{
  "index": "1986-B-1",
  "type": "GEO",
  "tag": [
    "GEO",
    "ALG"
  ],
  "difficulty": "",
  "question": "Inscribe a rectangle of base $b$ and height $h$ in a circle of radius\none, and inscribe an isosceles triangle in the region of the circle\ncut off by one base of the rectangle (with that side as the base of\nthe triangle).\nFor what\nvalue of $h$ do the rectangle and triangle have the same area?",
  "solution": "Solution. The radius \\( O X \\) (see Figure 3) has length equal to \\( h / 2 \\) plus the altitude of the triangle, so the altitude of the triangle is \\( 1-h / 2 \\). If the rectangle and triangle have the same area, then \\( b h=\\frac{1}{2} b(1-h / 2) \\). Cancel \\( b \\) and solve for \\( h \\) to get \\( h=2 / 5 \\).",
  "vars": [
    "h"
  ],
  "params": [
    "b",
    "O",
    "X"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "h": "heightvar",
        "b": "basewidth",
        "O": "centerpoint",
        "X": "edgepoint"
      },
      "question": "Inscribe a rectangle of base $basewidth$ and height $heightvar$ in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of $heightvar$ do the rectangle and triangle have the same area?",
      "solution": "Solution. The radius \\( centerpoint edgepoint \\) (see Figure 3) has length equal to \\( heightvar / 2 \\) plus the altitude of the triangle, so the altitude of the triangle is \\( 1-heightvar / 2 \\). If the rectangle and triangle have the same area, then \\( basewidth heightvar=\\frac{1}{2} basewidth(1-heightvar / 2) \\). Cancel \\( basewidth \\) and solve for \\( heightvar \\) to get \\( heightvar=2 / 5 \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "h": "wisterias",
        "b": "tangerine",
        "O": "gazeboing",
        "X": "windchime"
      },
      "question": "Inscribe a rectangle of base $tangerine$ and height $wisterias$ in a circle of radius\none, and inscribe an isosceles triangle in the region of the circle\ncut off by one base of the rectangle (with that side as the base of\nthe triangle).\nFor what\nvalue of $wisterias$ do the rectangle and triangle have the same area?",
      "solution": "Solution. The radius \\( gazeboing windchime \\) (see Figure 3) has length equal to \\( wisterias / 2 \\) plus the altitude of the triangle, so the altitude of the triangle is \\( 1-wisterias / 2 \\). If the rectangle and triangle have the same area, then \\( tangerine wisterias=\\frac{1}{2} tangerine(1-wisterias / 2) \\). Cancel \\( tangerine \\) and solve for \\( wisterias \\) to get \\( wisterias=2 / 5 \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "h": "depthvalue",
        "b": "apexwidth",
        "O": "periphery",
        "X": "centerpoint"
      },
      "question": "Inscribe a rectangle of base $apexwidth$ and height $depthvalue$ in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of $depthvalue$ do the rectangle and triangle have the same area?",
      "solution": "Solution. The radius \\( periphery centerpoint \\) (see Figure 3) has length equal to \\( depthvalue / 2 \\) plus the altitude of the triangle, so the altitude of the triangle is \\( 1-depthvalue / 2 \\). If the rectangle and triangle have the same area, then \\( apexwidth depthvalue=\\frac{1}{2} apexwidth(1-depthvalue / 2) \\). Cancel \\( apexwidth \\) and solve for \\( depthvalue \\) to get \\( depthvalue=2 / 5 \\)."
    },
    "garbled_string": {
      "map": {
        "h": "qzxwvtnp",
        "b": "hjgrksla",
        "O": "pwndlrta",
        "X": "mzcvqoba"
      },
      "question": "Inscribe a rectangle of base $hjgrksla$ and height $qzxwvtnp$ in a circle of radius\none, and inscribe an isosceles triangle in the region of the circle\ncut off by one base of the rectangle (with that side as the base of\nthe triangle).\nFor what\nvalue of $qzxwvtnp$ do the rectangle and triangle have the same area?",
      "solution": "Solution. The radius \\( pwndlrta mzcvqoba \\) (see Figure 3) has length equal to \\( qzxwvtnp / 2 \\) plus the altitude of the triangle, so the altitude of the triangle is \\( 1-qzxwvtnp / 2 \\). If the rectangle and triangle have the same area, then \\( hjgrksla qzxwvtnp=\\frac{1}{2} hjgrksla(1-qzxwvtnp / 2) \\). Cancel \\( hjgrksla \\) and solve for \\( qzxwvtnp \\) to get \\( qzxwvtnp=2 / 5 \\)."
    },
    "kernel_variant": {
      "question": "A sphere of radius 3 is centered at the origin.  A rectangular box whose horizontal cross-sections are squares is inscribed so that its vertical faces are parallel to the coordinate planes; let the square base have side b and let the box’s height be h.  In the lower spherical cap (below the box) a square pyramid is erected whose base is the lower face of the box and whose apex is the lowest point of the sphere.  For what value of h do the box and the pyramid have equal volume?",
      "solution": "(≈60 words)  \nPlace O (0,0,0).  Box vertices (±b/2, ±b/2, ±h/2) lie on x²+y²+z²=9, so  \n 2(b/2)²+(h/2)²=9 ⇒ 2b²+h²=36.  \nThe square pyramid’s apex is P (0,0,−3); its altitude from the base z=−h/2 is 3−h/2.  \nVolumes: box = b²h, pyramid = (1/3)b²(3−h/2).  \nEquate and cancel b²>0: h=(3−h/2)/3 ⇒ 3h=3−h/2 ⇒ 6h=6−h ⇒ 7h=6, hence  \n h = 6/7.",
      "_replacement_note": {
        "replaced_at": "2025-07-05T22:17:12.089083",
        "reason": "Original kernel variant was too easy compared to the original problem"
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}