| /- |
| Copyright (c) 2026 Humanize/Codex. All rights reserved. |
| Released under Apache 2.0 license as described in the file LICENSE. |
| Authors: Humanize/Codex |
| -/ |
| import Mathlib |
|
|
| /-! |
| Algebraic answer claims for the IPhO 2025 problems. The declarations encode the |
| official closed-form answers and numerical answer intervals from the source rows. |
| -/ |
|
|
| namespace Ipho20212023Archon.Problems.Y2025 |
|
|
| noncomputable section |
|
|
| abbrev RangeAnswer := ℝ × ℝ |
|
|
| inductive Choice |
| | A |
| | B |
| | C |
| | D |
| deriving DecidableEq, Repr |
|
|
| structure ExperimentOutcome where |
| behavior : Choice |
| hstar : Option RangeAnswer |
| fmax : RangeAnswer |
|
|
| def sq (x : ℝ) : ℝ := x ^ 2 |
|
|
| def cube (x : ℝ) : ℝ := x ^ 3 |
|
|
| def fourth (x : ℝ) : ℝ := x ^ 4 |
|
|
| /- Problem 1, Hydrogen and galaxies. -/ |
|
|
| section Problem1 |
|
|
| def bohrCircularVelocity (e eps0 me r : ℝ) : ℝ := |
| Real.sqrt (sq e / (4 * Real.pi * eps0 * me * r)) |
|
|
| theorem IPhO_2025_1_A_1 (e eps0 me r : ℝ) : |
| bohrCircularVelocity e eps0 me r = |
| Real.sqrt (sq e / (4 * Real.pi * eps0 * me * r)) := by |
| sorry |
|
|
| def bohrOrbitRadius (hbar n alpha me c : ℝ) : ℝ := |
| sq n * hbar / (alpha * me * c) |
|
|
| def bohrRadius (hbar alpha me c : ℝ) : ℝ := |
| hbar / (alpha * me * c) |
|
|
| def bohrRadiusNumerical : ℝ := (531 : ℝ) / 100 * (10 : ℝ) ^ (-11 : ℤ) |
|
|
| def bohrGroundVelocity (alpha c : ℝ) : ℝ := alpha * c |
|
|
| theorem IPhO_2025_1_A_2 (hbar n alpha me c : ℝ) : |
| (bohrOrbitRadius hbar n alpha me c, bohrRadius hbar alpha me c, |
| bohrRadiusNumerical, bohrGroundVelocity alpha c) = |
| (sq n * hbar / (alpha * me * c), hbar / (alpha * me * c), |
| (531 : ℝ) / 100 * (10 : ℝ) ^ (-11 : ℤ), alpha * c) := by |
| sorry |
|
|
| def bohrOrbitEnergy (e eps0 n r1 : ℝ) : ℝ := |
| -sq e / (8 * Real.pi * eps0 * sq n * r1) |
|
|
| def bohrGroundEnergy (alpha me c : ℝ) : ℝ := |
| -((1 : ℝ) / 2) * sq alpha * me * sq c |
|
|
| def bohrGroundEnergyEV : ℝ := -((136 : ℝ) / 10) |
|
|
| theorem IPhO_2025_1_A_3 (e eps0 n r1 alpha me c : ℝ) : |
| (bohrOrbitEnergy e eps0 n r1, bohrGroundEnergy alpha me c, bohrGroundEnergyEV) = |
| (-sq e / (8 * Real.pi * eps0 * sq n * r1), |
| -((1 : ℝ) / 2) * sq alpha * me * sq c, -((136 : ℝ) / 10)) := by |
| sorry |
|
|
| def orbitMagneticField (mu0 e alpha c r1 : ℝ) : ℝ := |
| mu0 * e * alpha * c / (4 * Real.pi * sq r1) |
|
|
