Olympiads Descriptive Vault & Companion PDFs

m_{\text{total}} = \rho_w \left(\frac{1}{2}\pi R^2 L\right) = 40\pi\text{ g} \approx 125.66\text{ g}\\
m_{\text{frog}} = \frac{40\pi}{3}\text{ g} \approx 41.89\text{ g}\\
\theta - \sin\theta = \frac{2\pi}{3} \implies x_2 = R\cos(\theta/2) \approx 0.53\text{ cm}\\
\omega = \sqrt{\frac{\rho_w g w L}{m_{\text{wood}}}} \approx 30.04\text{ rad/s}, \quad T \approx 0.21\text{ s}

\frac{dE}{dt} = q \vec{v} \cdot (\vec{v} \times \vec{B}) = 0 \implies v = \text{const}\\
R = \frac{p}{qB} = \frac{\sqrt{K(K + 2m_0 c^2)}}{qBc}\\
\omega_c = \frac{v}{R} = \frac{qB}{\gamma m_0} = \frac{qBc^2}{E}

M g \sin\theta - f_s = M a_{\text{cm}}, \quad f_s R = \beta M R^2 (a_{\text{cm}}/R)\\
a_{\text{cm}} = \frac{g \sin\theta}{1 + \beta}\\
f_s = \frac{\beta}{1 + \beta} M g \sin\theta \le \mu_s M g \cos\theta \implies \mu_s \ge \frac{\beta}{1 + \beta} \tan\theta

Q_H = nRT_H \ln(V_B/V_A), \quad Q_C = nRT_C \ln(V_C/V_D)\\
\frac{T_H}{T_C} = (V_C/V_B)^{\gamma-1} = (V_D/V_A)^{\gamma-1} \implies V_B/V_A = V_C/V_D\\
\eta = 1 - \frac{Q_C}{Q_H} = 1 - \frac{T_C}{T_H}, \quad \oint \frac{\delta Q}{T} = 0

L \frac{d^2 Q}{dt^2} + \frac{1}{C} Q = 0 \implies \omega_0 = \frac{1}{\sqrt{LC}}\\
Q(t) = Q_0 \cos(\omega_0 t), \quad I(t) = \frac{Q_0}{\sqrt{LC}} \sin(\omega_0 t)\\
U_{\text{total}} = \frac{Q^2}{2C} + \frac{1}{2}LI^2 = \frac{Q_0^2}{2C} = \text{const}

M = \int_0^R 4\pi r^2 \rho_0 (1 - r/R) dr = \frac{\pi \rho_0 R^3}{3} \implies \rho_0 = \frac{3M}{\pi R^3}\\
g(r) = \frac{4GM}{R^3}(r - 3r^2/(4R)) \implies r_{\text{max}} = \frac{2}{3}R, \quad g_{\text{max}} = \frac{4}{3}\frac{GM}{R^2}\\
\Phi(0) = -\frac{2GM}{R}, \quad v_{\text{esc}} = \sqrt{\frac{2GM}{R}}

\Delta L_{\text{film}} = (n - 1)t, \quad \Delta \phi = \frac{2\pi}{\lambda}(n - 1)t\\
\Delta r_{\text{net}} = d\frac{y}{D} - (n - 1)t = 0 \implies \Delta y = \frac{D}{d}(n - 1)t\\
t = \frac{m \lambda}{n - 1}

m \frac{dv}{dt} = \frac{4}{3}\pi r^3 (\rho_p - \rho_f)g - 6\pi \eta r v\\
v(t) = v_t \left(1 - e^{-t/\tau}\right), \quad \tau = \frac{2r^2\rho_p}{9\eta}\\
v_t = \frac{2 r^2 g (\rho_p - \rho_f)}{9 \eta}, \quad t_{99} = \tau \ln(100) \approx 4.61\tau

a_{\text{tidal}} = \frac{d g}{d r} r = \frac{2 G M r}{d^3}\\
\frac{2 G M \mu r}{d^3} = \frac{G m \mu}{r^2} \implies d_R = R \left(2 \frac{\rho_M}{\rho_m}\right)^{1/3} \approx 1.26 R \left(\frac{\rho_M}{\rho_m}\right)^{1/3}\\
d_{\text{fluid}} \approx 2.44 R \left(\frac{\rho_M}{\rho_m}\right)^{1/3}

