| [ |
| { |
| "id": 43, |
| "tag": "ELECTRICITY", |
| "answer": "$$\nI = A U_0 \\left[\\frac{\\sigma_1 \\sigma_2}{\\sigma_1 h_2 + \\sigma_2 h_1} + e^{-\\frac{\\sigma_1 h_2 + \\sigma_2 h_1}{\\varepsilon_1 h_2 + \\varepsilon_2 h_1} t} \\left(\\frac{\\varepsilon_2 \\sigma_1 + \\varepsilon_1 \\sigma_2}{\\varepsilon_1 h_2 + \\varepsilon_2 h_1} - \\frac{\\sigma_1 \\sigma_2}{\\sigma_1 h_2 + \\sigma_2 h_1} - \\frac{\\varepsilon_1 \\varepsilon_2 (\\sigma_1 h_2 + \\sigma_2 h_1)}{(\\varepsilon_1 h_2 + \\varepsilon_2 h_1)^2} \\right)\\right]\n$$", |
| "fixed_answer": "$$\nI = A U_0 [\\frac{\\sigma_1 \\sigma_2}{\\sigma_1 h_2 + \\sigma_2 h_1} + e^{-\\frac{\\sigma_1 h_2 + \\sigma_2 h_1}{\\varepsilon_1 h_2 + \\varepsilon_2 h_1} t} (\\frac{\\varepsilon_2 \\sigma_1 + \\varepsilon_1 \\sigma_2}{\\varepsilon_1 h_2 + \\varepsilon_2 h_1} - \\frac{\\sigma_1 \\sigma_2}{\\sigma_1 h_2 + \\sigma_2 h_1} - \\frac{\\varepsilon_1 \\varepsilon_2 (\\sigma_1 h_2 + \\sigma_2 h_1)}{(\\varepsilon_1 h_2 + \\varepsilon_2 h_1)^2} )]\n$$" |
| }, |
| { |
| "id": 45, |
| "tag": "THERMODYNAMICS", |
| "answer": "$$\\sqrt{\\frac{\\gamma\\mu}{\\mu\\rho - b\\rho^2}\\left(p + \\frac{a}{\\mu^2}\\rho^2\\right) - \\frac{2a\\rho}{\\mu^2}}$$", |
| "fixed_answer": "$$\\sqrt{\\frac{\\gamma\\mu}{\\mu\\rho - b\\rho^2}(p + \\frac{a}{\\mu^2}\\rho^2) - \\frac{2a\\rho}{\\mu^2}}$$" |
| }, |
| { |
| "id": 71, |
| "tag": "ELECTRICITY", |
| "answer": "$$\nv_E = \\frac{E \\times B}{B^2} \\left(1 - \\frac{1}{4} \\left(k \\frac{m v_\\perp}{|q| B}\\right)^2 \\right)\n$$", |
| "fixed_answer": "$$\nv_E = \\frac{E \\times B}{B^2} (1 - \\frac{1}{4} (k \\frac{m v_\\perp}{|q| B})^2 )\n$$" |
| }, |
| { |
| "id": 88, |
| "tag": "MODERN", |
| "answer": "$$\nw=\\frac{\\lambda N_0 E_0}{4\\pi}\\frac{\\left(1-\\left(\\frac{v}{c}\\right)^2\\right)^2}{\\left(1-\\frac{v}{c}\\cos\\theta\\right)^3}e^{-\\frac{4\\lambda}{7}}\n$$", |
| "fixed_answer": "$$\nw=\\frac{\\lambda N_0 E_0}{4\\pi}\\frac{(1-(\\frac{v}{c})^2)^2}{(1-\\frac{v}{c}\\cos\\theta)^3}e^{-\\frac{4\\lambda}{7}}\n$$" |
| }, |
| { |
| "id": 115, |
| "tag": "ADVANCED", |
| "answer": "$$\nP'=\\frac{n q^2 c}{6 \\varepsilon_0 \\sqrt{1-\\beta^2}} \\left[\\frac{\\omega^2 R^2 / c^2}{1-\\omega^2 R^2 / c^2}+\\ln\\left(1-\\frac{\\omega^2 R^2}{c^2}\\right)\\right]\n$$", |