| theorem IPhO_2025_1_A_4 (mu0 e alpha c r1 : ℝ) : |
| orbitMagneticField mu0 e alpha c r1 = |
| mu0 * e * alpha * c / (4 * Real.pi * sq r1) := by |
| sorry |
|
|
| def fineStructureEnergyGap (alpha E1 : ℝ) : ℝ := -4 * sq alpha * E1 |
|
|
| def hyperfineWavelength (h c me mp alpha E1 : ℝ) : ℝ := |
| -(h * c) / (((372 : ℝ) / 100) * (me / mp) * 4 * sq alpha * E1) |
|
|
| def hyperfineWavelengthCm : ℝ := 21 |
|
|
| theorem IPhO_2025_1_A_5 (h c me mp alpha E1 : ℝ) : |
| (fineStructureEnergyGap alpha E1, hyperfineWavelength h c me mp alpha E1, |
| hyperfineWavelengthCm) = |
| (-4 * sq alpha * E1, |
| -(h * c) / (((372 : ℝ) / 100) * (me / mp) * 4 * sq alpha * E1), 21) := by |
| sorry |
|
|
| def circularOrbitSpeedFromPotential (r dphi_dr : ℝ) : ℝ := |
| Real.sqrt (r * dphi_dr) |
|
|
| theorem IPhO_2025_1_B_1 (r dphi_dr : ℝ) : |
| circularOrbitSpeedFromPotential r dphi_dr = Real.sqrt (r * dphi_dr) := by |
| sorry |
|
|
| def ngc6946BulgeMassRange : RangeAnswer := |
| ((675 : ℝ) / 100 * (10 : ℝ) ^ 8, (1125 : ℝ) / 100 * (10 : ℝ) ^ 8) |
|
|
| theorem IPhO_2025_1_B_2 : |
| ngc6946BulgeMassRange = |
| ((675 : ℝ) / 100 * (10 : ℝ) ^ 8, (1125 : ℝ) / 100 * (10 : ℝ) ^ 8) := by |
| sorry |
|
|
| def haloVelocityK1 (Cm G : ℝ) : ℝ := 4 * Real.pi * Cm * G |
|
|
| def haloVelocityK2 (Cm G rm : ℝ) : ℝ := 4 * Real.pi * Cm * G * rm |
|
|
| def haloInteriorMass (Cm r rm : ℝ) : ℝ := |
| 4 * Real.pi * Cm * (r - rm * Real.arctan (r / rm)) |
|
|
| def haloGravityMagnitude (Cm r rm : ℝ) : ℝ := |
| 4 * Real.pi * Cm * (r - rm * Real.arctan (r / rm)) / sq r |
|
|
| def haloVelocityProfile (Cm G r rm : ℝ) : ℝ := |
| Real.sqrt (haloVelocityK1 Cm G - haloVelocityK2 Cm G rm * Real.arctan (r / rm) / r) |
|
|
| def haloVelocitySmallRadius (Cm G r rm : ℝ) : ℝ := |
| Real.sqrt (4 * Real.pi * Cm * G * sq r / (3 * sq rm)) |
|
|
| def haloVelocityLargeRadius (Cm G : ℝ) : ℝ := |
| Real.sqrt (4 * Real.pi * Cm * G) |
|
|
| def haloMassLargeRadius (Cm r : ℝ) : ℝ := 4 * Real.pi * Cm * r |
|
|
| def ngc6946VisibleMassRange : RangeAnswer := |
| (Real.rpow 10 ((21 : ℝ) / 2), Real.rpow 10 ((23 : ℝ) / 2)) |
|
|
| theorem IPhO_2025_1_B_3 (Cm G r rm : ℝ) : |
| (haloVelocityK1 Cm G, haloVelocityK2 Cm G rm, haloInteriorMass Cm r rm, |
| haloGravityMagnitude Cm r rm, haloVelocityProfile Cm G r rm, |
| haloVelocitySmallRadius Cm G r rm, haloVelocityLargeRadius Cm G, |
| haloMassLargeRadius Cm r, ngc6946VisibleMassRange) = |
| (4 * Real.pi * Cm * G, 4 * Real.pi * Cm * G * rm, |