Z(\omega) = R + j\left(\omega L - \frac{1}{\omega C}\right) \implies I_0 = \frac{V_0}{\sqrt{R^2 + (\omega L - 1/(\omega C))^2}}\\
\omega_0 = \frac{1}{\sqrt{LC}}, \quad Z(\omega_0) = R\\
\omega L - \frac{1}{\omega C} = \pm R \implies \Delta\omega = \omega_2 - \omega_1 = \frac{R}{L}, \quad Q = \frac{\omega_0 L}{R}

P(1): 1(1^2 + 5) = 6 \implies 6 \mid 6\\
(k+1)[(k+1)^2 + 5] = (k^3 + 5k) + 3k(k+1) + 6 = 6m + 6p + 6 = 6(m + p + 1)\\
n(n^2 + 5) = (n-1)n(n+1) + 6n \equiv 0 + 0 \equiv 0 \pmod 6

\text{Rotate } \Delta ABP \text{ by } 60^\circ \text{ about } B \implies BP' = y, \quad CP' = x\\
\Delta PBP' \text{ is equilateral } \implies PP' = y, \quad \angle B P' P = 60^\circ\\
z^2 = x^2 + y^2 \implies (PC)^2 = (CP')^2 + (PP')^2 \implies \angle P P' C = 90^\circ\\
s^2 = x^2 + y^2 - 2xy\cos(150^\circ) = x^2 + y^2 + \sqrt{3}xy \implies s = \sqrt{x^2 + y^2 + \sqrt{3}xy}

a \cdot S \equiv S \pmod p \implies a^{p-1} (p-1)! \equiv (p-1)! \pmod p \implies a^{p-1} \equiv 1 \pmod p\\
a^{\phi(m)} \prod r_i \equiv \prod r_i \pmod m \implies a^{\phi(m)} \equiv 1 \pmod m\\
\phi(100) = 40, \quad 7^4 \equiv 1 \pmod{100} \implies 7^{2026} \equiv 7^2 = 49 \pmod{100}

P(t) = \sum(a_i t - b_i)^2 \ge 0 \implies \Delta = 4(\sum a_i b_i)^2 - 4(\sum a_i^2)(\sum b_i^2) \le 0\\
u_i = \frac{x_i}{\sqrt{a_i}}, v_i = \sqrt{a_i} \implies \sum \frac{x_i^2}{a_i} \ge \frac{(\sum x_i)^2}{\sum a_i}\\
\frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} \ge \frac{(a+b+c)^2}{2(a+b+c)} = \frac{a+b+c}{2} = \frac{1}{2}

\angle ABK = \angle CBD \implies \Delta ABK \sim \Delta DBC \implies AK \cdot BD = AB \cdot CD\\
\angle KBC = \angle ABD \implies \Delta KBC \sim \Delta ABD \implies KC \cdot BD = BC \cdot AD\\
(AK + KC) \cdot BD = AC \cdot BD = AB \cdot CD + BC \cdot AD\\
\text{General: } AC \cdot BD \le AB \cdot CD + BC \cdot AD \text{ (Equality iff cyclic)}

x^3, y^3, z^3 \equiv 0, \pm 1 \pmod 9 \implies 3 \mid z, \quad 3 \nmid xy\\
(x+y)(x^2 - xy + y^2) = -z^3 \implies x+y = 3^{3k-1}u^3, \quad x^2-xy+y^2 = 3v^3\\
b^2 + 3c^2 = v^3 \implies c = 3q(p-q)(p+q) \implies 2q(p-q)(p+q) = 3^{3(k-1)}u^3\\
(p-q) + 2q = p+q \implies X^3 + (3^{k-1}Z)^3 = Y^3 \implies 0 < |z_1| < |z|