| "fixed_answer": "$$\nP'=\\frac{n q^2 c}{6 \\varepsilon_0 \\sqrt{1-\\beta^2}} [\\frac{\\omega^2 R^2 / c^2}{1-\\omega^2 R^2 / c^2}+\\ln(1-\\frac{\\omega^2 R^2}{c^2})]\n$$" |
| }, |
| { |
| "id": 121, |
| "tag": "MODERN", |
| "answer": "$$\nu = c^2 \\frac{\\left| \\sqrt{c^2 - v_1^2} - \\sqrt{c^2 - v_1^2} \\right|}{\\sqrt{c^2 (v_1^2 + v_2^2) - 2 v_1^2 v_2^2 - 2 v_1 v_2 \\cos \\alpha \\sqrt{(c^2 - v_1^2)(c^2 - v_2^2)}}}\n$$", |
| "fixed_answer": "$$\nu = c^2 \\frac{| \\sqrt{c^2 - v_1^2} - \\sqrt{c^2 - v_2^2} |}{\\sqrt{c^2 (v_1^2 + v_2^2) - 2 v_1^2 v_2^2 - 2 v_1 v_2 \\cos \\alpha \\sqrt{(c^2 - v_1^2)(c^2 - v_2^2)}}}\n$$" |
| }, |
| { |
| "id": 206, |
| "tag": "ELECTRICITY", |
| "answer": "$$\n\\frac{V_0}{\\ln\\left(\\frac{\\tan\\left(\\frac{\\alpha_2}{2}\\right)}{\\tan\\left(\\frac{\\alpha_1}{2}\\right)}\\right)}\\ln\\left(\\frac{\\tan\\left(\\frac{\\alpha}{2}\\right)}{\\tan\\left(\\frac{\\alpha_1}{2}\\right)}\\right)\n$$", |
| "fixed_answer": "$$\n\\frac{V_0}{\\ln(\\frac{\\tan(\\frac{\\alpha_2}{2})}{\\tan(\\frac{\\alpha_1}{2})})}\\ln(\\frac{\\tan(\\frac{\\alpha}{2})}{\\tan(\\frac{\\alpha_1}{2})})\n$$" |
| }, |
| { |
| "id": 221, |
| "tag": "MODERN", |
| "answer": "$$\nt=\\frac{1}{\\rho A}\\sqrt{\\frac{\\left(\\frac{k R_{0}^{2}}{m_{0}c^{2}}+\\sqrt{\\left(\\frac{k R_{0}^{2}}{m_{0}c^{2}}\\right)^{2}+4}\\right)m R_{0}^{4}}{2k}}\\left(\\frac{k}{c^{2}}\\left(-\\frac{1}{R}+\\frac{1}{R_{0}}\\right)+\\frac{4}{7}\\frac{\\left(\\frac{k R_{0}^{2}}{m_{0}c^{2}}+\\sqrt{\\left(\\frac{k R_{0}^{2}}{m_{0}c^{2}}\\right)^{2}+4}\\right)m R_{0}^{4}}{2}\\left(\\frac{1}{R^{7}}-\\frac{1}{R_{0}^{7}}\\right)\\right)\n$$", |
| "fixed_answer": "$$\nt=\\frac{1}{\\rho A}\\sqrt{\\frac{(\\frac{k R_{0}^{2}}{m_{0}c^{2}}+\\sqrt{(\\frac{k R_{0}^{2}}{m_{0}c^{2}})^{2}+4})m R_{0}^{4}}{2k}}(\\frac{k}{c^{2}}(-\\frac{1}{R}+\\frac{1}{R_{0}})+\\frac{4}{7}\\frac{(\\frac{k R_{0}^{2}}{m_{0}c^{2}}+\\sqrt{(\\frac{k R_{0}^{2}}{m_{0}c^{2}})^{2}+4})m R_{0}^{4}}{2}(\\frac{1}{R^{7}}-\\frac{1}{R_{0}^{7}}))\n$$" |
| }, |
| { |
| "id": 228, |
| "tag": "MECHANICS", |
