module
public import Mathlib.Data.Finset.Card
public import Mathlib.Data.Finset.Prod
public import Mathlib.Data.Finset.Sum
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Fintype.Prod
public import Mathlib.Combinatorics.Hall.Finite
public import Mathlib.GroupTheory.Perm.Sign
public import Mathlib.Logic.Equiv.Fintype
public import Mathlib.Tactic.FinCases
public import Mathlib.Tactic.GCongr
public import Mathlib.Tactic.Ring
public import Lean.Elab.Tactic.Omega
public section
/-!
# USA Mathematical Olympiad 2022, Problem 1
Let a and b be positive integers. The cells of an (a+b+1) × (a+b+1) grid
are colored amber and bronze such that there are at least a² + ab - b
amber cells and at least b² + ab - a bronze cells. Prove that it is
possible to choose a amber cells and b bronze cells such that no two
of the a + b chosen cells lie in the same row or column.
-/
namespace Usa2022P1
theorem usa2022_p1
(a b : ℕ)
(ha : 0 < a)
(hb : 0 < b)
(color : Fin (a + b + 1) × Fin (a + b + 1) → Fin 2)
(c0 : a ^ 2 + a * b - b ≤ Fintype.card {s // color s = 0})
(c1 : b ^ 2 + a * b - a ≤ Fintype.card {s // color s = 1}) :
∃ A B : Finset (Fin (a + b + 1) × Fin (a + b + 1)),
A.card = a ∧ B.card = b ∧
(∀ x ∈ A, color x = 0) ∧
(∀ y ∈ B, color y = 1) ∧
∀ x ∈ A ∪ B, ∀ y ∈ A ∪ B, x ≠ y → x.fst ≠ y.fst ∧ x.snd ≠ y.snd := sorry
end Usa2022P1
This problem has a complete formalized solution.