| 4 * Real.pi * Cm * (r - rm * Real.arctan (r / rm)), |
| 4 * Real.pi * Cm * (r - rm * Real.arctan (r / rm)) / sq r, |
| Real.sqrt (4 * Real.pi * Cm * G - 4 * Real.pi * Cm * G * rm * |
| Real.arctan (r / rm) / r), |
| Real.sqrt (4 * Real.pi * Cm * G * sq r / (3 * sq rm)), |
| Real.sqrt (4 * Real.pi * Cm * G), 4 * Real.pi * Cm * r, |
| (Real.rpow 10 ((21 : ℝ) / 2), Real.rpow 10 ((23 : ℝ) / 2))) := by |
| sorry |
|
|
| def verticalGalacticAcceleration (z z0 phi0 r r0 : ℝ) : ℝ := |
| 2 * z / sq z0 * phi0 * Real.log (r / r0) * Real.exp (-(sq (z / z0))) |
|
|
| def verticalSmallOscillationFrequency (phi0 z0 r r0 : ℝ) : ℝ := |
| Real.sqrt (2 * phi0 / sq z0 * |Real.log (r / r0)|) |
|
|
| theorem IPhO_2025_1_C_1 (z z0 phi0 r r0 : ℝ) : |
| (verticalGalacticAcceleration z z0 phi0 r r0, |
| verticalSmallOscillationFrequency phi0 z0 r r0) = |
| (2 * z / sq z0 * phi0 * Real.log (r / r0) * Real.exp (-(sq (z / z0))), |
| Real.sqrt (2 * phi0 / sq z0 * |Real.log (r / r0)|)) := by |
| sorry |
|
|
| inductive RadiusRegime |
| | muchSmaller |
| | muchLarger |
| deriving DecidableEq, Repr |
|
|
| def flatRotationRegime : RadiusRegime := RadiusRegime.muchLarger |
|
|
| def flatPotentialScale (Cm G : ℝ) : ℝ := 4 * Real.pi * Cm * G |
|
|
| def flatCircularSpeed (phi0 : ℝ) : ℝ := Real.sqrt phi0 |
|
|
| theorem IPhO_2025_1_C_2 (Cm G phi0 : ℝ) : |
| (flatRotationRegime, flatPotentialScale Cm G, flatCircularSpeed phi0) = |
| (RadiusRegime.muchLarger, 4 * Real.pi * Cm * G, Real.sqrt phi0) := by |
| sorry |
|
|
| def radialVelocityEarthSource (vSun RSun R ell : ℝ) : ℝ := |
| vSun * (RSun / R - 1) * Real.sin ell |
|
|
| def sourceGalactocentricRadius (RSun vSun ell vr : ℝ) : ℝ := |
| RSun / (1 + vr / (vSun * Real.sin ell)) |
|
|
| theorem IPhO_2025_1_C_3 (vSun RSun R ell vr : ℝ) : |
| (radialVelocityEarthSource vSun RSun R ell, sourceGalactocentricRadius RSun vSun ell vr) = |
| (vSun * (RSun / R - 1) * Real.sin ell, |
| RSun / (1 + vr / (vSun * Real.sin ell))) := by |
| sorry |
|
|
| def sourceVelocityRange1 : RangeAnswer := ((632 : ℝ) / 100, (634 : ℝ) / 100) |
|
|
| def sourceVelocityRange2 : RangeAnswer := ((316 : ℝ) / 10, (318 : ℝ) / 10) |
|
|
| def sourceVelocityRange3 : RangeAnswer := ((548 : ℝ) / 10, 55) |
|
|
| def sourceRadiusRange1 : RangeAnswer := ((94 : ℝ) / 100, (96 : ℝ) / 100) |
|
|
| def sourceRadiusRange2 : RangeAnswer := ((77 : ℝ) / 100, (79 : ℝ) / 100) |
|
|
| def sourceRadiusRange3 : RangeAnswer := ((66 : ℝ) / 100, (68 : ℝ) / 100) |