| "answer": "$$\n\\frac{3\\sin(\\theta_1-\\theta_2)\\left[2g\\cos\\theta_1+2l_1\\omega_1^2+l_2\\omega_2^2\\cos(\\theta_1-\\theta_2)\\right]}{l_2\\left[1+3\\sin^2(\\theta_1-\\theta_2)\\right]}\n$$", |
| "fixed_answer": "$$\n\\frac{3\\sin(\\theta_1-\\theta_2)[2g\\cos\\theta_1+2l_1\\omega_1^2+l_2\\omega_2^2\\cos(\\theta_1-\\theta_2)]}{l_2[1+3\\sin^2(\\theta_1-\\theta_2)]}\n$$" |
| }, |
| { |
| "id": 250, |
| "tag": "ELECTRICITY", |
| "answer": "$$\ni(t)=\\frac{U}{r_0\\left(1+\\frac{2\\arccosh(D/R)}{\\pi r_0 L(\\sigma_1+\\sigma_2)}\\right)}\\left(2+\\frac{2\\arccosh(D/R)}{\\pi r_0 L(\\sigma_1+\\sigma_2)}-\\exp\\left[-\\left(\\frac{\\sigma_1+\\sigma_2}{\\varepsilon_1+\\varepsilon_2}+\\frac{2\\arccosh(D/R)}{\\pi r_0 L(\\varepsilon_1+\\varepsilon_2)}\\right)t\\right]\\right)\n$$", |
| "fixed_answer": "$$\ni(t)=\\frac{U}{r_0(1+\\frac{2\\arccosh(D/R)}{\\pi r_0 L(\\sigma_1+\\sigma_2)})}(2+\\frac{2\\arccosh(D/R)}{\\pi r_0 L(\\sigma_1+\\sigma_2)}-\\exp[-(\\frac{\\sigma_1+\\sigma_2}{\\varepsilon_1+\\varepsilon_2}+\\frac{2\\arccosh(D/R)}{\\pi r_0 L(\\varepsilon_1+\\varepsilon_2)})t])\n$$" |
| }, |
| { |
| "id": 284, |
| "tag": "OPTICS", |
| "answer": "$$\\varphi=2\\arctan\\left(\\frac{\\sqrt{\\sin^2\\theta-\\left(1+\\frac{\\delta n}{n}\\right)^2}}{\\cos\\theta}\\right)$$", |
| "fixed_answer": "$$\\varphi=2\\arctan(\\frac{\\sqrt{\\sin^2\\theta-(1+\\frac{\\delta n}{n})^2}}{\\cos\\theta})$$" |
| }, |
| { |
| "id": 310, |
| "tag": "MODERN", |
| "answer": "$$\\frac{\\left(\\sqrt{\\frac{1+\\beta_0}{1-\\beta_0}}+\\frac{2E}{mc^2}\\right)^2 - 1}{\\left(\\sqrt{\\frac{1+\\beta_0}{1-\\beta_0}}+\\frac{2E}{mc^2}\\right)^2 + 1}$$", |
| "fixed_answer": "$$\\frac{(\\sqrt{\\frac{1+\\beta_0}{1-\\beta_0}}+\\frac{2E}{mc^2})^2 - 1}{(\\sqrt{\\frac{1+\\beta_0}{1-\\beta_0}}+\\frac{2E}{mc^2})^2 + 1}$$" |
| }, |
| { |
| "id": 332, |
| "tag": "MECHANICS", |
| "answer": "$$\n\\mu = \\frac{MR^2 \\sin\\varphi \\sin\\left(\\arccos\\frac{L^2 + d^2 - R^2}{2L d} + \\arccos\\frac{R^2 + d^2 - L^2}{2R d}\\right)}{\\sqrt{\\left(\\frac{L^2 + d^2 - R^2}{2 L d}\\right)^2 \\left[ M R^2 \\cos\\varphi + m L^2 \\sin^2\\left(\\arccos\\frac{L^2 + d^2 - R^2}{2L d}\\right) \\left(\\frac{3}{2} \\cos\\varphi - 1\\right)\\right]^2 + \\left[ M R^2 \\sin\\varphi \\cos\\left(\\arccos\\frac{L^2 + d^2 - R^2}{2L d} + \\arccos\\frac{R^2 + d^2 - L^2}{2R d}\\right)\\right]^2 }}\n$$", |
