There are 7 gold coins and 9 silver coins. Among them, there is one fake gold coin and one fake silver coin. You want to identify these fake coins using a magic bag. When you put coins into the magic bag and cast a spell, it emits a suspicious glow only if both fake coins are inside the bag. How many times do you need to cast the spell to determine both fake coins?
Let \( A \) and \( B \) be two points on a plane. You want to construct the line segment \( AB \), but the only ruler you have is shorter than the length of \( AB \). How can you construct the segment \( AB \) under these conditions? Note that measuring lengths with the ruler is not allowed.
Find all maps \(f\colon \mathbb{R}_{>0}\to \mathbb{R}_{>0}\) satisfying \(f(x^y)=f(x)^{f(y)}\).
Let \(P_1,P_2,\dots,P_n\) be points on a unit circle. Show that there is a point \(Q\) on the circle such that \[\prod_{i=1}^n P_iQ\geq 2.\]
Suppose that a family of arithmetic progressions \((a_i+b_i\mathbb{Z})_{i=1}^n\) covers \(\mathbb{Z}\). Show that there is a subset \(I\subset\{1,2,\dots,n\}\) such that \[\sum_{i\in I} \dfrac{1}{b_i}\] is a positive integer.