Olympiad problems

2020 USEMO, 1

Which positive integers can be written in the form \[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\] for positive integers $x$, $y$, $z$?

2022 USEMO, 3

Tags: geometry , incenter , USEMO , USEMO 2022

Point $P$ lies in the interior of a triangle $ABC$. Lines $AP$, $BP$, and $CP$ meet the opposite sides of triangle $ABC$ at $A$', $B'$, and $C'$ respectively. Let $P_A$ the midpoint of the segment joining the incenters of triangles $BPC'$ and $CPB'$, and define points $P_B$ and $P_C$ analogously. Show that if \[ AB'+BC'+CA'=AC'+BA'+CB' \] then points $P,P_A,P_B,$ and $P_C$ are concyclic. [i]Nikolai Beluhov[/i]

2020 USEMO, 3

Tags: radical axis , Euler , geometry , USEMO , USEMO 2020 , anant mudgal geo

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $\Gamma$ denote the circumcircle of triangle $ABC$, and $N$ the midpoint of $OH$. The tangents to $\Gamma$ at $B$ and $C$, and the line through $H$ perpendicular to line $AN$, determine a triangle whose circumcircle we denote by $\omega_A$. Define $\omega_B$ and $\omega_C$ similarly. Prove that the common chords of $\omega_A$,$\omega_B$ and $\omega_C$ are concurrent on line $OH$. Proposed by Anant Mudgal

2020 USEMO, 5

Tags: geometry , USEMO , USEMO 2020

The sides of a convex $200$-gon $A_1 A_2 \dots A_{200}$ are colored red and blue in an alternating fashion. Suppose the extensions of the red sides determine a regular $100$-gon, as do the extensions of the blue sides. Prove that the $50$ diagonals $\overline{A_1A_{101}},\ \overline{A_3A_{103}},\ \dots, \ \overline{A_{99}A_{199}}$ are concurrent. [i]Proposed by: [b]Ankan Bhattacharya[/b][/i]

2020 USEMO, 4

Tags: algebra , functional equation , USEMO , USEMO 2020

A function $f$ from the set of positive real numbers to itself satisfies $$f(x + f(y) + xy) = xf(y) + f(x + y)$$ for all positive real numbers $x$ and $y$. Prove that $f(x) = x$ for all positive real numbers $x$.

2019 USEMO, 5

Tags: USEMO , combinatorics

Let $\mathcal{P}$ be a regular polygon, and let $\mathcal{V}$ be its set of vertices. Each point in $\mathcal{V}$ is colored red, white, or blue. A subset of $\mathcal{V}$ is [i]patriotic[/i] if it contains an equal number of points of each color, and a side of $\mathcal{P}$ is [i]dazzling[/i] if its endpoints are of different colors. Suppose that $\mathcal{V}$ is patriotic and the number of dazzling edges of $\mathcal{P}$ is even. Prove that there exists a line, not passing through any point in $\mathcal{V}$, dividing $\mathcal{V}$ into two nonempty patriotic subsets. [i]Ankan Bhattacharya[/i]