{
  "index": "1947-A-4",
  "type": "ANA",
  "tag": [
    "ANA",
    "GEO"
  ],
  "difficulty": "",
  "question": "4. A coast artillery gun can fire at any angle of elevation between \\( 0^{\\circ} \\) and \\( 90^{\\circ} \\) in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant \\( \\left(=v_{0}\\right) \\), determine the set \\( H \\) of points in the plane and above the horizontal which can be hit.",
  "solution": "Solution. We take coordinates with origin at the gun, the \\( y \\)-axis vertical, and the \\( x \\)-axis horizontal in the direction of the fire. For a given angle \\( \\alpha \\) and the prescribed initial conditions the equations of motion\n\\[\n\\frac{d^{2} x}{d t^{2}}=0, \\quad \\frac{d^{2} y}{d t^{2}}=-g\n\\]\nlead to\n\\[\n\\begin{array}{l}\nx=v_{0} t \\cos \\alpha \\\\\ny=v_{0} t \\sin \\alpha-\\frac{1}{2} g t^{2}\n\\end{array}\n\\]\n\nElimination of \\( t \\) gives\n\\[\ny=x \\tan \\alpha-\\frac{g}{2 v_{0}{ }^{2}} x^{2} \\sec ^{2} \\alpha\n\\]\n\nFor a fixed positive \\( \\boldsymbol{x} \\) this can be written\n\\[\ny=\\frac{\\nu_{0}^{2}}{2 g}-\\frac{g x^{2}}{2 v_{0}{ }^{2}}-\\frac{g x^{2}}{2 v_{0}{ }^{2}}\\left(\\tan \\alpha-\\frac{\\nu_{0}{ }^{2}}{g x}\\right)^{2},\n\\]\nwhence it is clear that we can choose \\( \\alpha \\) so as to hit the point \\( (x, y) \\) if and only if\n\\[\ny \\leq \\frac{\\nu_{0}{ }^{2}}{2 g}-\\frac{g x^{2}}{2 \\nu_{0}{ }^{2}} .\n\\]\n\nTo hit a point \\( (0, y) \\), we fire straight up, i.e., take \\( \\alpha=90^{\\circ} \\); then the parametric equation for \\( y \\) can be written\n\\[\ny=\\frac{v_{0}{ }^{2}}{2 g}-\\frac{v_{0}{ }^{2}}{2 g}\\left(\\frac{g}{v_{0}} t-1\\right)^{2}\n\\]\nand it is clear that we can reach \\( (0, y) \\) if and only if \\( y \\leq \\nu_{0}{ }^{2} / 2 g \\). Therefore, the desired set \\( H \\) is defined by the inequalities (1) and \\( 0 \\leq x, 0<y \\).\n\nHistorical Note. The parabola \\( y_{\\text {max }}=\\left(\\nu_{0}{ }^{2} / 2 g\\right)-\\left(g x^{2} / 2 \\nu_{0}{ }^{2}\\right) \\) is sometimes called the \"parabola of safety.\" An airplane staying outside of this parabola cannot be hit by the artillery gun.\n\nSee \"Envelopes,\" by V. G. Boltyanskii (translated from the Russian by Robert B. Brown), vol. 12 in Popular Lectures in Mathematics. Pergamon Press, New York, 1964, where this problem is discussed.\n\nRemark. This problem is essentially the same as problem 6(ii) of the second competition. A different solution is given there.",
  "vars": [
    "x",
    "y",
    "t",
    "\\\\alpha",
    "y_max",
    "H"
  ],
  "params": [
    "g",
    "v_0",
    "\\\\nu_0"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "horizdist",
        "y": "vertheight",
        "t": "timevar",
        "\\alpha": "elevangle",
        "y_max": "maxheight",
        "H": "hitset",
        "g": "gravconst",
        "v_0": "muzzlevel",
        "\\nu_0": "iniveloc"
      },
      "question": "4. A coast artillery gun can fire at any angle of elevation between \\( 0^{\\circ} \\) and \\( 90^{\\circ} \\) in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant \\( (=\\muzzlevel) \\), determine the set \\( \\hitset \\) of points in the plane and above the horizontal which can be hit.",
