4. Mathematical Puzzles

1. Fake coins and a magic bag

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?

2. Short ruler

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.

3. Exponential-preserving map

Find all maps \(f\colon \mathbb{R}_{>0}\to \mathbb{R}_{>0}\) satisfying \(f(x^y)=f(x)^{f(y)}\).

4. Social distancing on a circle

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 \[\prod_{i=1}^n P_iQ\geq 2.\]

5. Covering integers by arithmetic progressions

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.

6. Social distancing in a square

Show that if there are \(10\) points inside a square of side length \(3\), then there exist two points whose distance is at most \(\sqrt{2}\).

7. Glasses on Glasses

Place \(n\) hollow cylinders of radius \(1\) and height \(1\) upright on a plane. On top of them, place more cylinders of the same size. Let \(f(n)\) be the maximum number of cylinders that can be placed in the second layer. Show that \[\lim_{n \to \infty} \frac{f(n)}{n} = \frac{5}{2}.\]

8. Addition and Multiplication

Does there exist a non-zero commutative ring \(R\) such that \((R,+)\) and \((R^\times, \cdot)\) are isomorphic as abelian groups?

9. Two-Colored Necklace

We make a necklace by connecting a total of \(n\) beads, each either white or black. Determine the number of initial arrangements from which, by repeating the following operation, we can end up with exactly two white beads. Necklaces that can be obtained from each other by rotation or reflection are considered distinct.

Operation: Choose one white bead and remove it, then flip the colors of the beads immediately to its left and right.

10. A Hexahedron Inscribed in a Sphere

Consider a hexahedron (think of a distorted rectangular box) in which all faces are quadrilaterals. Suppose that \(7\) of its \(8\) vertices lie on the same sphere. Show that the remaining vertex also lies on this sphere.

11. The Five-Pile Game

There are five piles of stones, and the total number of stones is even. Two players play a game with these piles. On each turn, a player takes exactly one stone from each of two different piles. The players alternate turns, and the player who cannot make a move loses. Determine the necessary and sufficient condition on the numbers of stones in the piles for the first player to have a winning strategy.