USAAAO 2025, I кръг — Задача 27

USAAAO 2025 - First Round · 8 февруари 2025 г.

Внимание: Бележка към темата

All 30 separately numbered original first-round multiple-choice questions and their matching official explanations/answer letters are retained. The printed explanations for Questions 2 and 3 stop before a promised formula; Question 26 refers to a following triangle absent from its native solution page. No missing expression or diagram is reconstructed. Native scientific discrepancies and ambiguous formula notation remain literal and are explicitly attributed in local source notes. A separate two-page common USAAAO archive constants/reference table is included with its own source identity; it is undated in the catalogue, and its use as the exact 2025 attachment is not established. No separate dated instructions, answer sheet or constants companion is present in the 2025 archive folder. No point total or time limit is printed in these two exam booklets.

Original first-round date and writers

USAAAO 2025 - First Round

February 8th, 2025

Writers: Abhay Bestrapalli, Sidhant Chaliha, Ferdinand, Ved Gund, Hagan Hensley, Yehong Jiang, Alexander Li, Joe McCarty, Lucas Carrit Delgado Pinheiro, Sahil Pontula.

Common archive reference — Table of Physical Constants

This separate common USAAAO archive reference is undated in the catalogue. Its use as the exact 2025 exam attachment is not established.

QuantitySymbolValue
Electron rest massmem_e9.109×10−31 kg9.109\times10^{-31}\ \mathrm{kg}
Proton rest massMpM_p1.6726×10−27 kg1.6726\times10^{-27}\ \mathrm{kg}
Electronic chargeee1.6022×10−19 C1.6022\times10^{-19}\ \mathrm C
Speed of light in free spacecc2.9979×108 m s−12.9979\times10^8\ \mathrm{m\ s^{-1}}
Permeability of free spaceμ0\mu_04π×10−7 H m−14\pi\times10^{-7}\ \mathrm{H\ m^{-1}}
Permittivity of free spaceϵ0\epsilon_08.854×10−12 F m−18.854\times10^{-12}\ \mathrm{F\ m^{-1}}
Planck’s constanthh6.626×10−34 J s6.626\times10^{-34}\ \mathrm{J\ s}
Reduced Planck’s constantℏ=h/2π\hbar=h/2\pi1.0546×10−34 J s1.0546\times10^{-34}\ \mathrm{J\ s}
ℏc\hbar c197.33 MeV fm197.33\ \mathrm{MeV\ fm}
Boltzmann’s constantkBk_B1.3807×10−23 J K−11.3807\times10^{-23}\ \mathrm{J\ K^{-1}}
Gas constantR=kB/mH\mathcal R=k_B/m_H8.250×103 J kg−1 K−18.250\times10^3\ \mathrm{J\ kg^{-1}\ K^{-1}}
Molar gas constantRR8.315 J mol−1 K−18.315\ \mathrm{J\ mol^{-1}\ K^{-1}}
Avogadro’s numberNAN_A6.022×1023 mol−16.022\times10^{23}\ \mathrm{mol^{-1}}
Standard molar volume22.414×10−3 m3 mol−122.414\times10^{-3}\ \mathrm{m^3\ mol^{-1}}
Unified atomic mass unit (12C^{12}\mathrm C scale)uu931.5 MeV/c2=1.660538×10−27 kg931.5\ \mathrm{MeV}/c^2=1.660538\times10^{-27}\ \mathrm{kg}
Mass of hydrogen atommHm_H1.0078u=1.6735×10−27 kg1.0078u=1.6735\times10^{-27}\ \mathrm{kg}
Bohr magnetonμB\mu_B9.274×10−24 A m29.274\times10^{-24}\ \mathrm{A\ m^2} or J T−1\mathrm{J\ T^{-1}}
Nuclear magnetonμN\mu_N5.051×10−27 A m25.051\times10^{-27}\ \mathrm{A\ m^2} or J T−1\mathrm{J\ T^{-1}}
Proton magnetic momentμp\mu_p2.7928μN2.7928\mu_N
Neutron magnetic momentμn\mu_n−1.9130μN-1.9130\mu_N
Bohr radiusa0a_05.292×10−11 m5.292\times10^{-11}\ \mathrm m
Fine structure constantα=e2/(4πϵ0ℏc)\alpha=e^2/(4\pi\epsilon_0\hbar c)(137.04)−1(137.04)^{-1}
Compton wavelength of electronλC=h/(mec)\lambda_C=h/(m_ec)2.4263×10−12 m2.4263\times10^{-12}\ \mathrm m
Rydberg’s constantR∞R_{\infty}1.0974×107 m−11.0974\times10^7\ \mathrm{m^{-1}}
R∞hcR_{\infty}hc13.606 eV13.606\ \mathrm{eV}
Stefan-Boltzmann constantσ\sigma5.671×10−8 W m−2 K−45.671\times10^{-8}\ \mathrm{W\ m^{-2}\ K^{-4}}
Radiation density constanta=4σ/ca=4\sigma/c7.561×10−16 J m−3 K−47.561\times10^{-16}\ \mathrm{J\ m^{-3}\ K^{-4}}
Gravitational constantGG6.673×10−11 N m2 kg−26.673\times10^{-11}\ \mathrm{N\ m^2\ kg^{-2}}

Rest masses of some leptons and hadrons in MeV/c2\mathrm{MeV}/c^2:

e±e^{\pm} 0.5110, μ±\mu^{\pm} 105.66, τ±\tau^{\pm} 1777, π0\pi^0 134.98, π±\pi^{\pm} 139.57, K±\mathrm K^{\pm} 493.7, K0\mathrm K^0 497.7, η\eta 547, D0\mathrm D^0 1865, D±\mathrm D^{\pm} 1869, p 938.3, n 939.6, Λ0\Lambda^0 1115.7, Σ+\Sigma^+ 1189, Σ0\Sigma^0 1193, Σ−\Sigma^- 1197, Ξ0\Xi^0 1315, Ξ−\Xi^- 1321, Ω−\Omega^- 1672, Z0\mathrm Z^0 91.187×10391.187\times10^3, W±\mathrm W^{\pm} 80.41×10380.41\times10^3.