| "fixed_answer": "$$\n\\mu = \\frac{MR^2 \\sin\\varphi \\sin(\\arccos\\frac{L^2 + d^2 - R^2}{2L d} + \\arccos\\frac{R^2 + d^2 - L^2}{2R d})}{\\sqrt{(\\frac{L^2 + d^2 - R^2}{2 L d})^2 [ M R^2 \\cos\\varphi + m L^2 \\sin^2(\\arccos\\frac{L^2 + d^2 - R^2}{2L d}) (\\frac{3}{2} \\cos\\varphi - 1)]^2 + [ M R^2 \\sin\\varphi \\cos(\\arccos\\frac{L^2 + d^2 - R^2}{2L d} + \\arccos\\frac{R^2 + d^2 - L^2}{2R d})]^2 }}\n$$" |
| }, |
| { |
| "id": 333, |
| "tag": "ELECTRICITY", |
| "answer": "$$\n\\beta = \\frac{2M}{mR^2 + 4\\mu_0 P_0^2 \\pi L \\left(\\frac{3}{8k^4} + \\left(\\frac{R^3}{2k} - \\frac{3R^2}{4k^2} + \\frac{3R}{4k^3} - \\frac{3}{8k^4}\\right)e^{2kR}\\right)}\n$$", |
| "fixed_answer": "$$\n\\beta = \\frac{2M}{mR^2 + 4\\mu_0 P_0^2 \\pi L (\\frac{3}{8k^4} + (\\frac{R^3}{2k} - \\frac{3R^2}{4k^2} + \\frac{3R}{4k^3} - \\frac{3}{8k^4})e^{2kR})}\n$$" |
| }, |
| { |
| "id": 336, |
| "tag": "MECHANICS", |
| "answer": "$$\nf=\\frac{1}{2\\pi}\\sqrt{\\frac{4g(b+R-\\frac{b^2R}{a^2})}{(a^2+5b^2)\\left(\\frac{bR}{a^2}-1\\right)}}\n$$", |
| "fixed_answer": "$$\nf=\\frac{1}{2\\pi}\\sqrt{\\frac{4g(b+R-\\frac{b^2R}{a^2})}{(a^2+5b^2)(\\frac{bR}{a^2}-1)}}\n$$" |
| }, |
| { |
| "id": 412, |
| "tag": "ELECTRICITY", |
| "answer": "$$\nT=2\\pi\\sqrt{\\frac{\\frac{1}{(z_0^{2}+R^{2})^{3/2}}(\\frac{\\mu_{0}n I\\pi r^{2}}{2L}(1-\\frac{z_0}{\\sqrt{z_0^{2}+R^{2}}})-I_{0})}{g\\left[(\\frac{\\mu_{0}n I\\pi r^{2}}{2L}-I_{0})\\frac{3z_{0}}{(z_{0}^{2}+R^{2})^{5/2}}-\\frac{\\mu_{0}n I\\pi r^{2}}{2L}\\frac{3z_{0}^{2}-R^{2}}{(z_{0}^{2}+R^{2})^{3}}\\right]}}\n$$\n\n", |
| "fixed_answer": "$$\nT=2\\pi\\sqrt{\\frac{\\frac{1}{(z_0^{2}+R^{2})^{3/2}}(\\frac{\\mu_{0}n I\\pi r^{2}}{2L}(1-\\frac{z_0}{\\sqrt{z_0^{2}+R^{2}}})-I_{0})}{g[(\\frac{\\mu_{0}n I\\pi r^{2}}{2L}-I_{0})\\frac{3z_{0}}{(z_{0}^{2}+R^{2})^{5/2}}-\\frac{\\mu_{0}n I\\pi r^{2}}{2L}\\frac{3z_{0}^{2}-R^{2}}{(z_{0}^{2}+R^{2})^{3}}]}}\n$$\n\n" |
| }, |
| { |
| "id": 419, |
| "tag": "OPTICS", |
