Positive integers $a,b,c$ satisfy that $a+b=b(a-c)$ and c+1 is a square of a prime. Prove that $a+b$ or $ab$ is a square.

There are $100$ people in the group. Is it possible that for each pair of people exist at least $50$ others, so every in that group knows exactly one person from the pair?

Positive reals $a,b,c,d$ satisfy $a+b+c+d=4$. Prove that $\sum_{cyc}\frac{a}{a^3 + 4} \le \frac{4}{5}$

In convex quadrilateral $ABCD$, where $AB=AD$, on $BD$ point $K$ is chosen. On $KC$ point $L$ is such that $\bigtriangleup BAD \sim \bigtriangleup BKL$. Line parallel to $DL$ and passes through $K$, intersect $CD$ at $P$. Prove that $\angle APK = \angle LBC$. @below edited

Prove that there exist 1000 consecutive numbers such that none of them is divisible by its sum of the digits

Some regular polygons are inscribed in a circle. Fedir turned some of them, so all polygons have a common vertice. Prove that the number of vertices did not increase.

In the acute triangle $ABC$ on $AC$ point $P$ is chosen such that $2AP=BC$. Points $X$ and $Y$ are symmetric to $P$ wrt $A$ and $C$ respectively. It turned out that $BX=BY$. Find angle $C$.

Rombus ABCD with acute angle $B$ is given. $O$ is a circumcenter of $ABC$. Point $P$ lies on line $OC$ beyond $C$. $PD$ intersect the line that goes through $O$ and parallel to $AB$ at $Q$. Prove that $\angle AQO=\angle PBC$.

There are 20 blue points on the circle and some red inside so no three are collinear. It turned out that there exists $1123$ triangles with blue vertices having 10 red points inside. Prove that all triangles have 10 red points inside

Prove for positive reals $a,b,c$ that $(ab+bc+ca+1)(a+b)(b+c)(c+a) \ge 2abc(a+b+c+1)^2$