\mathbf{1}_{\bigcup A_i} = 1 - \prod(1 - \mathbf{1}_{A_i}) \implies |\bigcup A_i| = \sum |A_i| - \sum |A_i \cap A_j| + \dots\\
D_n = n! - \sum_{k=1}^n (-1)^{k-1} \binom{n}{k}(n-k)! = n! \sum_{k=0}^n \frac{(-1)^k}{k!}\\
\left|D_n - \frac{n!}{e}\right| < \frac{1}{n+1} \le 0.5 \implies D_n = \left[\frac{n!}{e}\right]

a_i = \text{max increasing ending at } x_i, \quad b_i = \text{max decreasing ending at } x_i\\
j < k \implies (a_j, b_j) \ne (a_k, b_k) \text{ (Injective map)}\\
N = (r-1)(s-1) + 1 > (r-1)(s-1) \implies \exists i: a_i \ge r \text{ or } b_i \ge s

n = 2: \frac{x_1 + x_2}{2} - \sqrt{x_1 x_2} = \frac{(\sqrt{x_1} - \sqrt{x_2})^2}{2} \ge 0\\
n = 2^k \implies \text{Forward induction across pairs}\\
x_n = A = \frac{\sum_{1}^{n-1} x_i}{n-1} \implies A \ge (x_1 \dots x_{n-1} A)^{1/n} \implies A^{n-1} \ge \prod_{1}^{n-1} x_i

\frac{P'(x)}{P(x)} = \sum_{k=0}^\infty \frac{p_k}{x^{k+1}} \implies P'(x) = P(x) \sum_{k=0}^\infty p_k x^{-(k+1)}\\
p_k - e_1 p_{k-1} + e_2 p_{k-2} - \dots + (-1)^k k e_k = 0\\
e_1 = 2, e_2 = 3, e_3 = 4 \implies p_1 = 2, p_2 = -2, p_3 = 2, p_4 = 18

\text{Proplastid} \xrightarrow{\text{dark}} \text{Etioplast (PLB)} \xrightarrow{h\nu} \text{Chloroplast (Grana)}\\
\text{Chloroplast} \xrightarrow{\text{ripening}} \text{Chromoplast (Carotenoids)}\\
\text{Leucoplasts} = \begin{cases} \text{Amyloplasts: Starch storage \& gravitropism statoliths} \\ \text{Elaioplasts: Lipids, triacylglycerols \& pollenkitt} \\ \text{Proteinoplasts: Protein crystalline aleurone reserves} \end{cases}

\Psi_w = \Psi_s + \Psi_p + \Psi_g \approx \Psi_s + \Psi_p\\
\mu_w - \mu_w^0 = RT \ln a_w \approx -RT x_s \implies \Psi_s = -iCRT\\
\text{Equilibrium in pure water: } \Psi_{w,\text{final}} = 0 \implies \Psi_p = -\Psi_s = +0.7\text{ MPa}

R\text{H}_2 + \text{O}_2 \xrightarrow{\text{oxidase}} R + \text{H}_2\text{O}_2, \quad 2\text{H}_2\text{O}_2 \xrightarrow{\text{catalase}} 2\text{H}_2\text{O} + \text{O}_2\\
\text{Isocitrate} \xrightarrow{\text{isocitrate lyase}} \text{Succinate} + \text{Glyoxylate}\\
\text{Glyoxylate} + \text{Acetyl-CoA} \xrightarrow{\text{malate synthase}} \text{Malate}\\
\text{Net: } 2\text{ Acetyl-CoA} \longrightarrow \text{Succinate } (4\text{C}) \xrightarrow{\text{Gluconeogenesis}} \text{Glucose}

\text{V-ATPase: } \text{ATP} + \text{H}_2\text{O} + \text{H}^+_{\text{cytosol}} \longrightarrow \text{ADP} + \text{P}_i + \text{H}^+_{\text{lumen}}\\
\Delta \text{pH} = \text{pH}_{\text{cytosol}} - \text{pH}_{\text{lumen}} \approx 7.2 - 4.7 = 2.5 \text{ units (300-fold gradient)}\\
\text{Failsafe: } \text{Enzyme activity at pH 7.2} < 5\% \text{ of $V_{\text{max}}$ at pH 4.8}