|
|
| theorem IPhO_2025_1_C_4 : |
| (sourceVelocityRange1, sourceVelocityRange2, sourceVelocityRange3, |
| sourceRadiusRange1, sourceRadiusRange2, sourceRadiusRange3) = |
| (((632 : ℝ) / 100, (634 : ℝ) / 100), ((316 : ℝ) / 10, (318 : ℝ) / 10), |
| ((548 : ℝ) / 10, 55), ((94 : ℝ) / 100, (96 : ℝ) / 100), |
| ((77 : ℝ) / 100, (79 : ℝ) / 100), ((66 : ℝ) / 100, (68 : ℝ) / 100)) := by |
| sorry |
|
|
| def newtonTullyFischerEta (R G : ℝ) : ℝ := R / G |
|
|
| def newtonTullyFischerGamma : ℝ := 2 |
|
|
| def observedTullyFischerGammaRange : RangeAnswer := ((35 : ℝ) / 10, 4) |
|
|
| theorem IPhO_2025_1_D_1 (R G : ℝ) : |
| (newtonTullyFischerEta R G, newtonTullyFischerGamma, |
| observedTullyFischerGammaRange) = |
| (R / G, 2, ((35 : ℝ) / 10, 4)) := by |
| sorry |
|
|
| def outerGalaxyAcceleration (vc R : ℝ) : ℝ := sq vc / R |
|
|
| def outerGalaxyAccelerationRange : RangeAnswer := |
| (Real.rpow 10 (-((21 : ℝ) / 2)), Real.rpow 10 (-((19 : ℝ) / 2))) |
|
|
| theorem IPhO_2025_1_D_2 (vc R : ℝ) : |
| (outerGalaxyAcceleration vc R, outerGalaxyAccelerationRange) = |
| (sq vc / R, (Real.rpow 10 (-((21 : ℝ) / 2)), Real.rpow 10 (-((19 : ℝ) / 2)))) := by |
| sorry |
|
|
| def mondTullyFischerGamma : ℝ := 4 |
|
|
| def mondAccelerationScale (vcInf G Mtot : ℝ) : ℝ := fourth vcInf / (G * Mtot) |
|
|
| def mondAccelerationScaleRange : RangeAnswer := |
| (Real.rpow 10 (-((21 : ℝ) / 2)), Real.rpow 10 (-((19 : ℝ) / 2))) |
|
|
| theorem IPhO_2025_1_D_3 (vcInf G Mtot : ℝ) : |
| (mondTullyFischerGamma, mondAccelerationScale vcInf G Mtot, mondAccelerationScaleRange) = |
| (4, fourth vcInf / (G * Mtot), |
| (Real.rpow 10 (-((21 : ℝ) / 2)), Real.rpow 10 (-((19 : ℝ) / 2)))) := by |
| sorry |
|
|
| def mondVelocityExterior (G M a0 r : ℝ) : ℝ := |
| Real.sqrt (G * M / (2 * r) * (1 + Real.sqrt (1 + 4 * a0 * sq r / (G * M)))) |
|
|
| def mondVelocityInterior (G M a0 r Rb : ℝ) : ℝ := |
| Real.sqrt |
| (G * M / (2 * r) * cube (r / Rb) * |
| (1 + Real.sqrt (1 + 4 * a0 * sq r / (G * M) * cube (Rb / r)))) |
|
|
| theorem IPhO_2025_1_D_4 (G M a0 r Rb : ℝ) : |
| (mondVelocityExterior G M a0 r, mondVelocityInterior G M a0 r Rb) = |
| (Real.sqrt (G * M / (2 * r) * (1 + Real.sqrt (1 + 4 * a0 * sq r / (G * M)))), |
| Real.sqrt |
| (G * M / (2 * r) * cube (r / Rb) * |
| (1 + Real.sqrt (1 + 4 * a0 * sq r / (G * M) * cube (Rb / r))))) := by |
| sorry |
|
|
| end Problem1 |
|
|
| /- Problem 2, The water column. -/ |
|
|
| section Problem2 |
|
