Prove that for $n\ge 2$ the following inequality holds $$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left...
- Prove that for $n\ge 2$ the following inequality holds $$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$
- Solve the following equation in $\mathbb{R}$ $$\left(x-\frac{1}{x}\right)^\frac{1}{2}+\left(1-\frac{1}{x}\right)^\frac{1}{2}=x.$$
- From the point $P$ outside a circle $\omega$ with center $O$ draw the tangents $PA$ and $PB$ where $A$ and $B$ belong to $\omega$.In a random point $M$ in the chord $AB$ we draw the perpendicular to $OM$, which intersects $PA$ and $PB$ in $C$ and $D$. Prove that $M$ is the midpoint $CD$.
- Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x)f(y)=f(x+y)+xy$$ for all $x,y\in \mathbb{R}$.
- Find all non-negative integers $k,n$ which satisfy $$2^{2k+1} + 9\cdot 2^k+5=n^2.$$
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