Let \(n\) be a positive integer. Prove that \[\frac{20 \cdot 5^n-2}{3^n+47}\]is not an integer.

In a school with some students, for any three student, there exists at least one student who are friends with all these three students.(Friendships are mutual) For any friends $A$ and $B$, any two of their common friends are also friends with each other. It's not possible to partition these students into two groups, such that every student in each group are friends with all the students in the other gruop. Prove that any two people who aren't friends with each other, has the same number of common friends.(Each person is a friend of him/herself.)

A point $D$ is taken on the arc $BC$ of the circumcircle of triangle $ABC$ which does not contain $A$. A point $E$ is taken at the intersection of the interior region of the triangles $ABC$ and $ADC$ such that $m(\widehat{ABE})=m(\widehat{BCE})$. Let the circumcircle of the triangle $ADE$ meets the line $AB$ for the second time at $K$. Let $L$ be the intersection of the lines $EK$ and $BC$, $M$ be the intersection of the lines $EC$ and $AD$, $N$ be the intersection of the lines $BM$ and $DL$. Prove that $$m(\widehat{NEL})=m(\widehat{NDE})$$

In a fish shop with 28 kinds of fish, there are 28 fish sellers. In every seller, there exists only one type of each fish kind, depending on where it comes, Mediterranean or Black Sea. Each of the $k$ people gets exactly one fish from each seller and exactly one fish of each kind. For any two people, there exists a fish kind which they have different types of it (one Mediterranean, one Black Sea). What is the maximum possible number of $k$?

In a non isoceles triangle $ABC$, let the perpendicular bisector of $[BC]$ intersect $(ABC)$ at $M$ and $N$ respectively. Let the midpoints of $[AM]$ and $[AN]$ be $K$ and $L$ respectively. Let $(ABK)$ and $(ABL)$ intersect $AC$ again at $D$ and $E$ respectively, let $(ACK)$ and $(ACL)$ intersect $AB$ again at $F$ and $G$ respectively. Prove that the lines $DF$, $EG$ and $MN$ are concurrent.

For which positive integers $n$, one can find real numbers $x_1,x_2,\cdots ,x_n$ such that $$\dfrac{x_1^2+x_2^2+\cdots+x_n^2}{\left(x_1+2x_2+\cdots+nx_n\right)^2}=\dfrac{27}{4n(n+1)(2n+1)}$$and $i\leq x_i\leq 2i$ for all $i=1,2,\cdots ,n$ ?

Given a triangle $ABC$ with the circumcircle $\omega$ and incenter $I$. Let the line pass through the point $I$ and the intersection of exterior angle bisector of $A$ and $\omega$ meets the circumcircle of $IBC$ at $T_A$ for the second time. Define $T_B$ and $T_C$ similarly. Prove that the radius of the circumcircle of the triangle $T_AT_BT_C$ is twice the radius of $\omega$.

Let \(c\) be a real number. For all \(x\) and \(y\) real numbers we have, \[f(x-f(y))=f(x-y)+c(f(x)-f(y))\]and \(f(x)\) is not constant. \(a)\) Find all possible values of \(c\). \(b)\) Can \(f\) be periodic?

For which positive integer couples $(k,n)$, the equality $\Bigg|\Bigg\{{a \in \mathbb{Z}^+: 1\leq a\leq(nk)!, gcd \left(\binom{a}{k},n\right)=1}\Bigg\}\Bigg|=\frac{(nk)!}{6}$ holds?