QuarkChargeI3I_3SSCCBBTTMass (GeV/c2\mathrm{GeV}/c^2)
u+23+\frac2312\frac120000∼0.003\sim0.003
d−13-\frac13−12-\frac120000∼0.006\sim0.006
c+23+\frac2300+100∼1.25\sim1.25
s−13-\frac130-1000∼0.11\sim0.11
t+23+\frac230000+1174.3
b−13-\frac13000-104.2
Common archive reference — Astrophysical Data and identities

Astrophysical Data

QuantitySymbolValue
1 astronomical unitAU1.496×1011 m1.496\times10^{11}\ \mathrm m
1 parsecpc3.086×1016 m3.086\times10^{16}\ \mathrm m
Luminosity of SunL⊙L_{\odot}3.85×1026 W3.85\times10^{26}\ \mathrm W
Mass of SunM⊙M_{\odot}1.989×1030 kg1.989\times10^{30}\ \mathrm{kg}
Radius of SunR⊙R_{\odot}6.96×108 m6.96\times10^8\ \mathrm m
Mass of EarthMEM_E5.9742×1024 kg5.9742\times10^{24}\ \mathrm{kg}
Radius of EarthRER_E6.3781×106 m6.3781\times10^6\ \mathrm m

Other data and conversion factors

QuantitySymbolValue
1 ångstromÅ10−10 m10^{-10}\ \mathrm m
1 fermifm10−15 m10^{-15}\ \mathrm m
1 barnb10−28 m210^{-28}\ \mathrm{m^2}
1 pascalPa1 Nm−21\ \mathrm{Nm^{-2}}
1 standard atmosphere1.0132×105 Pa1.0132\times10^5\ \mathrm{Pa}
Standard acceleration due to gravitygg9.807 m s−29.807\ \mathrm{m\ s^{-2}}
1 electron volteV1.6022×10−19 J1.6022\times10^{-19}\ \mathrm J
eV/hc\mathrm{eV}/hc8.065×105 m−18.065\times10^5\ \mathrm{m^{-1}}
eV/kB\mathrm{eV}/k_B1.1604×104 K1.1604\times10^4\ \mathrm K
Wavelength of 1 eV photon1.2399×10−6 m1.2399\times10^{-6}\ \mathrm m

Trigonometrical identities

sin⁡(θ+ϕ)=sin⁡(θ)cos⁡(ϕ)+cos⁡(θ)sin⁡(ϕ)\sin(\theta+\phi)=\sin(\theta)\cos(\phi)+\cos(\theta)\sin(\phi)
cos⁡(θ+ϕ)=cos⁡(θ)cos⁡(ϕ)−sin⁡(θ)sin⁡(ϕ)\cos(\theta+\phi)=\cos(\theta)\cos(\phi)-\sin(\theta)\sin(\phi)
sin⁡α+sin⁡β=2sin⁡12(α+β)cos⁡12(α−β)\sin\alpha+\sin\beta=2\sin\frac12(\alpha+\beta)\cos\frac12(\alpha-\beta)
cos⁡α+cos⁡β=2cos⁡12(α+β)cos⁡12(α−β)\cos\alpha+\cos\beta=2\cos\frac12(\alpha+\beta)\cos\frac12(\alpha-\beta)
cos⁡α−cos⁡β=2sin⁡12(α+β)sin⁡12(β−α)\cos\alpha-\cos\beta=2\sin\frac12(\alpha+\beta)\sin\frac12(\beta-\alpha)

In a triangle ABC, a/sin⁡A=b/sin⁡B=c/sin⁡Ca/\sin A=b/\sin B=c/\sin C

and a2=b2+c2−2bccos⁡Aa^2=b^2+c^2-2bc\cos A

Prefixes

PrefixNameFactor
Ttera101210^{12}
Ggiga10910^9
Mmega10610^6
kkilo10310^3
ccenti10−210^{-2}
mmilli10−310^{-3}
μ\mumicro10−610^{-6}
nnano10−910^{-9}
ppico10−1210^{-12}
ffemto10−1510^{-15}

Source note: The common table prints g=9.807 m s−2g=9.807\ \mathrm{m\ s^{-2}} and RE=6.3781×106 mR_E=6.3781\times10^6\ \mathrm m. The 2025 question14/key15 use 9.81 and key7 uses 6371 km; those original exam values are preserved separately.

Условие

Cosmologists consider three possible universes: universes dominated by baryonic matter (i.e. ordinary matter), universes dominated by radiation, and universes dominated by dark energy (realistically, we consider some mix of the above). An equation written by Alexander Friedmann implies that for a matter-dominated universe, the Hubble parameter is proportional to the square root of that universe’s matter density. In a matter-dominated universe, how does the size of the universe evolve with time tt?

(a) constant

(b) t\sqrt t

(c) t2/3t^{2/3}

(d) tt

(e) ete^t

Отговори

Решение

Теми и трудност

Оценена трудност: 3/5.

Галактики и космология · Разширение на Вселената · Космология

Предпоставки: Гимназиална подготовка, Разширена олимпиадна подготовка

Оригинал в Архива: exam_2025.pdf · официални решения: sol_2025.pdf · допълнителни данни: physical-constants.pdf