      "solution": "Solution. We take coordinates with origin at the gun, the \\( vertheight \\)-axis vertical, and the \\( horizdist \\)-axis horizontal in the direction of the fire. For a given angle \\( elevangle \\) and the prescribed initial conditions the equations of motion\n\\[\n\\frac{d^{2} horizdist}{d timevar^{2}}=0, \\quad \\frac{d^{2} vertheight}{d timevar^{2}}=-gravconst\n\\]\nlead to\n\\[\n\\begin{array}{l}\nhorizdist= muzzlevel\\, timevar \\cos elevangle \\\\\nvertheight= muzzlevel\\, timevar \\sin elevangle-\\frac{1}{2} \\, gravconst\\, timevar^{2}\n\\end{array}\n\\]\n\nElimination of \\( timevar \\) gives\n\\[\nvertheight=horizdist \\tan elevangle-\\frac{gravconst}{2\\,muzzlevel^{2}}\\,horizdist^{2} \\sec ^{2} elevangle\n\\]\n\nFor a fixed positive \\( horizdist \\) this can be written\n\\[\nvertheight=\\frac{iniveloc^{2}}{2\\,gravconst}-\\frac{gravconst\\,horizdist^{2}}{2\\,muzzlevel^{2}}-\\frac{gravconst\\,horizdist^{2}}{2\\,muzzlevel^{2}}\\left(\\tan elevangle-\\frac{iniveloc^{2}}{gravconst\\,horizdist}\\right)^{2},\n\\]\nwhence it is clear that we can choose \\( elevangle \\) so as to hit the point \\( (horizdist,vertheight) \\) if and only if\n\\[\nvertheight \\le \\frac{iniveloc^{2}}{2\\,gravconst}-\\frac{gravconst\\,horizdist^{2}}{2\\,iniveloc^{2}}.\n\\]\n\nTo hit a point \\( (0,vertheight) \\), we fire straight up, i.e., take \\( elevangle=90^{\\circ} \\); then the parametric equation for \\( vertheight \\) can be written\n\\[\nvertheight=\\frac{muzzlevel^{2}}{2\\,gravconst}-\\frac{muzzlevel^{2}}{2\\,gravconst}\\left(\\frac{gravconst}{muzzlevel}\\,timevar-1\\right)^{2}\n\\]\nand it is clear that we can reach \\( (0,vertheight) \\) if and only if \\( vertheight \\le iniveloc^{2}/(2\\,gravconst) \\). Therefore, the desired set \\( \\hitset \\) is defined by the inequalities (1) and \\( 0 \\le horizdist,\\;0<vertheight \\).\n\nHistorical Note. The parabola \\( maxheight=\\left(iniveloc^{2}/(2\\,gravconst)\\right)-\\left(gravconst\\,horizdist^{2}/(2\\,iniveloc^{2})\\right) \\) is sometimes called the \"parabola of safety.\" An airplane staying outside of this parabola cannot be hit by the artillery gun.\n\nSee \"Envelopes,\" by V. G. Boltyanskii (translated from the Russian by Robert B. Brown), vol. 12 in Popular Lectures in Mathematics. Pergamon Press, New York, 1964, where this problem is discussed.\n\nRemark. This problem is essentially the same as problem 6(ii) of the second competition. A different solution is given there."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "blueberry",
        "y": "drumstick",
        "t": "horseback",
        "\\\\alpha": "staircase",
        "y_max": "panhandle",
        "H": "marshland",
        "g": "tortoise",
        "v_0": "cherrypie",
        "\\\\nu_0": "jellybean"
      },
      "question": "4. A coast artillery gun can fire at any angle of elevation between \\( 0^{\\circ} \\) and \\( 90^{\\circ} \\) in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant \\( \\left(=cherrypie\\right) \\), determine the set \\( marshland \\) of points in the plane and above the horizontal which can be hit.",
