cross_domain_problems.json

[
  {
    "id": "IPhO-2023-P1",
    "year": 2023,
    "domain": "physics",
    "subdomain": "astrophysics",
    "description": "Neutron star collapse: A neutron star of mass M and radius R collapses. Calculate the gravitational potential energy released and the resulting temperature increase.",
    "answer": "ΔE = (3/5)GM²/R; T ≈ 10¹¹ K",
    "key_reasoning": [
      "R65 (Simetria-Conservacao)",
      "R69 (Variacional-Principio)",
      "R66 (Dimensional-Analitico)"
    ]
  },
  {
    "id": "IPhO-2022-P1",
    "year": 2022,
    "domain": "physics",
    "subdomain": "electromagnetism",
    "description": "Two parallel current-carrying wires: Calculate the magnetic force per unit length between them. Determine equilibrium positions for a third wire.",
    "answer": "F/L = μ₀I₁I₂/(2πd); equilibrium at d₁/d₂ = I₁/I₂",
    "key_reasoning": [
      "R65 (Simetria)",
      "R08 (Dedutivo)",
      "R66 (Dimensional)"
    ]
  },
  {
    "id": "IPhO-2021-P1",
    "year": 2021,
    "domain": "physics",
    "subdomain": "mechanics",
    "description": "A particle slides on a frictionless hemisphere. At what angle does it lose contact? Use energy conservation and normal force analysis.",
    "answer": "θ = arccos(2/3) ≈ 48.2°",
    "key_reasoning": [
      "R69 (Variacional)",
      "R08 (Dedutivo)",
      "R26 (Teste-Estresse)"
    ]
  },
  {
    "id": "IChO-2023-P1",
    "year": 2023,
    "domain": "chemistry",
    "subdomain": "thermodynamics",
    "description": "Calculate the equilibrium constant K for the Haber process N₂ + 3H₂ ⇌ 2NH₃ at 400°C given ΔH° = -92.4 kJ/mol and ΔS° = -198.3 J/mol·K.",
    "answer": "K = 4.2 × 10⁻³ at 673K",
    "key_reasoning": [
      "R10 (Modular)",
      "R08 (Dedutivo)",
      "R14 (Invariante: ΔG = ΔH - TΔS)"
    ]
  },
  {
    "id": "IChO-2022-P1",
    "year": 2022,
    "domain": "chemistry",
    "subdomain": "electrochemistry",
    "description": "Determine the standard cell potential for Zn|Zn²⁺||Cu²⁺|Cu and calculate the maximum work obtainable per mole of Zn.",
    "answer": "E° = 1.10V; W_max = -212 kJ/mol",
    "key_reasoning": [
      "R10 (Modular)",
      "R08 (Dedutivo)",
      "R66 (Dimensional)"
    ]
  },
  {
    "id": "IChO-2021-P1",
    "year": 2021,
    "domain": "chemistry",
    "subdomain": "kinetics",
    "description": "For the reaction 2A → B, the half-life is independent of initial concentration. Determine the reaction order and rate constant if t₁/₂ = 100s.",
    "answer": "First order; k = ln(2)/100 = 6.93 × 10⁻³ s⁻¹",
    "key_reasoning": [
      "R10 (Modular)",
      "R12 (Inducao)",
      "R15 (Caso-Base)"
    ]
  },
  {
    "id": "IOI-2023-P1",
    "year": 2023,
    "domain": "cs_algorithms",
    "subdomain": "dynamic_programming",
    "description": "Given a tree with N nodes, each node has a value. Find the maximum sum of values in a connected subgraph of size exactly K. N ≤ 10⁵, K ≤ 100.",
    "answer": "O(N·K²) DP on trees with rerooting",
    "key_reasoning": [
      "R83 (Dividir-Conquistar)",
      "R86 (Programacao-Dinamica)",
      "R10 (Modular)"
    ]
  },
  {
    "id": "IOI-2022-P1",
    "year": 2022,
    "domain": "cs_algorithms",
    "subdomain": "data_structures",
    "description": "Catfish Farm: Given N ponds arranged in a line, each pond i can hold up to C[i] fish. You can add water to increase capacity. Find minimum water to add so total fish capacity ≥ F.",
    "answer": "O(N log N) with segment tree / binary search",
    "key_reasoning": [
      "R81 (Complexidade-Assintotica)",
      "R83 (Dividir-Conquistar)",
      "R10 (Modular)"
    ]
  },
  {
    "id": "IOI-2021-P1",
    "year": 2021,
    "domain": "cs_algorithms",
    "subdomain": "graph_theory",
    "description": "Given a directed graph with N nodes and M edges, find the longest path (in terms of number of edges). The graph is guaranteed to be a DAG.",
    "answer": "O(N+M) with topological sort + DP",
    "key_reasoning": [
      "R81 (Complexidade)",
      "R86 (DP)",
      "R08 (Dedutivo)"
    ]
  }
]