`Compfiles/Usa2010P5.lean`

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3 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 2010, Problem 5

Let $q = \dfrac{3p-5}{2}$ where $p$ is an odd prime, and let

\[S_q = \frac{1}{2\cdot 3 \cdot 4} + \frac{1}{5\cdot 6 \cdot 7} + \cdots + \frac{1}{q\cdot (q+1) \cdot (q+2)}.\]

Prove that if $\dfrac{1}{p}-2S_q = \dfrac{m}{n}$ for integers $m$ and $n$, then $m-n$ is divisible by $p$.

Imports (2)

Imported by 1 file

Compfiles.lean

lemma Usa2010P5.sum_range_triple Summation of three consecutive terms in blocks of three: the sum of f (3i+2) + f (3i+3) + f (3i+4) over i ∈ range t equals the sum of f j over j ∈ Icc 2 (3t+1) line 37 source
Source documentation
Summation of three consecutive terms in blocks of three: the sum of f (3i+2) + f (3i+3) + f (3i+4) over i ∈ range t equals the sum of f j over j ∈ Icc 2 (3t+1).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat, Eq, Finset.sum, Finset.range, HAdd.hAdd, HMul.hMul, OfNat.ofNat, Finset.Icc
Proof uses: Nat.recAux, of_eq_true, Eq.trans, True, congrArg, Finset.sum_congr, EmptyCollection.emptyCollection, Finset, congrFun', MulZeroClass.mul_zero, zero_add, Finset.Icc_eq_empty_of_lt, LT.lt, LE.le, Membership.mem, Eq.refl, eq_self, Eq.mpr, id, Finset.sum_range_succ and 6 more
56 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
```
lemma sum_range_triple (f : ℕ → ℚ) (t : ℕ) :
    ∑ i ∈ Finset.range t, (f (3*i+2) + f (3*i+3) + f (3*i+4)) =
    ∑ j ∈ Finset.Icc 2 (3*t+1), f j := by
…
```
Mentions: f
lemma Usa2010P5.sum_one_div_eq_div_prod A sum of reciprocals as a single fraction with the product of all denominators as denominator line 51 source
Source documentation
A sum of reciprocals as a single fraction with the product of all denominators as denominator.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat, Rat, Membership.mem, Ne, OfNat.ofNat, Eq, Finset.sum, HDiv.hDiv, Finset.prod, Finset.erase
Proof uses: HMul.hMul, Eq.mpr, id, congrArg, Finset.mul_sum, Finset.sum_congr, rfl, Eq.symm, Finset.mul_prod_erase, mul_right_comm, mul_one_div_cancel, one_mul, Eq.refl, Iff.mpr, Finset.prod_ne_zero_iff, EuclideanDomain.eq_div_of_mul_eq_right
66 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
lemma sum_one_div_eq_div_prod {s : Finset ℕ} {c : ℕ → ℚ} (hc : ∀ i ∈ s, c i ≠ 0) :
    ∑ i ∈ s, 1 / c i = (∑ i ∈ s, ∏ j ∈ s.erase i, c j) / ∏ i ∈ s, c i := by
…
```
problem Usa2010P5.usa2010_p5 line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Odd, Eq, HDiv.hDiv, HSub.hSub, HMul.hMul, OfNat.ofNat, Int, Ne, Rat, Finset.sum, Finset.filter, HMod.hMod, Finset.Icc, HAdd.hAdd, Dvd.dvd
Proof uses: LE.le, Nat.Prime.two_le, Finset, Finset.image, Finset.range, Finset.ext, Eq.mpr, Iff, Membership.mem, And, Exists, LT.lt, id, congr, congrArg, Eq.trans, congrFun', funext, Iff.intro, Exists.intro and 227 more
297 instances, coercions and auxiliary constants left out.

Tactic trace: 209 steps (first 200 of 209)
Open to load the checked before-and-after goal states.