      "solution": "Solution. We take coordinates with origin at the gun, the \\( drumstick \\)-axis vertical, and the \\( blueberry \\)-axis horizontal in the direction of the fire. For a given angle \\( staircase \\) and the prescribed initial conditions the equations of motion\n\\[\n\\frac{d^{2} blueberry}{d horseback^{2}}=0, \\quad \\frac{d^{2} drumstick}{d horseback^{2}}=-tortoise\n\\]\nlead to\n\\[\n\\begin{array}{l}\nblueberry=cherrypie horseback \\cos staircase \\\\\ndrumstick=cherrypie horseback \\sin staircase-\\frac{1}{2} tortoise horseback^{2}\n\\end{array}\n\\]\n\nElimination of \\( horseback \\) gives\n\\[\ndrumstick=blueberry \\tan staircase-\\frac{tortoise}{2 cherrypie^{2}} blueberry^{2} \\sec ^{2} staircase\n\\]\n\nFor a fixed positive \\( \\boldsymbol{blueberry} \\) this can be written\n\\[\ndrumstick=\\frac{jellybean^{2}}{2 tortoise}-\\frac{tortoise blueberry^{2}}{2 cherrypie^{2}}-\\frac{tortoise blueberry^{2}}{2 cherrypie^{2}}\\left(\\tan staircase-\\frac{jellybean^{2}}{tortoise blueberry}\\right)^{2},\n\\]\nwhence it is clear that we can choose \\( staircase \\) so as to hit the point \\( (blueberry, drumstick) \\) if and only if\n\\[\ndrumstick \\leq \\frac{jellybean^{2}}{2 tortoise}-\\frac{tortoise blueberry^{2}}{2 jellybean^{2}} .\n\\]\n\nTo hit a point \\( (0, drumstick) \\), we fire straight up, i.e., take \\( staircase=90^{\\circ} \\); then the parametric equation for \\( drumstick \\) can be written\n\\[\ndrumstick=\\frac{cherrypie^{2}}{2 tortoise}-\\frac{cherrypie^{2}}{2 tortoise}\\left(\\frac{tortoise}{cherrypie} horseback-1\\right)^{2}\n\\]\nand it is clear that we can reach \\( (0, drumstick) \\) if and only if \\( drumstick \\leq jellybean^{2} / 2 tortoise \\). Therefore, the desired set \\( marshland \\) is defined by the inequalities (1) and \\( 0 \\leq blueberry, 0<drumstick \\).\n\nHistorical Note. The parabola \\( panhandle=\\left(jellybean^{2} / 2 tortoise\\right)-\\left(tortoise blueberry^{2} / 2 jellybean^{2}\\right) \\) is sometimes called the \"parabola of safety.\" An airplane staying outside of this parabola cannot be hit by the artillery gun.\n\nSee \"Envelopes,\" by V. G. Boltyanskii (translated from the Russian by Robert B. Brown), vol. 12 in Popular Lectures in Mathematics. Pergamon Press, New York, 1964, where this problem is discussed.\n\nRemark. This problem is essentially the same as problem 6(ii) of the second competition. A different solution is given there."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "verticalpos",
        "y": "horizontalpos",
        "t": "distance",
        "\\alpha": "depression",
        "y_max": "horizontalmin",
        "H": "unreachable",
        "g": "antigrav",
        "v_0": "slowrate",
        "\\nu_0": "slowspeed"
      },
      "question": "4. A coast artillery gun can fire at any angle of elevation between \\( 0^{\\circ} \\) and \\( 90^{\\circ} \\) in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant \\( \\left(=slowrate\\right) \\), determine the set \\( unreachable \\) of points in the plane and above the horizontal which can be hit.",
