| /- |
| 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 2022 theoretical problems. |
| The definitions below deliberately encode the official solution formulas, |
| while omitting large physical models such as diagrams, material response, |
| and graphical construction from thermodynamic charts. |
| -/ |
|
|
| namespace Ipho20212023Archon.Problems.Y2022 |
|
|
| noncomputable section |
|
|
| def cylinderDipoleMoment (diameter height magnetization : Real) : Real := |
| Real.pi / 4 * diameter ^ 2 * height * magnetization |
|
|
| def coaxialDipoleEnergy (mu0 moment separation : Real) : Real := |
| -mu0 * moment ^ 2 / (2 * Real.pi * separation ^ 3) |
|
|
| def coaxialDipoleForce (mu0 moment separation : Real) : Real := |
| 3 * mu0 * moment ^ 2 / (2 * Real.pi * separation ^ 4) |
|
|
| theorem IPhO_2022_1_A_1 |
| (mu0 diameter height magnetization separation : Real) : |
| coaxialDipoleForce mu0 (cylinderDipoleMoment diameter height magnetization) separation = |
| 3 * mu0 * (Real.pi / 4 * diameter ^ 2 * height * magnetization) ^ 2 / |
| (2 * Real.pi * separation ^ 4) := by |
| sorry |
|
|
| def ringDipoleMoment (diameter current : Real) : Real := |
| Real.pi / 4 * diameter ^ 2 * current |
|
|
| def equivalentRingCurrent (magnetization height : Real) : Real := |
| magnetization * height |
|
|
| theorem IPhO_2022_1_A_2 (diameter height magnetization : Real) : |
| ringDipoleMoment diameter (equivalentRingCurrent magnetization height) = |
| cylinderDipoleMoment diameter height magnetization := by |
| sorry |
|
|
| def parallelCurrentForce (mu0 current diameter separation : Real) : Real := |
| mu0 * current ^ 2 * diameter / (2 * separation) |
|
|
| theorem IPhO_2022_1_A_3 |
| (mu0 magnetization height diameter separation : Real) : |
| parallelCurrentForce mu0 (equivalentRingCurrent magnetization height) diameter separation = |
| mu0 * (magnetization * height) ^ 2 * diameter / (2 * separation) := by |
| sorry |
|
|
| def sphereMagnetMass (density diameter : Real) : Real := |
| Real.pi / 6 * density * diameter ^ 3 |
|
|
| def sphereDipoleMoment (magnetization diameter : Real) : Real := |
| Real.pi / 6 * magnetization * diameter ^ 3 |
|
|
| def chainMaximumLength (mu0 moment mass gravity diameter : Real) : Real := |
| mu0 * moment ^ 2 * Real.pi ^ 3 / (60 * mass * gravity * diameter ^ 3) |
|
|
| theorem IPhO_2022_1_A_4 (mu0 moment mass gravity diameter : Real) : |
| chainMaximumLength mu0 moment mass gravity diameter = |
| mu0 * moment ^ 2 * Real.pi ^ 3 / (60 * mass * gravity * diameter ^ 3) := by |
| sorry |
|
|
| def endpointChainField (mu0 magnetization diameter radius : Real) : Real := |
| magnetization * mu0 * diameter ^ 2 / (24 * radius ^ 2) |
|
|
| theorem IPhO_2022_1_A_5 (mu0 magnetization diameter radius : Real) : |
| endpointChainField mu0 magnetization diameter radius = |
| magnetization * mu0 * diameter ^ 2 / (24 * radius ^ 2) := by |
| sorry |
|
|
| def outsideFieldLineTangent (relativePermeability insideAngle : Real) : Real := |
| Real.tan insideAngle / relativePermeability |
|
|
| theorem IPhO_2022_1_B_1 |
| (relativePermeability insideAngle : Real) (hmu : relativePermeability ≠ 0) : |
| relativePermeability * outsideFieldLineTangent relativePermeability insideAngle = |
| Real.tan insideAngle := by |
| sorry |
|
|
| inductive PlateOrientation |
| | perpendicular |
| | parallel |
| deriving DecidableEq, Repr |
|
|
| def stablePlateOrientation : PlateOrientation := |
| PlateOrientation.perpendicular |
|
|
| def imagePlateNormalForce (mu0 moment diameter : Real) : Real := |
| 3 * mu0 * moment ^ 2 / (2 * Real.pi * diameter ^ 4) |
|
|
| theorem IPhO_2022_1_B_2 (mu0 moment diameter : Real) : |
