Determine the minimum value of the expression $$\frac {a+1}{a(a+2)}+ \frac {b+1}{b(b+2)}+\frac {c+1}{c(c+2)}$$ for positive real numbers $a,b,c$ such that $a+b+c \leq 3$.

Let $D$ be an arbitrary point on side $AB$ of triangle $ABC$. Circumcircles of triangles $BCD$ and $ACD$ intersect sides $AC$ and $BC$ at points $E$ and $F$, respectively. Perpendicular bisector of $EF$ cuts $AB$ at point $M$, and line perpendicular to $AB$ at $D$ at point $N$. Lines $AB$ and $EF$ intersect at point $T$, and the second point of intersection of circumcircle of triangle $CMD$ and line $TC$ is $U$. Prove that $NC=NU$

Prove that there exist infinitely many composite positive integers $n$ such that $n$ divides $3^{n-1}-2^{n-1}$.

Let $X$ be a set which consists from $8$ consecutive positive integers. Set $X$ is divided on two disjoint subsets $A$ and $B$ with equal number of elements. If sum of squares of elements from set $A$ is equal to sum of squares of elements from set $B$, prove that sum of elements of set $A$ is equal to sum of elements of set $B$.

Let $N$ be a positive integer. It is given set of weights which satisfies following conditions: $i)$ Every weight from set has some weight from $1,2,...,N$; $ii)$ For every $i\in {1,2,...,N}$ in given set there exists weight $i$; $iii)$ Sum of all weights from given set is even positive integer. Prove that set can be partitioned into two disjoint sets which have equal weight

Let $D$, $E$ and $F$ be points in which incircle of triangle $ABC$ touches sides $BC$, $CA$ and $AB$, respectively, and let $I$ be a center of that circle.Furthermore, let $P$ be a foot of perpendicular from point $I$ to line $AD$, and let $M$ be midpoint of $DE$. If $\{N\}=PM\cap{AC}$, prove that $DN \parallel EF$