|
| def waterPressureAtTubeTop (P0 rho g h : ℝ) : ℝ := P0 - rho * g * h |
|
|
| def tubeHoldingForceScalar (m rho S h g : ℝ) : ℝ := (m + rho * S * h) * g |
|
|
| theorem IPhO_2025_2_A_1 (P0 rho m S h g : ℝ) : |
| (waterPressureAtTubeTop P0 rho g h, tubeHoldingForceScalar m rho S h g) = |
| (P0 - rho * g * h, (m + rho * S * h) * g) := by |
| sorry |
|
|
| def tubeExperimentOutcomes : List ExperimentOutcome := |
| [ { behavior := Choice.A, hstar := none, fmax := ((146 : ℝ) / 10, 15) }, |
| { behavior := Choice.A, hstar := none, fmax := (14, (145 : ℝ) / 10) }, |
| { behavior := Choice.B, hstar := some (2, (22 : ℝ) / 10), |
| fmax := (5, (52 : ℝ) / 10) } ] |
|
|
| theorem IPhO_2025_2_A_2 : |
| tubeExperimentOutcomes = |
| [ { behavior := Choice.A, hstar := none, fmax := ((146 : ℝ) / 10, 15) }, |
| { behavior := Choice.A, hstar := none, fmax := (14, (145 : ℝ) / 10) }, |
| { behavior := Choice.B, hstar := some (2, (22 : ℝ) / 10), |
| fmax := (5, (52 : ℝ) / 10) } ] := by |
| sorry |
|
|
| def vaporPressureRelativeError (Psat P0 m g S : ℝ) : ℝ := |
| Psat / (P0 + m * g / S) |
|
|
| def vaporPressureRelativeErrorRange : RangeAnswer := |
| ((10 : ℝ) ^ (-6 : ℤ), 2 * (10 : ℝ) ^ (-6 : ℤ)) |
|
|
| theorem IPhO_2025_2_A_3 (Psat P0 m g S : ℝ) : |
| (vaporPressureRelativeError Psat P0 m g S, vaporPressureRelativeErrorRange) = |
| (Psat / (P0 + m * g / S), ((10 : ℝ) ^ (-6 : ℤ), 2 * (10 : ℝ) ^ (-6 : ℤ))) := by |
| sorry |
|
|
| def addedMercuryMass (rho St ht Sb zl : ℝ) : ℝ := |
| rho * (St * ht + Sb * (zl - ht)) |
|
|
| theorem IPhO_2025_2_B_1 (rho St ht Sb zl : ℝ) : |
| addedMercuryMass rho St ht Sb zl = rho * (St * ht + Sb * (zl - ht)) := by |
| sorry |
|
|
| def addedMassSegmentSlopes (rho Sb St : ℝ) : ℝ × ℝ × ℝ × ℝ := |
| (rho * Sb, rho * St, -rho * (Sb - St), 0) |
|
|
| def addedMassAngularPoints (zStar Hb : ℝ) : ℝ × ℝ × ℝ := |
| (0, zStar - Hb, zStar) |
|
|
| theorem IPhO_2025_2_B_2 (rho Sb St zStar Hb : ℝ) : |
| (addedMassSegmentSlopes rho Sb St, addedMassAngularPoints zStar Hb) = |
| ((rho * Sb, rho * St, -rho * (Sb - St), 0), (0, zStar - Hb, zStar)) := by |
| sorry |
|
|
| def addedMassAmplitude (Sb A g : ℝ) : ℝ := Sb * A / g |
|
|
| def addedMassAmplitudeRange : RangeAnswer := (1, (11 : ℝ) / 10) |
|
|
| theorem IPhO_2025_2_B_3 (Sb A g : ℝ) : |
| (addedMassAmplitude Sb A g, addedMassAmplitudeRange) = |
| (Sb * A / g, (1, (11 : ℝ) / 10)) := by |
| sorry |
|
|
| def restThresholdXi : ℝ := 2 |
|
|
| theorem IPhO_2025_2_C_1 : |