| stablePlateOrientation = PlateOrientation.perpendicular |
| ∧ imagePlateNormalForce mu0 moment diameter = |
| 3 * mu0 * moment ^ 2 / (2 * Real.pi * diameter ^ 4) := by |
| sorry |
|
|
| def ferromagneticPlateForce (mu0 magnetization diameter : Real) : Real := |
| 3 * Real.pi / 32 * magnetization ^ 2 * mu0 * diameter ^ 2 |
|
|
| theorem IPhO_2022_1_B_3 (mu0 magnetization diameter : Real) : |
| ferromagneticPlateForce mu0 magnetization diameter = |
| 3 * Real.pi / 32 * magnetization ^ 2 * mu0 * diameter ^ 2 := by |
| sorry |
|
|
| inductive MagneticOrder |
| | ferromagnetic |
| | antiferromagnetic |
| deriving DecidableEq, Repr |
|
|
| def squareLatticeOrder : MagneticOrder := |
| MagneticOrder.antiferromagnetic |
|
|
| def squareLatticePullEnergy (mu0 moment diameter : Real) : Real := |
| 3 * mu0 * moment ^ 2 / (2 * Real.pi * diameter ^ 3) |
|
|
| theorem IPhO_2022_1_C_1 (mu0 moment diameter : Real) : |
| squareLatticeOrder = MagneticOrder.antiferromagnetic |
| ∧ squareLatticePullEnergy mu0 moment diameter = |
| 3 * mu0 * moment ^ 2 / (2 * Real.pi * diameter ^ 3) := by |
| sorry |
|
|
| def triangularLatticeOrder : MagneticOrder := |
| MagneticOrder.ferromagnetic |
|
|
| def triangularLatticePullEnergy (mu0 moment diameter : Real) : Real := |
| 3 * mu0 * moment ^ 2 / (4 * Real.pi * diameter ^ 3) |
|
|
| theorem IPhO_2022_1_C_2 (mu0 moment diameter : Real) : |
| triangularLatticeOrder = MagneticOrder.ferromagnetic |
| ∧ triangularLatticePullEnergy mu0 moment diameter = |
| 3 * mu0 * moment ^ 2 / (4 * Real.pi * diameter ^ 3) := by |
| sorry |
|
|
| def imageDiameter (objectDiameter focalLength objectDistance : Real) : Real := |
| objectDiameter * focalLength / objectDistance |
|
|
| theorem IPhO_2022_2_A_1 (objectDiameter focalLength objectDistance : Real) : |
| imageDiameter objectDiameter focalLength objectDistance = |
| objectDiameter * focalLength / objectDistance := by |
| sorry |
|
|
| def apertureDiameterFromArea (area : Real) : Real := |
| 2 * Real.sqrt (area / Real.pi) |
|
|
| def diffractionCentralDiameter (wavelength focalLength apertureDiameter : Real) : Real := |
| (244 / 100 : Real) * wavelength * focalLength / apertureDiameter |
|
|
| theorem IPhO_2022_2_A_2 (wavelength focalLength apertureDiameter : Real) : |
| diffractionCentralDiameter wavelength focalLength apertureDiameter = |
| (244 / 100 : Real) * wavelength * focalLength / apertureDiameter := by |
| sorry |
|
|
| def ccdTemperatureNoDiffraction |
| (mirrorArea focalLength starTemperature : Real) : Real := |
| (mirrorArea / (Real.pi * focalLength ^ 2)) ^ (1 / 4 : Real) * starTemperature |
|
|
| def ccdTemperatureWithDiffraction |
| (mirrorArea focalLength starTemperature areaFactor : Real) : Real := |
| (mirrorArea / (areaFactor * Real.pi * focalLength ^ 2)) ^ (1 / 4 : Real) * |
| starTemperature |
|
|
| theorem IPhO_2022_2_A_3 |
| (mirrorArea focalLength starTemperature areaFactor : Real) : |
| ccdTemperatureWithDiffraction mirrorArea focalLength starTemperature areaFactor = |
| (mirrorArea / (areaFactor * Real.pi * focalLength ^ 2)) ^ (1 / 4 : Real) * |
| starTemperature := by |
| sorry |
|
|
| def darkCurrentGraphSlope : Real := |
| ((3 : Real) - (-1)) / ((111 : Real) / 1000 - (151 : Real) / 1000) |
|
|
| def thermalSourceTemperatureFromSlope (slope : Real) : Real := |
| 6 * Real.log 10 * (-slope) |
|
|
| theorem IPhO_2022_2_B_1 : |
| darkCurrentGraphSlope = -100 |
| ∧ thermalSourceTemperatureFromSlope darkCurrentGraphSlope = 600 * Real.log 10 := by |
| sorry |
|
|
| def totalCountVariance |
| (readoutNoise darkCurrent photonRate exposureTime : Real) : Real := |
| readoutNoise ^ 2 + (darkCurrent + photonRate) * exposureTime |
