Problems

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 $(a-b)(b-c)(c-a)=1$.

A five-digit number is called indecomposable if it is not decomposed into the product of two three-digit numbers. What is the largest number of indecomposable five-digit numbers that can come in a row?

Find all solutions of the equation: $x^2+y^2+z^2+t^2=x(y+z+t)$.

Let $a$ and $b$ be real numbers. Find a representation of $a^3+b^3$ as a product.

• Find a representation of the number $117=121-4$ as a product.

• Let $a$ and $b$ be real numbers. Find a representation of $a^2-b^2$ as a product.

Solve the system of equations in real numbers: $$\{\begin{array}{r}x+y=2\\ xy-z^2=1\end{array}$$

Find all solutions of the equation: $xy+1=x+y$.

Find all solutions of the system of equations: $$\{\begin{array}{r}(x+y{)}^3=z\\ (x+z{)}^3=y\\ (y+z{)}^3=x\end{array}$$

Sometimes different areas in mathematics are more related than they seem to be. A lot of algebraic expressions have geometric interpretation, and a lot of them can be used to solve problems in number theory.

Today we will solve several logic problems that revolve about a very simple idea. Imagine you are in a room in a dungeon and you can see doors leading out of the room. Some of them lead to the treasure and some of them lead to traps. It is possible that all doors lead to treasure or all lead to traps, but it is also possible that one door leads to treasure and all other lead to traps. Unless specified, there is always something behind the door.
Each door has a sign with a statement on it, but those statements are not always true. You have a dungeon guide, who is always honest with you and will tell you something about the truthfulness of the statements on the doors, but it will be up to you to put it all together and pick the correct door... or walk away, if you believe there is no treasure.