Problems

Solve the equation: $|x-2005|+|2005-x|=2006$.

Does there exist a function $f(x)$ defined for all $x\in \mathbb{R}$ and for all $x,y\in \mathbb{R}$ satisfying the inequality $|f(x+y)+\sin x+\sin y|<2$?

The quadratic trinomials $f(x)$ and $g(x)$ are such that $f^{\prime }(x)g^{\prime }(x)\ge |f(x)|+|g(x)|$ for all real $x$. Prove that the product $f(x)g(x)$ is equal to the square of some trinomial.

Two people play a game with the following rules: one of them guesses a set of integers $(x_1,x_2,\dots ,x_n)$ which are single-valued digits and can be either positive or negative. The second person is allowed to ask what is the sum $a_1{x}_1+\cdots +a_n{x}_n$, where $(a_1,\dots ,a_n)$ is any set. What is the smallest number of questions for which the guesser recognizes the intended set?

Solve the inequality: $|x+2000|<|x-2001|$.

For any real number $x$, the absolute value of $x$, written $\left|x\right|$, is defined to be $x$ if $x>0$ and $-x$ if $x\le 0$. What are $\left|3\right|$, $|-4.3|$ and $\left|0\right|$?

Let $x$ and $y$ be real numbers. Prove that $x\le \left|x\right|$ and $0\le \left|x\right|$. Then prove that the following inequality holds $|x+y|\le \left|x\right|+\left|y\right|$.

Prove that $|x|\ge x$. It may be helpful to compare each of $|3|$, $|-4.3|$ and $|0|$ with $3$, $-4.3$ and $0$ respectively.