Problems

Solve the equation: $$x+\frac{x}{x}+\frac{x}{x+\frac{x}{x}}=1$$

Suppose that: $$\frac{x+y}{x-y}+\frac{x-y}{x+y}=3.$$ Find the value of the following expression: $$\frac{x^2+y^2}{x^2-y^2}+\frac{x^2-y^2}{x^2+y^2}.$$

Solve this equation: $$(x+2010)(x+2011)(x+2012)=(x+2011)(x+2012)(x+2013).$$

Let $x$ be a natural number. Among the statements:

$2x$ is more than 70;

$x$ is less than 100;

$3x$ is greater than 25;

$x$ is not less than 10;

$x$ is greater than 5;

three are true and two are false. What is $x$?

A broken calculator carries out only one operation “asterisk”: $a\ast b=1-a/b$. Prove that using this calculator it is possible to carry out all four arithmetic operations (addition, subtraction, multiplication, division).