| "answer": "\\[\nR_{s}=\\left(1+\\left(\\frac{2\\mathrm{e}^{-\\frac{2\\pi d}{\\lambda}\\sqrt{\\mathrm{sin}i_2^2-n^2}}}{1 - \\mathrm{e}^{-\\frac{4\\pi d}{\\lambda}\\sqrt{\\mathrm{sin}i_2^2-n^2}}}\\frac{2\\sqrt{(1 - \\sin^{2}i_{2})(\\sin^{2}i_{2}-n^{2})}}{1 - n^{2}}\\right)^{2}\\right)^{-1}\n\\] ", |
| "fixed_answer": "$$\nR_{s}=(1+(\\frac{2e^{-\\frac{2\\pi d}{\\lambda}\\sqrt{\\sin^{2}(i_2)-n^2}}}{1 - e^{-\\frac{4\\pi d}{\\lambda}\\sqrt{\\sin^{2}(i_2)-n^2}}}\\frac{2\\sqrt{(1 - \\sin^{2}(i_2))(\\sin^{2}(i_{2})-n^{2})}}{1 - n^{2}})^{2})^{-1}\n$$" |
| }, |
| { |
| "id": 423, |
| "tag": "THERMODYNAMICS", |
| "answer": "$$\nS_{D}-S_{A}=\\frac{3p_Av_A}{T_A}\\left(\\left(\\frac{p_Dv_D^2}{p_Av_A^2}\\right)^{\\frac{1}{3}}-1\\right)\n$$\n", |
| "fixed_answer": "$$\nS_{D}-S_{A}=\\frac{3p_Av_A}{T_A}((\\frac{p_Dv_D^2}{p_Av_A^2})^{\\frac{1}{3}}-1)\n$$\n" |
| }, |
| { |
| "id": 497, |
| "tag": "MECHANICS", |
| "answer": "\\[\n\\Delta A = \\frac{m A_0}{4M}\\left[\\frac{1}{3}(\\gamma_1 - 1) + \\gamma_2\\right]\n\\] ", |
| "fixed_answer": "$$\n\\Delta A = \\frac{m A_0}{4M}[\\frac{1}{3}(\\mathrm{\\gamma}_1 - 1) + \\mathrm{\\gamma}_2]\n$$" |
| }, |
| { |
| "id": 498, |
| "tag": "OPTICS", |
| "answer": "\\[ \\varphi = \\frac{2\\pi d}{\\lambda (n_b - n_a)} \\left[ n_b\\sqrt{n_b^2 - \\sin^2 i} - n_a\\sqrt{n_a^2 - \\sin^2 i} - \\sin^2 i \\ln \\left( \\frac{n_b + \\sqrt{n_b^2 - \\sin^2 i}}{n_a + \\sqrt{n_a^2 - \\sin^2 i}} \\right) \\right] \\]", |
| "fixed_answer": "$$\n\\varphi = \\frac{2\\pi d}{\\lambda (n_b - n_a)} [ n_b\\sqrt{n_b^2 - \\sin^2 i} - n_a\\sqrt{n_a^2 - \\sin^2 i} - \\sin^2 i \\ln ( \\frac{n_b + \\sqrt{n_b^2 - \\sin^2 i}}{n_a + \\sqrt{n_a^2 - \\sin^2 i}})]\n$$" |
| }, |
| { |
| "id": 648, |
| "tag": "MECHANICS", |
| "answer": "$$\n\\sqrt{\\frac{2kQq}{mR} \\cdot \\frac{1-\\cos\\left(\\frac{\\pi}{n}\\right)}{2\\cos\\left(\\frac{\\pi}{n}\\right)-1}}\n$$", |
| "fixed_answer": "$$\\sqrt{\\frac{2kQq}{mR} \\cdot \\frac{1-\\cos(\\frac{\\pi}{n})}{2\\cos(\\frac{\\pi}{n})-1}}$$" |
| }, |
| { |
| "id": 654, |
| "tag": "MECHANICS", |
| "answer": "$$\n\\frac{2nSmc(v_{1}-v_{2})^{2}}{(1-\\left(\\frac{v_1}{c}\\right)^2)\\sqrt{1-\\left(\\frac{v_2}{c}\\right)^2}h\\nu_{0}}\n$$", |
| "fixed_answer": "$$\n\\frac{2nSmc(v_{1}-v_{2})^{2}}{(1-(\\frac{v_1}{c})^2)\\sqrt{1-(\\frac{v_2}{c})^2}h\\nu_{0}}\n$$" |
| } |
| ] |
