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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
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