      "solution": "Solution. We take coordinates with origin at the gun, the \\( horizontalpos \\)-axis vertical, and the \\( verticalpos \\)-axis horizontal in the direction of the fire. For a given angle \\( depression \\) and the prescribed initial conditions the equations of motion\n\\[\n\\frac{d^{2} verticalpos}{d distance^{2}}=0, \\quad \\frac{d^{2} horizontalpos}{d distance^{2}}=-antigrav\n\\]\nlead to\n\\[\n\\begin{array}{l}\nverticalpos=slowrate\\ distance \\cos depression \\\\\nhorizontalpos=slowrate\\ distance \\sin depression-\\frac{1}{2} antigrav\\ distance^{2}\n\\end{array}\n\\]\n\nElimination of \\( distance \\) gives\n\\[\nhorizontalpos=verticalpos \\tan depression-\\frac{antigrav}{2 slowrate^{2}} verticalpos^{2} \\sec ^{2} depression\n\\]\n\nFor a fixed positive \\( \\boldsymbol{verticalpos} \\) this can be written\n\\[\nhorizontalpos=\\frac{slowspeed^{2}}{2 antigrav}-\\frac{antigrav\\ verticalpos^{2}}{2 slowrate^{2}}-\\frac{antigrav\\ verticalpos^{2}}{2 slowrate^{2}}\\left(\\tan depression-\\frac{slowspeed^{2}}{antigrav\\ verticalpos}\\right)^{2},\n\\]\nwhence it is clear that we can choose \\( depression \\) so as to hit the point \\( (verticalpos, horizontalpos) \\) if and only if\n\\[\nhorizontalpos \\leq \\frac{slowspeed^{2}}{2 antigrav}-\\frac{antigrav\\ verticalpos^{2}}{2 slowspeed^{2}} .\n\\]\n\nTo hit a point \\( (0, horizontalpos) \\), we fire straight up, i.e., take \\( depression=90^{\\circ} \\); then the parametric equation for \\( horizontalpos \\) can be written\n\\[\nhorizontalpos=\\frac{slowrate^{2}}{2 antigrav}-\\frac{slowrate^{2}}{2 antigrav}\\left(\\frac{antigrav}{slowrate}\\ distance-1\\right)^{2}\n\\]\nand it is clear that we can reach \\( (0, horizontalpos) \\) if and only if \\( horizontalpos \\leq slowspeed^{2} / 2 antigrav \\). Therefore, the desired set \\( unreachable \\) is defined by the inequalities (1) and \\( 0 \\leq verticalpos, 0<horizontalpos \\).\n\nHistorical Note. The parabola \\( horizontalmin=\\left(slowspeed^{2} / 2 antigrav\\right)-\\left(antigrav\\ verticalpos^{2} / 2 slowspeed^{2}\\right) \\) is sometimes called the \"parabola of safety.\" An airplane staying outside of this parabola cannot be hit by the artillery gun.\n\nSee \"Envelopes,\" by V. G. Boltyanskii (translated from the Russian by Robert B. Brown), vol. 12 in Popular Lectures in Mathematics. Pergamon Press, New York, 1964, where this problem is discussed.\n\nRemark. This problem is essentially the same as problem 6(ii) of the second competition. A different solution is given there."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "y": "hjgrksla",
        "t": "skdmvplq",
        "\\alpha": "rjfndkhs",
        "y_max": "jclqzuep",
        "H": "ugpnxwle",
        "g": "zsoqlmnr",
        "v_0": "dbcialyo",
        "\\nu_0": "oqjfdvhw"
      },
      "question": "Problem:\n<<<\n4. A coast artillery gun can fire at any angle of elevation between \\( 0^{\\circ} \\) and \\( 90^{\\circ} \\) in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant \\( (=dbcialyo) \\), determine the set \\( ugpnxwle \\) of points in the plane and above the horizontal which can be hit.\n>>>\n",
