TSTST 2024

For every ordered pair of integers $(i,j)$ , not necessarily positive, we wish to select a point $P_{i,j}$ in the Cartesian plane whose coordinates lie inside the unit square defined by

$$i< x< i + 1, j< y< j + 1.$$

Find all real numbers $c > 0$ for which it's possible to choose these points such that for all integers $i$ and $j$ , the (possibly concave or degenerate) quadrilateral $P_{i,j}P_{i + 1,j}P_{i + 1,j + 1}P_{i,j + 1}$ has perimeter strictly less than $c$ .

Let $p$ be an odd prime number. Suppose $P$ and $Q$ are polynomials with integer coefficients such that $P(0) = Q(0) = 1$ , there is no nonconstant polynomial dividing both $P$ and $Q$ , and

$$1 + \frac{x}{1 + \frac{2x}{1 + \frac{\cdots}{1 + (p - 1)x}}} = \frac{P(x)}{Q(x)}.$$

Show that all coefficients of $P$ except for the constant coefficient are divisible by $p$ , and all coefficients of $Q$ are not divisible by $p$ .

Let $A = \{a_{1},\ldots ,a_{2024}\}$ be a set of 2024 pairwise distinct real numbers. Assume that there exist positive integers $b_{1},b_{2},\ldots ,b_{2024}$ such that

$$a_{1}b_{1} + a_{2}b_{2} + \cdot \cdot \cdot +a_{2024}b_{2024} = 0.$$

Prove that one can choose $a_{2025},a_{2026},a_{2027},\ldots$ such that $a_{k}\in A$ for all $k\geq 2025$ and, for every positive integer $d$ , there exist infinitely many positive integers $n$ satisfying

$$\sum_{k = 1}^{n}a_{k}k^{d} = 0.$$

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$ and $E$ be the intersection of segments $AC$ and $BD$ . Let $\omega_{1}$ be the circumcircle of $ADE$ and $\omega_{2}$ be the circumcircle of $BCE$ . The tangent to $\omega_{1}$ at $A$ and the tangent to $\omega_{2}$ at $C$ meet at $P$ . The tangent to $\omega_{1}$ at $D$ and the tangent to $\omega_{2}$ at $B$ meet at $Q$ . Show that $OP = OQ$ .

For a positive integer $k$ , let $s(k)$ denote the number of 1s in the binary representation of $k$ . Prove that for any positive integer $n$ ,

$$\sum_{i = 1}^{n}(-1)^{s(3i)} > 0.$$

Determine whether there exists a function $f\colon \mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ such that for all positive integers $m$ and $n$ ,

$$f(m + n f(m)) = f(n)^{m} + 2024!\cdot m.$$

An infinite sequence $a_{1},a_{2},a_{3},\ldots$ of real numbers satisfies

$$a_{2n - 1} + a_{2n} > a_{2n + 1} + a_{2n + 2} \mathrm{and} a_{2n} + a_{2n + 1}< a_{2n + 2} + a_{2n + 3}$$

for every positive integer $n$ . Prove that there exists a real number $C$ such that $a_{n}a_{n + 1}< C$ for every positive integer $n$ .

Let $ABC$ be a scalene triangle, and let $D$ be a point on side $BC$ satisfying $\angle BAD = \angle DAC$ . Suppose that $X$ and $Y$ are points inside $ABC$ such that triangles $ABX$ and $ACY$ are similar and quadrilaterals $ACDX$ and $ABDY$ are cyclic. Let lines $BX$ and $CY$ meet at $S$ and lines $BY$ and $CX$ meet at $T$ . Prove that lines $DS$ and $AT$ are parallel.

Let $n \geq 2$ be a fixed integer. The cells of an $n \times n$ table are filled with the integers from 1 to $n^2$ with each number appearing exactly once. Let $N$ be the number of unordered quadruples of cells on this board which form an axis-aligned rectangle, with the two smaller integers being on opposite vertices of this rectangle. Find the largest possible value of $N$ .