the fog was thick, and Frodo could not see where everyone was, so he asked each of his friends to introduce themselves.
The one who from Frodo’s perspective was on the left, said: "Merry is next to me."
The one on Frodo’s right said: "The person who just spoke is Pippin."
Finally, the one in the center announced, "On my left is Sam."
Identify who stood where, knowing that Sam always lies, Pippin sometimes lies, and Merry never lies?

Using areas of squares and rectangles, show that for any positive real numbers $a$ and $b$, $(a+b{)}^2=a^2+2ab+b^2$.
The identity above is true for any real numbers, not necessarily positive, in fact in order to prove it the usual way one only needs to remember that multiplication is commutative and the distributive property of addition and multiplication:

• $a\times b=b\times a$;

• $(a+b)\times c=a\times c+b\times c$.

Annie found a prime number $p$ to which you can add $4$ to make it a perfect square. What is the value of $p$?

Let $a$ and $b$ be positive real numbers. Using areas of rectangles and squares, show that $a^2-b^2=(a-b)\times (a+b)$.
Try to prove it in two ways, one geometric and one algebraic.

Let $a$ and $b$ be positive real numbers. Using volumes of cubes and parallelepipeds, show that $(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$.
Hint: Place the cubes with sides $a$ and $b$ along the same diagonal.

The real numbers $a,b,c$ are non-zero and satisfy the following equations: $$\{\begin{array}{l}{a}^2+a=b^2\\ b^2+b=c^2\\ c^2+c=a^2.\end{array}$$ Show that $(a-b)(b-c)(c-a)=1$.

Find the representation of $(a+b{)}^n$ as the sum of $X_{n,k}{a}^k{b}^{n-k}$ for general $n$. Here by $X_{n,k}$ we denote coefficients that depend only on $k$ and $n$.