      "solution": "Solution:\n<<<\nSolution. We take coordinates with origin at the gun, the \\( hjgrksla \\)-axis vertical, and the \\( qzxwvtnp \\)-axis horizontal in the direction of the fire. For a given angle \\( rjfndkhs \\) and the prescribed initial conditions the equations of motion\n\\[\n\\frac{d^{2} qzxwvtnp}{d skdmvplq^{2}}=0, \\quad \\frac{d^{2} hjgrksla}{d skdmvplq^{2}}=-zsoqlmnr\n\\]\nlead to\n\\[\n\\begin{array}{l}\nqzxwvtnp = dbcialyo skdmvplq \\cos rjfndkhs \\\\\nhjgrksla = dbcialyo skdmvplq \\sin rjfndkhs - \\frac{1}{2} zsoqlmnr skdmvplq^{2}\n\\end{array}\n\\]\n\nElimination of \\( skdmvplq \\) gives\n\\[\nhjgrksla = qzxwvtnp \\tan rjfndkhs - \\frac{zsoqlmnr}{2 dbcialyo^{2}} qzxwvtnp^{2} \\sec^{2} rjfndkhs\n\\]\n\nFor a fixed positive \\( \\boldsymbol{qzxwvtnp} \\) this can be written\n\\[\nhjgrksla = \\frac{oqjfdvhw^{2}}{2 zsoqlmnr} - \\frac{zsoqlmnr qzxwvtnp^{2}}{2 dbcialyo^{2}} - \\frac{zsoqlmnr qzxwvtnp^{2}}{2 dbcialyo^{2}} \\left( \\tan rjfndkhs - \\frac{oqjfdvhw^{2}}{zsoqlmnr qzxwvtnp} \\right)^{2},\n\\]\nwhence it is clear that we can choose \\( rjfndkhs \\) so as to hit the point \\( (qzxwvtnp, hjgrksla) \\) if and only if\n\\[\nhjgrksla \\leq \\frac{oqjfdvhw^{2}}{2 zsoqlmnr} - \\frac{zsoqlmnr qzxwvtnp^{2}}{2 oqjfdvhw^{2}} .\n\\]\n\nTo hit a point \\( (0, hjgrksla) \\), we fire straight up, i.e., take \\( rjfndkhs=90^{\\circ} \\); then the parametric equation for \\( hjgrksla \\) can be written\n\\[\nhjgrksla = \\frac{dbcialyo^{2}}{2 zsoqlmnr} - \\frac{dbcialyo^{2}}{2 zsoqlmnr} \\left( \\frac{zsoqlmnr}{dbcialyo} skdmvplq - 1 \\right)^{2}\n\\]\nand it is clear that we can reach \\( (0, hjgrksla) \\) if and only if \\( hjgrksla \\leq oqjfdvhw^{2} / 2 zsoqlmnr \\). Therefore, the desired set \\( ugpnxwle \\) is defined by the inequalities (1) and \\( 0 \\leq qzxwvtnp, 0< hjgrksla \\).\n\nHistorical Note. The parabola \\( jclqzuep = ( oqjfdvhw^{2} / 2 zsoqlmnr ) - ( zsoqlmnr qzxwvtnp^{2} / 2 oqjfdvhw^{2} ) \\) is sometimes called the \"parabola of safety.\" An airplane staying outside of this parabola cannot be hit by the artillery gun.\n\nSee \"Envelopes,\" by V. G. Boltyanskii (translated from the Russian by Robert B. Brown), vol. 12 in Popular Lectures in Mathematics. Pergamon Press, New York, 1964, where this problem is discussed.\n\nRemark. This problem is essentially the same as problem 6(ii) of the second competition. A different solution is given there.\n>>>\n"
    },
    "kernel_variant": {
      "question": "A rail-gun has been emplaced on the air-less asteroid $(433)$ Eros at its local south pole.  \nChoose a right-handed Cartesian frame  \n\n$OX$ points toward local east,  \n$OY$ points toward local north,  \n$OZ$ is the outward local vertical,\n\nand put the breech of the gun at the origin $O$.  \n\nThe barrel can be steered in two independent angular directions  \n\n• elevation $\\beta$ (measured from the local horizontal) with $-70^{\\circ}\\le \\beta\\le 90^{\\circ}$;  \n $\\beta=90^{\\circ}$ gives a vertical shot, $\\beta<0^{\\circ}$ corresponds to depression below the horizontal;\n\n• azimuth $\\varphi$ (measured in the horizontal plane from the negative $X$-axis toward the $+Y$-axis) with $|\\varphi|\\le\\varphi_{0}$, where the fixed constant $\\varphi_{0}$ satisfies $0<\\varphi_{0}<90^{\\circ}$.  \n Thus $\\varphi=0$ means “due west’’ and admissible shots are confined to the horizontal wedge $-\\varphi_{0}\\le\\varphi\\le\\varphi_{0}$.\n\nEvery projectile leaves the muzzle with the same speed $U$ (independent of $\\beta$ and $\\varphi$).\n\nAfter exit the only forces acting are  \n\n1. the practically uniform asteroid gravity $\\mathbf g=(0,0,-\\gamma)$ with $\\gamma>0$,  \n2. a constant tangential “tidal-stress’’ acceleration $\\mathbf p=(\\kappa,0,0)$ of known magnitude $\\kappa\\ge0$ that points along (and is fixed in) the $X$-direction.\n\nAir drag is negligible, the rotation of Eros is ignored, and the local surface of the asteroid is approximated by the plane $Z=0$.  Only points with $Z>0$ are of interest.\n\nFor $t>0$ introduce  \n\n\\[\nQ(t;X,Y,Z)=\\bigl[X-\\tfrac12\\kappa t^{2}\\bigr]^{2}+Y^{2}+\\bigl[Z+\\tfrac12\\gamma t^{2}\\bigr]^{2},\\qquad\nC(t)=Z+\\tfrac12\\gamma t^{2},\\qquad\nT(t)=\\frac{Y}{-X+\\tfrac12\\kappa t^{2}}\\;(\\text{denominator}\\neq0).\n\\]\n\nPut  \n\n\\[\ns_{1}=\\sin(-70^{\\circ})=-\\sin70^{\\circ},\\qquad\ns_{2}=1,\\qquad\n\\tau=\\tan\\varphi_{0}.\n\\]\n\nLet $R$ be the set of all points $P=(X,Y,Z)$ with $Z>0$ that can be struck by an ideal projectile under the above restrictions.  Prove that  \n\n\\[\n\\boxed{\\,R=\\bigcup_{t>0}R(t)\\,},\n\\]\n\nwhere for each fixed flight time $t>0$\n\n\\[\n\\begin{aligned}\nR(t)=\\{(X,Y,Z)\\in\\mathbb R^{3}:\\;&Z>0,\\; \\text{and}\\\\\n&\\text{(i)}\\; Q(t;X,Y,Z)=U^{2}t^{2},\\\\\n&\\text{(ii)}\\; C(t)/(Ut)\\in[s_{1},s_{2}],\\\\\n&\\text{(iii)}\\;\\text{either }[-X+\\tfrac12\\kappa t^{2}>0\\ \\text{and}\\ |T(t)|\\le\\tau]\\\\\n&\\hphantom{\\text{(iii)}}\\text{or (vertical–shot case) }[-X+\\tfrac12\\kappa t^{2}=0\\ \\text{and}\\ Y=0]\\}.\n\\end{aligned}\n\\]\n\nFinally, describe geometrically how the simultaneous conditions (i)–(iii) carve the attainable region out of the one-parameter family of “shifted safety spheres’’ of radius $Ut$ whose centres travel along the space parabola $\\bigl(\\tfrac12\\kappa t^{2},0,-\\tfrac12\\gamma t^{2}\\bigr)$.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "Step 1.  Equations of motion.  \nBecause both $\\mathbf g$ and $\\mathbf p$ are constant, the trajectory of any fired projectile is  \n\n\\[\n\\mathbf r(t)=\\mathbf v_{0}t+\\tfrac12(\\mathbf p+\\mathbf g)t^{2}\\qquad(t\\ge0),\n\\]\n\nwith initial velocity  \n\n\\[\n\\mathbf v_{0}=U\\bigl(-\\cos\\beta\\cos\\varphi,\\; \\cos\\beta\\sin\\varphi,\\; \\sin\\beta\\bigr).\n\\]\n\nWriting the components and setting the impact time equal to the flight parameter $t$ gives\n\n\\[\n\\begin{aligned}\nX&=-Ut\\cos\\beta\\cos\\varphi+\\tfrac12\\kappa t^{2},\\\\\nY&= Ut\\cos\\beta\\sin\\varphi,\\\\\nZ&= Ut\\sin\\beta-\\tfrac12\\gamma t^{2}.\n\\end{aligned}\\tag{1}\n\\]\n\nConversely, if $(X,Y,Z)$ and $(\\beta,\\varphi,t)$ satisfy (1) with $Z>0$ and the admissible ranges for $\\beta$ and $\\varphi$, the projectile indeed reaches $P=(X,Y,Z)$.