{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"研究水準の物理問題で、科学的理解と多段階の推論・計算を評価するベンチマークです。公開データには70の課題があり、問題文とコード雛形を組み合わせて解答を構成します。\n\nCritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex"},"task_id":"1d5de58e-6b53-5db1-aa62-b00cd23784f9","task_key":"train--Challenge~5f33~5fmain","task_revision_id":"1","upstream_id":"Challenge_33_main","short_description":"Consider the Hamiltonian for three kinds of particles labeled by $A$, $B$, $C$:","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return the value of a + 10b + 100c and s, and the kinds of particles that will form crystal state.\\n\\n    Inputs\\n    ----------\\n    None\\n\\n    Outputs\\n    ----------\\n    crystal_particles: set[str], the kinds of particles that will form crystal state when $r > r_o$; choose from {'A', 'B', 'C'}\\n    a_plus_10b_plus_100c: float, the value of a + 10b + 100c; $r_o$ scales with $v^a w^b z^c$\\n    s: float, the value of s; $r_o \\\\geq 10^s$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    crystal_particles = ...\\n    a_plus_10b_plus_100c = ...\\n    s = ...\\n    # ---------------------------------------------------------------\\n\\n    return crystal_particles, a_plus_10b_plus_100c, s\",\"problem_description\":\"# Problem setup:\\n\\nConsider the Hamiltonian for three kinds of particles labeled by $A$, $B$, $C$:\\n\\n$\\nH = -\\\\frac{1}{2} \\\\sum_i \\\\left( v\\\\nabla_{A_i}^{\\\\alpha} + w\\\\nabla_{B_i}^{\\\\beta} + w\\\\nabla_{C_i}^{\\\\beta} \\\\right)\\n+ \\\\sum_{i<j} \\\\frac{z}{|\\\\mathbf{r}_{A_i} - \\\\mathbf{r}_{A_j}|^{\\\\gamma}}\\n+ \\\\sum_{i<j} \\\\frac{z}{|\\\\mathbf{r}_{B_i} - \\\\mathbf{r}_{B_j}|^{\\\\eta}}\\n+ \\\\sum_{i<j} \\\\frac{z}{|\\\\mathbf{r}_{C_i} - \\\\mathbf{r}_{C_j}|^{\\\\eta}}\\n- \\\\sum_{i,j} \\\\frac{z^2}{\\\\sqrt{|\\\\mathbf{r}_{A_i} - \\\\mathbf{r}_{B_j}|^{\\\\xi} + d^2}}\\n- \\\\sum_{i,j} \\\\frac{z^4}{\\\\sqrt{|\\\\mathbf{r}_{A_i} - \\\\mathbf{r}_{C_j}|^{2\\\\eta} + f^3}}\\n+ \\\\sum_{i,j} \\\\frac{z}{\\\\sqrt{|\\\\mathbf{r}_{C_i} - \\\\mathbf{r}_{B_j}|^{2\\\\eta} + g^4}}.\\n$\\n\\nHere, $v$ and $w$ are kinetic coefficients; $z$ is the interaction strength; $f$, $g$ and $d$ are particle layer separations; $\\\\alpha$ and $\\\\beta$ are dispersion powers; $\\\\gamma$, $\\\\eta$ and $\\\\xi$ are interaction dimensions; and $i$ and $j$ are particle indices.\\n\\nSuppose\\n\\n\\\\begin{cases}\\n(\\\\alpha + z^5 + \\\\xi^{2.1})\\\\ln (g+\\\\alpha^2-3\\\\eta) + \\\\alpha^z \\\\ln(\\\\beta^3 - \\\\xi^2 + \\\\alpha\\\\eta + 6^{\\\\alpha-2}+1) = 0,\\\\\\\\[8pt]\\n((\\\\log_{10} v)^2 - 2 \\\\log_{10}z \\\\log_{10} v + 5^2 -81)^{4} + 2^{d}\\\\,g^4 = 0,\\\\\\\\[8pt]\\nv^2(\\\\ln z)^v (9^{\\\\log_{10} (w/z)} -3^4)^v + \\\\dfrac{\\\\ln(1+f^2)}{f^3} = 0,\\\\\\\\[8pt]\\n(0.25(\\\\log_{10} z + 1)^3 + 3^2 +\\\\ln e^3)^4+2^8)^8 + 1249e^{-d} = 0,\\\\\\\\[8pt]\\ng^{3.5} + (\\\\alpha + g + \\\\log_{10}(v/w) - 3)^{10+v} = 0,\\\\\\\\[8pt]\\n(w\\\\,v - 10)^2\\\\,g^6 + (2\\\\xi - \\\\alpha^{2+g})^2 = 0,\\\\\\\\[8pt]\\n3^{-\\\\frac{z\\\\,v^2}{w}} \\\\dfrac{g \\\\alpha}{\\\\xi} + (\\\\alpha\\\\beta\\\\eta - 2^{2+\\\\alpha}+3\\\\xi)^4 = 0.\\\\\\\\\\n\\\\end{cases}\\n\\n\\nSuppose that a phase transition appears when the average distance between particles is less than\\na critical value $r_o$.\\n\\n# Main problem:\\n\\nWhich kinds of particles will form a crystal state when $r > r_o$? Suppose $r_o$ scales with $v^a w^b z^c$ and $r_o \\\\geq 10^s$. Please compute a + 10b + 100c and s.\\n\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}