`Compfiles/Usa2021P5.lean`

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3 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 2021, Problem 5

Let n ≥ 4 be an integer. Find all positive real solutions to the following system of 2n equations:

a₁ = 1/a₂ₙ + 1/a₂, a₂ = a₁ + a₃, a₃ = 1/a₂ + 1/a₄, a₄ = a₃ + a₅, a₅ = 1/a₄ + 1/a₆, a₆ = a₅ + a₇, ⋮ a₂ₙ₋₁ = 1/a₂ₙ₋₂ + 1/a₂ₙ, a₂ₙ = a₂ₙ₋₁ + a₁.

Imports (3)

Imported by 1 file

Compfiles.lean

lemma Usa2021P5.even_relation Eliminating the odd-indexed variables from the system yields a relation among the even-indexed variables alone: a k = 1 / a (k - 1) + 2 / a k + 1 / a (k + 1) line 41 source
Source documentation
Eliminating the odd-indexed variables from the system yields a relation among the even-indexed variables alone: a k = 1 / a (k - 1) + 2 / a k + 1 / a (k + 1).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, ZMod, Real, Eq, HAdd.hAdd, OfNat.ofNat, HDiv.hDiv, HSub.hSub
Proof uses: Eq.mpr, id, Eq.rec, Eq.refl, Eq.trans, congrArg, Eq.mp, add_sub_cancel_right, Mathlib.Tactic.Ring.of_eq, HMul.hMul, HPow.hPow, Inv.inv, Nat.rawCast, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.div_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_one, Mathlib.Tactic.Ring.Common.atom_pf, rfl and 23 more
89 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
lemma even_relation {n : ℕ} {a b : ZMod n → ℝ}
    (h1 : ∀ k, a k = b k + b (k + 1))
    (h2 : ∀ k, b k = 1 / a (k - 1) + 1 / a k) (k : ZMod n) :
    a k = 1 / a (k - 1) + 2 / a k + 1 / a (k + 1) := by
…
```
determine Usa2021P5.solution The unique solution of the system: the even-indexed terms equal 2 and the odd-indexed terms equal 1, i.e. (a₁, a₂, a₃, a₄, …) = (1, 2, 1, 2, …) line 56 source
Source documentation
The unique solution of the system: the even-indexed terms equal 2 and the odd-indexed terms equal 1, i.e. (a₁, a₂, a₃, a₄, …) = (1, 2, 1, 2, …). The first component is the sequence of even-indexed terms and the second component is the sequence of odd-indexed terms.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Prod, ZMod, Real
Proof uses: Prod.mk, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
determine solution (n : ℕ) : (ZMod n → ℝ) × (ZMod n → ℝ) :=
…
```
problem Usa2021P5.usa2021_p5 line 59 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, ZMod, Real, LT.lt, Iff, And, Eq, HAdd.hAdd, HDiv.hDiv, HSub.hSub, Prod, Prod.mk, Usa2021P5.solution
Proof uses: NeZero, NeZero.mk, Iff.intro, Usa2021P5.even_relation, Membership.mem, Finset, Finset.univ, Exists, Finset.exists_min_image, Iff.mpr, Finset.Nonempty, Nonempty, Finset.univ_nonempty_iff, Nonempty.intro, Finset.exists_max_image, Finset.mem_univ, one_div_le_one_div_of_le, Trans.trans, Mathlib.Tactic.Ring.of_eq, HMul.hMul and 147 more
165 instances, coercions and auxiliary constants left out.

Tactic trace: 57 steps
Open to load the checked before-and-after goal states.