\n\nStep 2.  A useful shift of origin.  \nIntroduce the shifted coordinates  \n\n\\[\nA:=X-\\tfrac12\\kappa t^{2},\\qquad B:=Y,\\qquad C:=Z+\\tfrac12\\gamma t^{2}.\n\\]\n\nWith these, system (1) becomes  \n\n\\[\nA=-Ut\\cos\\beta\\cos\\varphi,\\qquad\nB= Ut\\cos\\beta\\sin\\varphi,\\qquad\nC= Ut\\sin\\beta. \\tag{2}\n\\]\n\nSquaring and adding the three equalities yields  \n\n\\[\nA^{2}+B^{2}+C^{2}=U^{2}t^{2}\\bigl(\\cos^{2}\\beta\\cos^{2}\\varphi+\\cos^{2}\\beta\\sin^{2}\\varphi+\\sin^{2}\\beta\\bigr)=U^{2}t^{2}. \\tag{3}\n\\]\n\nThus every feasible shot satisfies  \n\n\\[\nQ(t;X,Y,Z)=U^{2}t^{2}. \\tag{4}\n\\]\n\nStep 3.  Translating the mechanical constraints.  \n\n(a) Elevation.  From (2) we get  \n\n\\[\n\\sin\\beta=\\frac{C}{Ut}.\n\\]\n\nBecause $-70^{\\circ}\\le \\beta\\le 90^{\\circ}$, this is equivalent to  \n\n\\[\ns_{1}\\le \\frac{C}{Ut}\\le s_{2}. \\tag{5}\n\\]\n\n(b) Azimuth.  Suppose $-X+\\tfrac12\\kappa t^{2}\\neq0$.  Dividing the second equality of (2) by $-\\, $the first gives  \n\n\\[\n\\tan\\varphi=\\frac{B}{-A}=\\frac{Y}{-X+\\tfrac12\\kappa t^{2}}.\n\\]\n\nSince $|\\varphi|\\le\\varphi_{0}$ one must have  \n\n\\[\n|T(t)|=\\left|\\frac{Y}{-X+\\tfrac12\\kappa t^{2}}\\right|\\le\\tau\n\\quad\\text{and}\\quad\n-X+\\tfrac12\\kappa t^{2}>0. \\tag{6}\n\\]\n\n(c) Vertical-shot exception.  \nIf $-X+\\tfrac12\\kappa t^{2}=0$, the first two rows of (2) give $\\cos\\beta\\cos\\varphi=0$ and $Y=0$.  Because $|\\varphi|<90^{\\circ}$ we have $\\cos\\varphi\\neq0$, hence $\\cos\\beta=0$ and $\\beta=90^{\\circ}$.  The point must therefore lie on the ray  \n\n\\[\nX=\\tfrac12\\kappa t^{2},\\qquad Y=0,\\qquad Z=Ut-\\tfrac12\\gamma t^{2},\\qquad Z>0. \\tag{7}\n\\]\n\nFor definiteness we may choose $\\varphi=0$, which respects $|\\varphi|\\le\\varphi_{0}$.\n\nStep 4.  Definition of $R(t)$.  \nFor each $t>0$ set  \n\n\\[\n\\begin{aligned}\nR(t)=\\{(X,Y,Z)\\in\\mathbb R^{3}:\\;&Z>0,\\\\\n&\\text{(i)}\\;Q(t;X,Y,Z)=U^{2}t^{2},\\\\\n&\\text{(ii)}\\;C(t)/(Ut)\\in[s_{1},s_{2}],\\\\\n&\\text{(iii)}\\;\\text{either }[-X+\\tfrac12\\kappa t^{2}>0\\ \\text{and}\\ |T(t)|\\le\\tau]\\\\\n&\\hphantom{\\text{(iii)}}\\text{or the vertical case (7)}\\}.\n\\end{aligned}\n\\]\n\nStep 5.  Necessity and sufficiency of (i)–(iii).  \n\nNecessity.  \nFor any admissible shot the derivations (4), (5), (6) and (7) imply that its impact point belongs to $R(t)$.\n\nSufficiency.  \nConversely, suppose $(X,Y,Z)\\in R(t)$.  \n\n• From (i) we have $A^{2}+B^{2}+C^{2}=U^{2}t^{2}$.  \n\n• By (ii) we may define  \n\n\\[\n\\beta=\\arcsin\\!\\bigl[C/(Ut)\\bigr]\\in[-70^{\\circ},90^{\\circ}].\n\\]\n\n• If $-X+\\tfrac12\\kappa t^{2}>0$, (iii) supplies $|T(t)|\\le\\tau$, whence we can choose  \n\n\\[\n\\varphi=\\arctan\\!\\bigl[T(t)\\bigr]\\in[-\\varphi_{0},\\varphi_{0}],\n\\]\n\nand the three equalities (2) are fulfilled, so (1) holds and the projectile hits $(X,Y,Z)$ after flight time $t$.\n\n• If instead $-X+\\tfrac12\\kappa t^{2}=0$, (iii) forces $Y=0$ and (7) gives $\\beta=90^{\\circ}$; taking $\\varphi=0$ satisfies all constraints and (1).\n\nThus the conditions (i)–(iii) are jointly necessary and sufficient; hence  \n\n\\[\nR=\\bigcup_{t>0}R(t).