In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color.

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. South Africa

Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other

Let $ABC$ be a triangle with $AB > AC$. Let $D$ be a point on the side $AB$ such that $DB = DC$ and let $M$ be the midpoint of $AC$. The line parallel to $BC$ passing through $D$ intersects the line $BM$ in $K$. Show that $\angle KCD = \angle DAC.$

Find all polynomials $P$ with real coefficients having no repeated roots, such that for any complex number $z$, the equation $zP(z) = 1$ holds if and only if $P(z-1)P(z + 1) = 0$. Remark: Remember that the roots of a polynomial are not necessarily real numbers.

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: $(i)$ $f(n) \neq 0$ for at least one $n$; $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.