Subcontests
(6)Sequence of positive real numbers
The sequence a1,a2,…,an of positive real numbers satisfies the following conditions:
\begin{align*}
\sum_{i=1}^n \frac{1}{a_i} \le 1 \ \ \ \ \hbox{and} \ \ \ \ a_i \le a_{i-1}+1
\end{align*}
for all i∈{1,2,…,n}, where a0 is an integer. Prove that
\begin{align*}
n \le 4a_0 \cdot \sum_{i=1}^n \frac{1}{a_i}
\end{align*} Injection and divisibility
Let n,k,ℓ be positive integers and σ:{1,2,…,n}→{1,2,…,n} an injection such that σ(x)−x∈{k,−ℓ} for all x∈{1,2,…,n}. Prove that k+ℓ∣n. Guests at a party
n≥3 guests met at a party. Some of them know each other but there is no quartet of different guests a,b,c,d such that in pairs {a,b},{b,c},{c,d},{d,a} guests know each other but in pairs {a,c},{b,d} guests don't know each other. We say a nonempty set of guests X is an ingroup, when guests from X know each other pairwise and there are no guests not from X knowing all guests from X. Prove that there are at most 2n(n−1) different ingroups at that party. Concyclic points
Let ABC be an acute triangle. Points X and Y lie on the segments AB and AC, respectively, such that AX=AY and the segment XY passes through the orthocenter of the triangle ABC. Lines tangent to the circumcircle of the triangle AXY at points X and Y intersect at point P. Prove that points A,B,C,P are concyclic.