US-TST 2024

Find the smallest constant $C > 1$ such that the following statement holds: for every integer $n\geq 2$ and sequence of non-integer positive real numbers $a_{1}$ , $a_{2}$ , ..., $a_{n}$ satisfying

$$\frac{1}{a_{1}} +\frac{1}{a_{2}} +\dots +\frac{1}{a_{n}} = 1,$$

it's possible to choose positive integers $b_{i}$ such that

(i) for each $i = 1,2,\ldots ,n$ , either $b_{i} = \lfloor a_{i}\rfloor$ or $b_{i} = \lfloor a_{i}\rfloor +1$ ; and

(ii) we have

$$1< \frac{1}{b_{1}} +\frac{1}{b_{2}} +\dots +\frac{1}{b_{n}}\leq C.$$

Let $A B C$ be a triangle with incenter $I$ . Let segment $A I$ intersect the incircle of triangle $A B C$ at point $D$ . Suppose that line $B D$ is perpendicular to line $A C$ . Let $P$ be a point such that $\angle B P A = \angle P A I = 90^{\circ}$ . Point $Q$ lies on segment $B D$ such that the circumcircle of triangle $A B Q$ is tangent to line $B I$ . Point $X$ lies on line $P Q$ such that $\angle I A X = \angle X A C$ . Prove that $\angle A X P = 45^{\circ}$ .

Let $n > k\geq 1$ be integers and let $p$ be a prime dividing ${\binom{n}{k}}$ . Prove that the $k$ -element subsets of $\{1,\ldots ,n\}$ can be split into $p$ classes of equal size, such that any two subsets with the same sum of elements belong to the same class.

Find all integers $n\geq 2$ for which there exists a sequence of $2n$ pairwise distinct points $(P_{1},\ldots ,P_{n},Q_{1},\ldots ,Q_{n})$ in the plane satisfying the following four conditions:

(i) no three of the $2n$ points are collinear;

(ii) $P_{i}P_{i + 1}\geq 1$ for all $i = 1,2,\ldots ,n$ , where $P_{n + 1} = P_{1}$

(iii) $Q_{i}Q_{i + 1}\geq 1$ for all $i = 1,2,\ldots ,n$ , where $Q_{n + 1} = Q_{1}$ ; and

(iv) $P_{i}Q_{j}\leq 1$ for all $i = 1,2,\ldots ,n$ and $j = 1,2,\ldots ,n$

Suppose $a_{1}< a_{2}< \dots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1}< b_{2}< \dots < b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.

Solve over $\mathbb{R}$ the functional equation

$$f(x f(y)) + f(y) = f(x + y) + f(x y).$$