\text{Gen 0: } ^{15}\text{N} \to \text{Gen 1: } 100\%\; ^{15}\text{N-}^{14}\text{N (Hybrid)} \to \text{Gen 2: } 50\%\; ^{15}\text{N-}^{14}\text{N} + 50\%\; ^{14}\text{N-}^{14}\text{N}\\
\text{Pol I Nick Translation: } 5'\to 3' \text{ Exonuclease (RNA Degradation)} + 5'\to 3' \text{ Polymerase (DNA Fill)}\\
\text{Ligase: } \text{Enzyme} + \text{ATP} \to \text{Enzyme-AMP} \to 5'\text{-P-AMP} \xrightarrow{3'\text{-OH}} 3'\text{-}5' \text{ Phosphodiester} + \text{AMP}

\text{Daughter } aa \implies \text{Mother } Aa \times \text{Father } (Aa \text{ or } aa)\\
\text{If Father normal } (Aa): Aa \times Aa \implies 1 AA : 2 Aa : 1 aa \implies P(\text{Albino}) = \frac{1}{4}\\
P(Aa \mid \text{Normal}) = \frac{P(Aa)}{P(AA) + P(Aa)} = \frac{1/2}{1/4 + 1/2} = \frac{2}{3} \approx 66.7\%

\text{Complete Hoagland vs Minus-N: } \text{Replace } \text{KNO}_3 \to \text{KCl}, \; \text{Ca(NO}_3)_2 \to \text{CaCl}_2\\
\text{Nitrogen is phloem-mobile: } \text{Rubisco breakdown in old leaves} \xrightarrow{\text{phloem}} \text{Young apex}\\
\text{NO}_3^- + \text{NADH} + \text{H}^+ \xrightarrow{\text{NR}} \text{NO}_2^- + \text{NAD}^+ + \text{H}_2\text{O}\\
\text{NO}_2^- + 6\,Fd_{\text{red}} + 8\text{H}^+ \xrightarrow{\text{NiR}} \text{NH}_4^+ + 6\,Fd_{\text{ox}} + 2\text{H}_2\text{O}

\text{Lumen (pH } 1.5\text{)} \xrightarrow{\text{Mucus + HCO}_3^-} \text{Epithelial Surface (pH } 7.0\text{)}\\
\text{Pepsinogen (42.5 kDa)} \xrightarrow{\text{pH } < 3.0} \text{Pepsin (35 kDa)} + \text{44-residue prosegment}\\
\text{NSAIDs} \xrightarrow{\text{inhibit}} \text{COX-1} \implies \downarrow \text{PGE}_2 \implies \downarrow \text{Mucus/HCO}_3^- \implies \text{Peptic Ulcer}

\text{ipRGCs (Melanopsin } 480\,\text{nm)} \xrightarrow{\text{RHT}} \text{SCN} \xrightarrow{\text{PVN \& SCG}} \text{Pineal Gland}\\
\text{Positive: } \text{CLOCK:BMAL1} \xrightarrow{\text{E-box}} \text{PER + CRY} \implies \text{Negative: } \text{PER:CRY translocate} \dashv \text{CLOCK:BMAL1}\\
\text{Tryptophan} \to \text{Serotonin} \xrightarrow{\text{AANAT (rate-limiting)}} N\text{-Acetylserotonin} \xrightarrow{\text{ASMT}} \text{Melatonin}

\text{GI Lumen} \xrightarrow{f_a} \text{Enterocytes} \xrightarrow{1-E_G} \text{Portal Vein} \xrightarrow{1-E_H} \text{Vena Cava (Systemic)}\\
F = f_a \times (1 - E_G) \times (1 - E_H), \quad E_H = \frac{CL_{\text{hepatic}}}{Q_H}\\
\text{Phase I: } \text{Drug-H} + \text{O}_2 + \text{NADPH} + \text{H}^+ \xrightarrow{\text{CYP}} \text{Drug-OH} + \text{NADP}^+ + \text{H}_2\text{O}