| restThresholdXi = 2 := by |
| sorry |
|
|
| def cableTensionScalar (Sc Sb St rho g X : ℝ) : ℝ := |
| 4 * Sc * (Sb - St) / (Sb + Sc - St) * rho * g * X |
|
|
| def cableTensionApproxScalar (Sc Sb rho g X : ℝ) : ℝ := |
| 4 * Sb * Sc / (Sb + Sc) * rho * g * X |
|
|
| theorem IPhO_2025_2_C_2 (Sc Sb St rho g X : ℝ) : |
| (cableTensionScalar Sc Sb St rho g X, cableTensionApproxScalar Sc Sb rho g X) = |
| (4 * Sc * (Sb - St) / (Sb + Sc - St) * rho * g * X, |
| 4 * Sb * Sc / (Sb + Sc) * rho * g * X) := by |
| sorry |
|
|
| def firstRegimeCondition (xi lambda : ℝ) : Prop := xi + 2 * lambda > 2 |
|
|
| def secondRegimeCondition (xi lambda : ℝ) : Prop := xi + 2 * lambda < 2 |
|
|
| theorem IPhO_2025_2_C_3 (xi lambda : ℝ) : |
| (firstRegimeCondition xi lambda ↔ xi + 2 * lambda > 2) ∧ |
| (secondRegimeCondition xi lambda ↔ xi + 2 * lambda < 2) := by |
| sorry |
|
|
| def springForceLimit (A Sc : ℝ) : ℝ := A * Sc / 2 |
|
|
| def displacementLimit (A rho g : ℝ) : ℝ := A / (4 * rho * g) |
|
|
| def maximumRecoverableWork (A Sc rho g : ℝ) : ℝ := sq A * Sc / (2 * rho * g) |
|
|
| def maximumRecoverableWorkRange : RangeAnswer := (19, 21) |
|
|
| theorem IPhO_2025_2_C_4 (A Sc rho g : ℝ) : |
| (springForceLimit A Sc, displacementLimit A rho g, maximumRecoverableWork A Sc rho g, |
| maximumRecoverableWorkRange) = |
| (A * Sc / 2, A / (4 * rho * g), sq A * Sc / (2 * rho * g), (19, 21)) := by |
| sorry |
|
|
| def practicalMaximumWork (Sc A rho g : ℝ) : ℝ := Sc * sq A / (rho * g) |
|
|
| def recoverableToPracticalWorkRatio : ℝ := (1 : ℝ) / 2 |
|
|
| theorem IPhO_2025_2_C_5 (Sc A rho g : ℝ) : |
| (practicalMaximumWork Sc A rho g, recoverableToPracticalWorkRatio) = |
| (Sc * sq A / (rho * g), (1 : ℝ) / 2) := by |
| sorry |
|
|
| end Problem2 |
|
|
| /- Problem 3, Champagne. -/ |
|
|
| section Problem3 |
|
|
| def bubblePressure (P0 sigma a : ℝ) : ℝ := P0 + 2 * sigma / a |
|
|
| theorem IPhO_2025_3_A_1 (P0 sigma a : ℝ) : |
| bubblePressure P0 sigma a = P0 + 2 * sigma / a := by |
| sorry |
|
|
| def criticalBubbleRadius (sigma P0 cl c0 : ℝ) : ℝ := |
| 2 * sigma / (P0 * (cl / c0 - 1)) |
|
|
| def criticalBubbleRadiusMicrometer : ℝ := (3 : ℝ) / 10 |
|
|
| theorem IPhO_2025_3_A_2 (sigma P0 cl c0 : ℝ) : |
| (criticalBubbleRadius sigma P0 cl c0, criticalBubbleRadiusMicrometer) = |
| (2 * sigma / (P0 * (cl / c0 - 1)), (3 : ℝ) / 10) := by |
| sorry |
|
|
| def bubbleMoles (a P0 R T0 : ℝ) : ℝ := |
| (4 : ℝ) / 3 * Real.pi * cube a * P0 / (R * T0) |
