Assume that for pairwise distinct real numbers $a,b,c,d$ holds: $$ (a^2+b^2-1)(a+b)=(b^2+c^2-1)(b+c)=(c^2+d^2-1)(c+d).$$Prove that $ a+b+c+d=0.$

Let $n$ be a positive integer. Jadzia has to write all integers from $1$ to $2n-1$ on a board, and she writes each integer in blue or red color. We say that pair of numbers $i,j\in \{1,2,3,...,2n-1\}$, where $i\leqslant j$, is $\textit{good}$ if and only if number of blue numbers among $i,i+1,...,j$ is odd. Determine, in terms of $n$, maximal number of good pairs.

Let $M$ be the midpoint of the side $BC$ of a acute triangle $ABC$. Incircle of the triangle $ABM$ is tangent to the side $AB$ at the point $D$. Incircle of the triangle $ACM$ is tangent to the side $AC$ at the point $E$. Let $F$ be the such point, that the quadrilateral $DMEF$ is a parallelogram. Prove that $F$ lies on the bisector of $\angle BAC$.

Let $ABCDEF$ be a such convex hexagon that $$ AB=CD=EF\; \text{and} \; BC=DE=.FA$$Prove that if $\sphericalangle FAB + \sphericalangle ABC=\sphericalangle FAB + \sphericalangle EFA = 240^{\circ}$, then $\sphericalangle FAB+\sphericalangle CDE=240^{\circ}$.

Let $p>$ be a prime number and $S$ be a set of $p+1$ integers. Prove that there exist pairwise distinct numbers $a_1,a_2,...,a_{p-1}\in S$ that $$ a_1+2a_2+3a_3+...+(p-1)a_{p-1}$$is divisible by $p$.

Let $(a_0,a_1,a_2,...)$ and $(b_0,b_1,b_2,...)$ be such sequences of non-negative real numbers, that for every integer $i\geqslant 1$ holds $a_i^2\leqslant a_{i-1}a_{i+1}$ and $b_i^2\leqslant b_{i-1}b_{i+1}$. Define sequence $c_0,c_1,c_2,...$ as $$c_0=a_0b_0, \; c_n=\sum_{i=0}^{n} {{n}\choose{i}} a_ib_{n-i}.$$Prove that for every integer $k\geqslant 1$ holds $c_{k}^2\leqslant c_{k-1}c_{k+1}$.