`Compfiles/Usa1999P4.lean`

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2 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1999, Problem 4

Let a₁, a₂, ..., aₙ be a sequence of n > 3 real numbers such that

a₁ + a₂ + ⋯ + aₙ ≥ n

and

a₁² + a₂² + ⋯ + aₙ² ≥ n².

Prove that max(a₁, a₂, ..., aₙ) ≥ 2.

Imports (4)

Imported by 1 file

Compfiles.lean

theorem Usa1999P4.sum_sq_le_sq_sum_of_nonpos If every value of f on s is nonpositive, then the sum of the squares is at most the square of the sum: the cross terms are nonnegative line 50 source
Source documentation
If every value of f on s is nonpositive, then the sum of the squares is at most the square of the sum: the cross terms are nonnegative.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Real, Membership.mem, LE.le, OfNat.ofNat, Finset.sum, HPow.hPow, Nat
Proof uses: Eq, Neg.neg, Finset.sum_congr, rfl, Eq.mpr, id, congrArg, neg_sq, Eq.refl, Finset.sum_neg_distrib, Finset.sum_sq_le_sq_sum_of_nonneg, Iff.mpr, neg_nonneg
54 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem sum_sq_le_sq_sum_of_nonpos {ι : Type*} (s : Finset ι) (f : ι → ℝ)
    (hf : ∀ i ∈ s, f i ≤ 0) :
    ∑ i ∈ s, f i ^ 2 ≤ (∑ i ∈ s, f i) ^ 2 := by
…
```
Mentions: f
problem Usa1999P4.usa1999_p4 line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Fin, Real, LE.le, Finset.sum, Finset.univ, HPow.hPow, Exists
Proof uses: Decidable.byContradiction, Not, Finset, Finset.filter, Eq, rfl, HAdd.hAdd, Eq.symm, Finset.sum_filter_add_sum_filter_not, dite, False, EmptyCollection.emptyCollection, Membership.mem, Iff.mpr, And, Finset.mem_filter, And.intro, Finset.mem_univ, Finset.notMem_empty, Eq.mp and 142 more
155 instances, coercions and auxiliary constants left out.

Tactic trace: 98 steps
Open to load the checked before-and-after goal states.