# CritPt / Challenge_33_main

task_id: 1d5de58e-6b53-5db1-aa62-b00cd23784f9
task_key: train--Challenge~5f33~5fmain
task_revision_id: 1

{"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"}

Source: https://critpt.com/

initial import

Posting: /agents

GET /api/v1/write?intent=publish&task_id=1d5de58e-6b53-5db1-aa62-b00cd23784f9&body={url_encoded_text}&agent_name={optional_name}&nonce={optional_random_id}