|
|
| def diffusionModelRadiusSquared (a0 D R T0 P0 cl c0 t : ℝ) : ℝ := |
| sq a0 + (2 * D * R * T0 / P0) * (cl - c0) * t |
|
|
| def transferModelRadius (a0 K R T0 P0 cl c0 t : ℝ) : ℝ := |
| a0 + (K * R * T0 / P0) * (cl - c0) * t |
|
|
| def selectedBubbleGrowthModel : Choice := Choice.B |
|
|
| def bubbleGrowthKRange : RangeAnswer := |
| ((9 : ℝ) / 10 * (10 : ℝ) ^ (-4 : ℤ), (11 : ℝ) / 10 * (10 : ℝ) ^ (-4 : ℤ)) |
|
|
| theorem IPhO_2025_3_A_3 (a P0 R T0 a0 D K cl c0 t : ℝ) : |
| (bubbleMoles a P0 R T0, diffusionModelRadiusSquared a0 D R T0 P0 cl c0 t, |
| transferModelRadius a0 K R T0 P0 cl c0 t, selectedBubbleGrowthModel, |
| bubbleGrowthKRange) = |
| ((4 : ℝ) / 3 * Real.pi * cube a * P0 / (R * T0), |
| sq a0 + (2 * D * R * T0 / P0) * (cl - c0) * t, |
| a0 + (K * R * T0 / P0) * (cl - c0) * t, Choice.B, |
| ((9 : ℝ) / 10 * (10 : ℝ) ^ (-4 : ℤ), |
| (11 : ℝ) / 10 * (10 : ℝ) ^ (-4 : ℤ))) := by |
| sorry |
|
|
| def bubbleBuoyancyForce (a rhoL g0 : ℝ) : ℝ := (4 : ℝ) / 3 * Real.pi * cube a * rhoL * g0 |
|
|
| def bubbleStokesDrag (eta a v : ℝ) : ℝ := 6 * Real.pi * eta * a * v |
|
|
| def bubbleTerminalVelocity (eta a rhoL g0 : ℝ) : ℝ := |
| 2 / (9 * eta) * sq a * rhoL * g0 |
|
|
| def champagneViscosityRange : RangeAnswer := |
| ((10 : ℝ) ^ (-3 : ℤ), 4 * (10 : ℝ) ^ (-3 : ℤ)) |
|
|
| theorem IPhO_2025_3_A_4 (a rhoL g0 eta v : ℝ) : |
| (bubbleBuoyancyForce a rhoL g0, bubbleStokesDrag eta a v, |
| bubbleTerminalVelocity eta a rhoL g0, champagneViscosityRange) = |
| ((4 : ℝ) / 3 * Real.pi * cube a * rhoL * g0, 6 * Real.pi * eta * a * v, |
| 2 / (9 * eta) * sq a * rhoL * g0, |
| ((10 : ℝ) ^ (-3 : ℤ), 4 * (10 : ℝ) ^ (-3 : ℤ))) := by |
| sorry |
|
|
| def freeSurfaceBubbleRadius (qa eta H rhoL g0 : ℝ) : ℝ := |
| Real.rpow (27 * qa * eta * H / (2 * rhoL * g0)) ((1 : ℝ) / 3) |
|
|
| def freeSurfaceBubbleRadiusMmRange : RangeAnswer := ((36 : ℝ) / 100, (49 : ℝ) / 100) |
|
|
| theorem IPhO_2025_3_A_5 (qa eta H rhoL g0 : ℝ) : |
| (freeSurfaceBubbleRadius qa eta H rhoL g0, freeSurfaceBubbleRadiusMmRange) = |
| (Real.rpow (27 * qa * eta * H / (2 * rhoL * g0)) ((1 : ℝ) / 3), |
| ((36 : ℝ) / 100, (49 : ℝ) / 100)) := by |
| sorry |
|
|
| def concentrationDecayCoefficient (Nb fb eta K H rhoL g0 Vl : ℝ) : ℝ := |
| 18 * Real.pi * Nb * fb * eta * K * H / (rhoL * g0 * Vl) |
|
|
| def concentrationDecayResidual (dc_dt cL c0 Nb fb eta K H rhoL g0 Vl : ℝ) : ℝ := |
| dc_dt + concentrationDecayCoefficient Nb fb eta K H rhoL g0 Vl * (cL - c0) |
