`Compfiles/Imo2003P1.lean`

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3 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 2003, Problem 1

Let A be a 101-element subset of S = {1,2,...10⁶}. Prove that there exist numbers t₁, t₂, ..., t₁₀₀ in S such that the sets

Aⱼ = {x + tⱼ | x ∈ A}, j = 1,2, ..., 100

are pairwise disjoint.

Imports (2)

Imported by 1 file

Compfiles.lean

abbrev Imo2003P1.S line 30 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat
Proof uses: Finset.Icc, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
theorem Imo2003P1.induction_lemma Proof by induction on the number of tᵢ's line 35 source
Source comment
Proof by induction on the number of tᵢ's

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat, LE.le, Imo2003P1.S, Eq, Finset.card, OfNat.ofNat, Exists, And, Membership.mem, Ne, Disjoint, Set, Set.ofPred, HAdd.hAdd
Proof uses: Nat.recAux, Exists.intro, EmptyCollection.emptyCollection, of_eq_true, Eq.trans, True, congr, congrArg, eq_self, Not, forall_congr, False, implies_congr, Eq.refl, implies_true, and_self, Finset.filter, Classical.propDecidable, Finset.Nonempty, Iff.mp and 58 more
99 instances, coercions and auxiliary constants left out.

Tactic trace: 90 steps
Open to load the checked before-and-after goal states.
```
theorem induction_lemma (A : Finset ℕ) (_AS : A ⊆ S) (Acard : A.card = 101)
    {k : ℕ} (hk : k ≤ 100) :
    ∃ t ⊆ S, t.card = k ∧ ∀ x ∈ t, ∀ y ∈ t, x ≠ y → Disjoint {x + a | a ∈ A} {y + a | a ∈ A} := by
…
```
Mentions: Imo2003P1.S
problem Imo2003P1.imo2003_p1 line 131 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat, LE.le, Imo2003P1.S, Eq, Finset.card, OfNat.ofNat, Exists, And, Membership.mem, Ne, Disjoint, Set, Set.ofPred, HAdd.hAdd
Proof uses: Imo2003P1.induction_lemma, Nat.le_refl
21 instances, coercions and auxiliary constants left out.