\text{Alkene: } \text{C}_6\text{H}_{10} + \text{Br}_2 \longrightarrow \text{trans-1,2-dibromocyclohexane (Decolorization)}\\
\text{Benzene: } \text{C}_6\text{H}_6 + \text{Br}^+ \xrightarrow{\text{slow}} [\text{C}_6\text{H}_6\text{Br}]^+ \xrightarrow{\text{fast, } -H^+} \text{C}_6\text{H}_5\text{Br}\\
\Delta H_{\text{resonance}} \approx 152\,\text{kJ/mol} \implies \text{Substitution preserves aromaticity, addition destroys it.}

G(-5, 0), F(0, 5) \implies y = x + 5\\
C(7, 0), D(4, 3) \implies y = -x + 7\\
E = (1, 6), \quad GC = 12, \quad GE = CE = 6\sqrt{2}\\
\text{Perimeter} = 12 + 12\sqrt{2} = 12(1 + \sqrt{2}) \approx 28.97

d\mu = V_m dP = \frac{RT}{P} dP \implies \mu_i = \mu_i^\circ + RT \ln(P_i/P^\circ)\\
\Delta_r G = \sum \nu_i \mu_i = \sum \nu_i \mu_i^\circ + RT \sum \nu_i \ln a_i = \Delta_r G^\circ + RT \ln Q\\
\text{At equilibrium, } \Delta_r G = 0, Q = K \implies \Delta_r G^\circ = -RT \ln K \iff K = e^{-\Delta_r G^\circ / RT}

\text{0th Order: } [A]_t = [A]_0 - k_0 t, \quad t_{1/2} = \frac{[A]_0}{2k_0}\\
\text{1st Order: } \ln[A]_t = \ln[A]_0 - k_1 t, \quad t_{1/2} = \frac{\ln 2}{k_1}\\
\text{2nd Order: } \frac{1}{[A]_t} = \frac{1}{[A]_0} + k_2 t, \quad t_{1/2} = \frac{1}{k_2 [A]_0}\\
\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)

K_a = \frac{[H^+][A^-]}{[HA]} \implies \text{pH} = pK_a + \log_{10}\left(\frac{[A^-]}{[HA]}\right)\\
\beta = \frac{d C_b}{d(\text{pH})} = 2.303 \left([H^+] + [OH^-] + C \frac{K_a [H^+]}{(K_a + [H^+])^2}\right)\\
\frac{d\beta}{d[H^+]} = 0 \implies [H^+] = K_a \implies \text{pH} = pK_a, \quad \beta_{\text{max}} = \frac{2.303}{4}C \approx 0.576 C

2 E(e_g) - 3 E(t_{2g}) = 0 \implies E(e_g) = +0.6\Delta_o, \quad E(t_{2g}) = -0.4\Delta_o\\
\Delta_t = \left(\frac{4}{6}\text{ ligands}\right) \times \left(\frac{2}{3}\text{ directional overlap}\right) \Delta_o = \frac{4}{9}\Delta_o\\
\Delta_t \ll P \implies \text{Tetrahedral complexes are almost universally high-spin.}

R\text{COL} + \text{Nu}^- \xrightleftharpoons{\text{Addition, } sp^2 \to sp^3} [R\text{-C(O}^-\text{)(Nu)(L)}] \xrightarrow{\text{Elimination, } sp^3 \to sp^2} R\text{CONu} + L^-\\
\text{Reactivity: } R\text{COCl} > (R\text{CO})_2O > R\text{COOR}' > R\text{CONH}_2 > R\text{COO}^-\\
\text{Leaving group } pK_a(\text{conj acid}): \text{HCl } (-7) < \text{RCOOH } (4.8) < \text{ROH } (16) < \text{NH}_3 \; (38)