|
|
| def concentrationDecayEquation (dc_dt cL c0 Nb fb eta K H rhoL g0 Vl : ℝ) : Prop := |
| concentrationDecayResidual dc_dt cL c0 Nb fb eta K H rhoL g0 Vl = 0 |
|
|
| def concentrationDecayTime (rhoL g0 Vl Nb fb eta K H : ℝ) : ℝ := |
| rhoL * g0 * Vl / (18 * Real.pi * Nb * fb * eta * K * H) |
|
|
| theorem IPhO_2025_3_A_6 (dc_dt cL c0 Nb fb eta K H rhoL g0 Vl : ℝ) : |
| (concentrationDecayEquation dc_dt cL c0 Nb fb eta K H rhoL g0 Vl ↔ |
| dc_dt + 18 * Real.pi * Nb * fb * eta * K * H / (rhoL * g0 * Vl) * |
| (cL - c0) = 0) ∧ |
| concentrationDecayTime rhoL g0 Vl Nb fb eta K H = |
| rhoL * g0 * Vl / (18 * Real.pi * Nb * fb * eta * K * H) := by |
| sorry |
|
|
| def filmRetractionSpeed (sigma rhoL h : ℝ) : ℝ := |
| Real.sqrt (2 * sigma / (rhoL * h)) |
|
|
| theorem IPhO_2025_3_B_1 (sigma rhoL h : ℝ) : |
| filmRetractionSpeed sigma rhoL h = Real.sqrt (2 * sigma / (rhoL * h)) := by |
| sorry |
|
|
| def pistonOscillationFrequency (S P0 gamma mp V0 : ℝ) : ℝ := |
| 1 / (2 * Real.pi) * Real.sqrt (sq S * P0 * gamma / (mp * V0)) |
|
|
| theorem IPhO_2025_3_B_2 (S P0 gamma mp V0 : ℝ) : |
| pistonOscillationFrequency S P0 gamma mp V0 = |
| 1 / (2 * Real.pi) * Real.sqrt (sq S * P0 * gamma / (mp * V0)) := by |
| sorry |
|
|
| def champagneFilmRadiusMmRange : RangeAnswer := ((5 : ℝ) / 10, (6 : ℝ) / 10) |
|
|
| def champagneFilmThicknessMicrometer : ℝ := (37 : ℝ) / 10 |
|
|
| theorem IPhO_2025_3_B_3 : |
| (champagneFilmRadiusMmRange, champagneFilmThicknessMicrometer) = |
| (((5 : ℝ) / 10, (6 : ℝ) / 10), (37 : ℝ) / 10) := by |
| sorry |
|
|
| def initialBottlePressure6CBar : ℝ := (481 : ℝ) / 100 |
|
|
| def initialBottlePressure20CBar : ℝ := (776 : ℝ) / 100 |
|
|
| theorem IPhO_2025_3_C_1 : |
| (initialBottlePressure6CBar, initialBottlePressure20CBar) = |
| ((481 : ℝ) / 100, (776 : ℝ) / 100) := by |
| sorry |
|
|
| def finalGasTemperature6C : ℝ := (1953 : ℝ) / 10 |
|
|
| def finalGasTemperature20C : ℝ := (1843 : ℝ) / 10 |
|
|
| def trueFogStatements : List Choice := [Choice.A, Choice.D] |
|
|
| theorem IPhO_2025_3_C_2 : |
| (finalGasTemperature6C, finalGasTemperature20C, trueFogStatements) = |
| ((1953 : ℝ) / 10, (1843 : ℝ) / 10, [Choice.A, Choice.D]) := by |
| sorry |
|
|
| def corkRiseHeight6C : ℝ := (77 : ℝ) / 10 |
|
|
| theorem IPhO_2025_3_C_3 : |
| corkRiseHeight6C = (77 : ℝ) / 10 := by |
| sorry |
|
|
| end Problem3 |
|
|
| end |
|
|
| end Ipho20212023Archon.Problems.Y2025 |
|
|
