Olympiad problems

2019 USA TSTST, 1

Find all binary operations $\diamondsuit: \mathbb R_{>0}\times \mathbb R_{>0}\to \mathbb R_{>0}$ (meaning $\diamondsuit$ takes pairs of positive real numbers to positive real numbers) such that for any real numbers $a, b, c > 0$, [list] [*] the equation $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ holds; and [*] if $a\ge 1$ then $a\,\diamondsuit\, a\ge 1$. [/list] [i]Evan Chen[/i]

2019 USA TSTST, 2

Tags: geometry , tstst 2019

Let $ABC$ be an acute triangle with circumcircle $\Omega$ and orthocenter $H$. Points $D$ and $E$ lie on segments $AB$ and $AC$ respectively, such that $AD = AE$. The lines through $B$ and $C$ parallel to $\overline{DE}$ intersect $\Omega$ again at $P$ and $Q$, respectively. Denote by $\omega$ the circumcircle of $\triangle ADE$. [list=a] [*] Show that lines $PE$ and $QD$ meet on $\omega$. [*] Prove that if $\omega$ passes through $H$, then lines $PD$ and $QE$ meet on $\omega$ as well. [/list] [i]Merlijn Staps[/i]

2019 USA TSTST, 5

Tags: geometry , orthocenter , tstst 2019 , moving points

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. A line through $H$ intersects segments $AB$ and $AC$ at $E$ and $F$, respectively. Let $K$ be the circumcenter of $\triangle AEF$, and suppose line $AK$ intersects $\Gamma$ again at a point $D$. Prove that line $HK$ and the line through $D$ perpendicular to $\overline{BC}$ meet on $\Gamma$. [i]Gunmay Handa[/i]

2019 USA TSTST, 9

Tags: geometry , tstst 2019 , USA TSTST , tangent circles , projective geometry , homothety , prism lemma

Let $ABC$ be a triangle with incenter $I$. Points $K$ and $L$ are chosen on segment $BC$ such that the incircles of $\triangle ABK$ and $\triangle ABL$ are tangent at $P$, and the incircles of $\triangle ACK$ and $\triangle ACL$ are tangent at $Q$. Prove that $IP=IQ$. [i]Ankan Bhattacharya[/i]

2019 USA TSTST, 6

Tags: algebra , tstst 2019 , Integer Polynomial , Fibonacci

Suppose $P$ is a polynomial with integer coefficients such that for every positive integer $n$, the sum of the decimal digits of $|P(n)|$ is not a Fibonacci number. Must $P$ be constant? (A [i]Fibonacci number[/i] is an element of the sequence $F_0, F_1, \dots$ defined recursively by $F_0=0, F_1=1,$ and $F_{k+2} = F_{k+1}+F_k$ for $k\ge 0$.) [i]Nikolai Beluhov[/i]

2019 USA TSTST, 7

Tags: number theory , greatest common divisor , algebra , functional equation , tstst 2019

Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying $$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$ for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$. [i]Ankan Bhattacharya[/i]

2019 USA TSTST, 4

Tags: tstst 2019 , combinatorics

Consider coins with positive real denominations not exceeding 1. Find the smallest $C>0$ such that the following holds: if we have any $100$ such coins with total value $50$, then we can always split them into two stacks of $50$ coins each such that the absolute difference between the total values of the two stacks is at most $C$. [i]Merlijn Staps[/i]

2019 USA TSTST, 3

Tags: tstst 2019 , combinatorics

On an infinite square grid we place finitely many [i]cars[/i], which each occupy a single cell and face in one of the four cardinal directions. Cars may never occupy the same cell. It is given that the cell immediately in front of each car is empty, and moreover no two cars face towards each other (no right-facing car is to the left of a left-facing car within a row, etc.). In a [i]move[/i], one chooses a car and shifts it one cell forward to a vacant cell. Prove that there exists an infinite sequence of valid moves using each car infinitely many times. [i]Nikolai Beluhov[/i]

2019 USA TSTST, 8

Tags: tstst 2019 , combinatorics , Plane Geometry , geometry , combinatorial geometry

Let $\mathcal S$ be a set of $16$ points in the plane, no three collinear. Let $\chi(S)$ denote the number of ways to draw $8$ lines with endpoints in $\mathcal S$, such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of $\chi(\mathcal S)$ across all such $\mathcal S$. [i]Ankan Bhattacharya[/i]