|
|
| theorem IPhO_2022_2_B_2 |
| (readoutNoise darkCurrent photonRate exposureTime : Real) : |
| totalCountVariance readoutNoise darkCurrent photonRate exposureTime = |
| readoutNoise ^ 2 + (darkCurrent + photonRate) * exposureTime := by |
| sorry |
|
|
| def signalToNoiseTenEquation |
| (readoutNoise darkCurrent exposureTime photonCount : Real) : Prop := |
| photonCount ^ 2 = 100 * (readoutNoise ^ 2 + darkCurrent * exposureTime + photonCount) |
|
|
| def photonRateFromCount (photonCount exposureTime : Real) : Real := |
| photonCount / exposureTime |
|
|
| theorem IPhO_2022_2_B_3 |
| (readoutNoise darkCurrent exposureTime photonCount : Real) : |
| signalToNoiseTenEquation readoutNoise darkCurrent exposureTime photonCount = |
| (photonCount ^ 2 = |
| 100 * (readoutNoise ^ 2 + darkCurrent * exposureTime + photonCount)) := by |
| sorry |
|
|
| def photonIntensity |
| (photonRate photonEnergy mirrorArea : Real) : Real := |
| photonRate * photonEnergy / mirrorArea |
|
|
| theorem IPhO_2022_2_B_4 (photonRate photonEnergy mirrorArea : Real) : |
| photonIntensity photonRate photonEnergy mirrorArea = |
| photonRate * photonEnergy / mirrorArea := by |
| sorry |
|
|
| def sunshieldK (alpha beta : Real) : Real := |
| 1 |
| + 1 / (alpha + beta) |
| + alpha / (alpha + beta) ^ 2 |
| + alpha ^ 2 / (alpha + beta) ^ 3 |
| + alpha ^ 3 / (alpha + beta) ^ 4 |
| + alpha ^ 4 / (alpha + beta) ^ 4 |
|
|
| def firstSheetTemperature |
| (solarIntensity stefanBoltzmann alpha beta : Real) : Real := |
| (solarIntensity * (sunshieldK alpha beta - 1) / |
| (sunshieldK alpha beta * stefanBoltzmann)) ^ (1 / 4 : Real) |
|
|
| def fifthSheetTemperature |
| (solarIntensity stefanBoltzmann alpha beta : Real) : Real := |
| alpha / (alpha + beta) * |
| (solarIntensity / (sunshieldK alpha beta * stefanBoltzmann)) ^ (1 / 4 : Real) |
|
|
| theorem IPhO_2022_2_C_1 |
| (solarIntensity stefanBoltzmann alpha beta : Real) : |
| firstSheetTemperature solarIntensity stefanBoltzmann alpha beta = |
| (solarIntensity * (sunshieldK alpha beta - 1) / |
| (sunshieldK alpha beta * stefanBoltzmann)) ^ (1 / 4 : Real) |
| ∧ fifthSheetTemperature solarIntensity stefanBoltzmann alpha beta = |
| alpha / (alpha + beta) * |
| (solarIntensity / (sunshieldK alpha beta * stefanBoltzmann)) ^ |
| (1 / 4 : Real) := by |
| sorry |
|
|
| def sunshieldGapArea (separation sheetArea : Real) : Real := |
| 4 * separation * Real.sqrt sheetArea |
|
|
| def sunshieldAlphaEstimate (emissivity sheetArea gapArea : Real) : Real := |
| emissivity * sheetArea / (2 * emissivity * sheetArea + gapArea) |
|
|
| def sunshieldBetaPrimeEstimate (emissivity sheetArea gapArea : Real) : Real := |
| gapArea / (2 * emissivity * sheetArea + gapArea) |
|
|
| def sunshieldBetaEstimateFromAlpha (alpha : Real) : Real := |
| 5 / 3 - 10 / 3 * alpha |
|
|
| theorem IPhO_2022_2_C_2 (emissivity sheetArea gapArea alpha : Real) : |
| sunshieldAlphaEstimate emissivity sheetArea gapArea = |
| emissivity * sheetArea / (2 * emissivity * sheetArea + gapArea) |
| ∧ sunshieldBetaPrimeEstimate emissivity sheetArea gapArea = |
| gapArea / (2 * emissivity * sheetArea + gapArea) |
| ∧ sunshieldBetaEstimateFromAlpha alpha = 5 / 3 - 10 / 3 * alpha := by |
| sorry |
|
|
| def sunshieldNumericalTemperatureEstimate : Real × Real := |
| (380, 102) |
|
|
| theorem IPhO_2022_2_C_3 : |
| sunshieldNumericalTemperatureEstimate = ((380 : Real), (102 : Real)) := by |
| sorry |
|
|
| inductive Comparison |
| | less |
| | equal |
| | greater |
| | unknown |
| deriving DecidableEq, Repr |
|
|
| structure PlugStateComparisons where |
| pressure : Comparison |
| internalEnergy : Comparison |
| entropy : Comparison |
