`Compfiles/Imo1959P3.lean`

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6 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 1959, Problem 3

Let a, b, c be real numbers. Given the equation for cos x:

a cos²x + b cos x + c = 0,

form a quadratic equation in cos 2x whose roots are the same values of x. Compare the equations in cos x and cos 2x for a = 4, b = 2, c = -1.

Imports (2)

Imported by 1 file

Compfiles.lean

determine Imo1959P3.formedQuadratic Documented declaration: formedQuadratic line 30 source
Source documentation
The coefficients of the quadratic equation in cos (2 * x) formed from a * cos x ^ 2 + b * cos x + c = 0: writing cos (2 * x) = 2 * cos x ^ 2 - 1 and eliminating the odd power of cos x by squaring.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Prod
Proof uses: Prod.mk, HPow.hPow, Nat, OfNat.ofNat, HSub.hSub, HAdd.hAdd, HMul.hMul
15 instances, coercions and auxiliary constants left out.
```
determine formedQuadratic (a b c : ℝ) : ℝ × ℝ × ℝ :=
…
```
def Imo1959P3.formedQuadraticEval The formed quadratic evaluated at t, i.e. the quadratic with coefficients formedQuadratic a b c applied to t line 35 source
Source documentation
The formed quadratic evaluated at t, i.e. the quadratic with coefficients formedQuadratic a b c applied to t.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: HAdd.hAdd, HMul.hMul, Prod.fst, Prod, Imo1959P3.formedQuadratic, HPow.hPow, Nat, OfNat.ofNat, Prod.snd
9 instances, coercions and auxiliary constants left out.
```
def formedQuadraticEval (a b c : ℝ) (t : ℝ) : ℝ :=
…
```
lemma Imo1959P3.sq_b_mul_cos Squaring the rearranged equation b * cos x = -(a * cos x ^ 2 + c) eliminates the odd power of cos x line 43 source
Source documentation
Squaring the rearranged equation b * cos x = -(a * cos x ^ 2 + c) eliminates the odd power of cos x.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, HAdd.hAdd, HMul.hMul, HPow.hPow, Nat, Real.cos, OfNat.ofNat
Proof uses: Neg.neg, Mathlib.Tactic.Linarith.eq_of_not_lt_of_not_gt, Not.intro, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, HSub.hSub, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Meta.NormNum.instAddMonoidWithOne, Mathlib.Tactic.Ring.Common.neg_congr, Mathlib.Tactic.Ring.Common.sub_congr, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.atom_pf and 59 more
70 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
lemma sq_b_mul_cos (a b c x : ℝ) (h : a * cos x ^ 2 + b * cos x + c = 0) :
    b ^ 2 * cos x ^ 2 = (a * cos x ^ 2 + c) ^ 2 := by
…
```
Mentions: h
lemma Imo1959P3.formedQuadratic_cos_two_mul The formed quadratic evaluated at cos (2 * x), expressed back in terms of cos x via cos (2 * x) = 2 * cos x ^ 2 - 1 line 51 source
Source documentation
The formed quadratic evaluated at cos (2 * x), expressed back in terms of cos x via cos (2 * x) = 2 * cos x ^ 2 - 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, Imo1959P3.formedQuadraticEval, Real.cos, HMul.hMul, OfNat.ofNat, Nat, HSub.hSub, HPow.hPow, HAdd.hAdd
Proof uses: id, Prod.fst, Prod, Imo1959P3.formedQuadratic, Prod.snd, Eq.mpr, Prod.mk, congrArg, Real.cos_two_mul, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Meta.NormNum.instAddMonoidWithOne, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.Common.atom_pf and 52 more
61 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
lemma formedQuadratic_cos_two_mul (a b c x : ℝ) :
    formedQuadraticEval a b c (cos (2 * x)) =
      4 * ((a * cos x ^ 2 + c) ^ 2 - b ^ 2 * cos x ^ 2) := by
…
```
Mentions: Imo1959P3.formedQuadraticEval
problem Imo1959P3.imo1959_p3 line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, HAdd.hAdd, HMul.hMul, HPow.hPow, Nat, Real.cos, OfNat.ofNat, Imo1959P3.formedQuadraticEval
Proof uses: Eq.mpr, HSub.hSub, id, congrArg, Imo1959P3.formedQuadratic_cos_two_mul, Imo1959P3.sq_b_mul_cos, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.mul_congr, Nat.rawCast, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Mathlib.Tactic.Ring.Common.sub_congr, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.symm, Mathlib.Tactic.Ring.Common.pow_add and 48 more
57 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
problem Imo1959P3.imo1959_p3_comparison For a = 4, b = 2, c = -1 the formed equation in cos 2x is four times the original equation in cos x: the two equations are the same line 67 source
Source documentation
For a = 4, b = 2, c = -1 the formed equation in cos 2x is four times the original equation in cos x: the two equations are the same.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, Imo1959P3.formedQuadraticEval, OfNat.ofNat, Nat, Neg.neg, HMul.hMul, HSub.hSub, HAdd.hAdd, HPow.hPow
Proof uses: id, Prod.fst, Prod, Imo1959P3.formedQuadratic, Prod.snd, Mathlib.Tactic.Ring.of_eq, Prod.mk, Int.rawCast, inferInstance, Ring, Int.negOfNat, Nat.rawCast, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Mathlib.Tactic.Ring.Common.pow_add and 49 more
65 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
problem imo1959_p3_comparison (t : ℝ) :
    formedQuadraticEval 4 2 (-1) t = 4 * (4 * t ^ 2 + 2 * t - 1) := by
…
```
Mentions: Imo1959P3.formedQuadraticEval