| volume : Comparison |
| temperature : Comparison |
| deriving Repr |
|
|
| def jouleThomsonComparisons : PlugStateComparisons := |
| { pressure := Comparison.greater |
| internalEnergy := Comparison.less |
| entropy := Comparison.less |
| volume := Comparison.less |
| temperature := Comparison.unknown } |
|
|
| theorem IPhO_2022_2_D_1 : |
| jouleThomsonComparisons.pressure = Comparison.greater |
| ∧ jouleThomsonComparisons.internalEnergy = Comparison.less |
| ∧ jouleThomsonComparisons.entropy = Comparison.less |
| ∧ jouleThomsonComparisons.volume = Comparison.less |
| ∧ jouleThomsonComparisons.temperature = Comparison.unknown := by |
| sorry |
|
|
| def enthalpyLikeQuantity (internalEnergy pressure volume : Real) : Real := |
| internalEnergy + pressure * volume |
|
|
| theorem IPhO_2022_2_D_2 |
| (U1 U2 P1 P2 V1 V2 : Real) |
| (h : U2 - U1 = P1 * V1 - P2 * V2) : |
| enthalpyLikeQuantity U2 P2 V2 = enthalpyLikeQuantity U1 P1 V1 := by |
| sorry |
|
|
| def graphPressureFromIsentropeSlope (dUdV : Real) : Real := |
| -dUdV |
|
|
| def graphEnthalpyEstimate : Real := |
| 407 / 10 |
|
|
| theorem IPhO_2022_2_D_3 : |
| graphPressureFromIsentropeSlope (-41) = 41 |
| ∧ graphEnthalpyEstimate = 407 / 10 := by |
| sorry |
|
|
| def maximumInletTemperatureEstimate : Real := |
| 11 |
|
|
| theorem IPhO_2022_2_D_4 : |
| maximumInletTemperatureEstimate = 11 := by |
| sorry |
|
|
| def graphPressureMPa (enthalpy internalEnergy volume : Real) : Real := |
| (enthalpy - internalEnergy) / volume / 1000 |
|
|
| theorem IPhO_2022_2_D_5 : |
| graphPressureMPa 41 10 (17 / 1000) = 31 / 17 := by |
| sorry |
|
|
| def spaghettiMaximumLength (oldLength oldDiameter newDiameter : Real) : Real := |
| oldLength * Real.sqrt (newDiameter / oldDiameter) |
|
|
| theorem IPhO_2022_3_A_1 (oldLength oldDiameter newDiameter : Real) : |
| spaghettiMaximumLength oldLength oldDiameter newDiameter = |
| oldLength * Real.sqrt (newDiameter / oldDiameter) := by |
| sorry |
|
|
| def fineSandBreakingForce |
| (coarseBreakingForce coarseToFineVolumeRatio : Real) : Real := |
| coarseBreakingForce * coarseToFineVolumeRatio ^ (1 / 3 : Real) |
|
|
| theorem IPhO_2022_3_B_1 |
| (coarseBreakingForce coarseToFineVolumeRatio : Real) : |
| fineSandBreakingForce coarseBreakingForce coarseToFineVolumeRatio = |
| coarseBreakingForce * coarseToFineVolumeRatio ^ (1 / 3 : Real) := by |
| sorry |
|
|
| def constantProperAccelerationDistance |
| (lightSpeed acceleration properTime : Real) : Real := |
| lightSpeed ^ 2 / acceleration * |
| (Real.cosh (acceleration * properTime / lightSpeed) - 1) |
|
|
| def interstellarDistanceRatio |
| (lightSpeed oldAcceleration newAcceleration quarterProperTime : Real) : Real := |
| constantProperAccelerationDistance lightSpeed newAcceleration quarterProperTime / |
| constantProperAccelerationDistance lightSpeed oldAcceleration quarterProperTime |
|
|
| theorem IPhO_2022_3_C_1 |
| (lightSpeed oldAcceleration newAcceleration quarterProperTime : Real) : |
| interstellarDistanceRatio lightSpeed oldAcceleration newAcceleration quarterProperTime = |
| constantProperAccelerationDistance lightSpeed newAcceleration quarterProperTime / |
| constantProperAccelerationDistance lightSpeed oldAcceleration quarterProperTime := by |
| sorry |
|
|
| def minimumOscillatingRadius (referenceRadius frequencyRatio : Real) : Real := |
| referenceRadius * (1 - frequencyRatio ^ 2) ^ (1 / 3 : Real) |
|
|
| theorem IPhO_2022_3_D_1 (referenceRadius : Real) : |
| minimumOscillatingRadius referenceRadius (99 / 100) = |
| referenceRadius * (199 / 10000) ^ (1 / 3 : Real) := by |
| sorry |
|
|
| end |
|
|
| end Ipho20212023Archon.Problems.Y2022 |
|
|