\n\\]\n\nStep 6.  Geometry of the attainable region.  \nFor each $t$ the equality (i) represents the sphere of radius $Ut$ centred at  \n\n\\[\nC_{t}=\\bigl(\\tfrac12\\kappa t^{2},\\, 0,\\,-\\tfrac12\\gamma t^{2}\\bigr).\n\\]\n\nAs $t$ varies, these “safety spheres’’ slide along the space parabola $C_{t}$.  \nCondition (ii) clips each sphere to a horizontal angular slab corresponding to the admissible elevations, while (iii) cuts away the parts outside the double wedge of azimuthal aperture $2\\varphi_{0}$.  The union over all $t>0$ of the thus-trimmed spherical caps is a three-dimensional solid whose envelope is the “shifted safety paraboloid’’—the three-dimensional analogue, in the presence of a steady horizontal acceleration, of the classical parabola of safety.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.410624",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimensionality.  \n   The original problem is planar; the current kernel variant stays planar though angles may be negative.  The enhanced version is fully three-dimensional, forcing the solver to manage vectors, azimuth and elevation simultaneously.\n\n2. Multiple constant accelerations.  \n   Besides uniform gravity a second, independent, horizontal acceleration κ complicates the kinematics, destroying axial symmetry and producing a sheared envelope rather than a simple paraboloid.\n\n3. Dual angular constraints.  \n   Both elevation and azimuth are bounded, so one must translate two independent angular windows into nonlinear inequalities (15) and (17) after eliminating β and φ.\n\n4. Elimination of two internal variables.  \n   The solver must remove β and φ from (7)–(9) by an algebraic trick (squaring-and-adding) and then recover them to verify feasibility, rather than dealing with one angle as in the original.\n\n5. Nontrivial geometry of the boundary.  \n   The final region is not a surface of revolution; its boundary is the union of a 1-parameter family of tilted paraboloids intersected with two moving vertical half-planes and two moving horizontal slabs—considerably more intricate than a single 2-D parabola.\n\n6. More steps and deeper insight.  \n   One must recognise the need for the translation (10), work out three coupled inequalities, and interpret the geometry in 3-D—steps absent from the original planar case.\n\nAltogether these extra dimensions, forces, and constraints demand linear-algebraic manipulation, multivariable inequality handling and spatial geometric reasoning well beyond the original problem’s scope."
      }
    },
    "original_kernel_variant": {
      "question": "A rail-gun has been emplaced on the air-less asteroid (433) Eros at its local south pole.  \nChoose a right–handed Cartesian frame  \n\n OX points toward local east,  \n OY points toward local north,  \n OZ is the outward local vertical,\n\nand put the breech of the gun at the origin O.  \nThe barrel can be steered in two independent angular directions\n\n• elevation β (measured from the local horizontal) with −70° ≤ β ≤ 90°;  \n  β = 90° gives a vertical shot, β<0° corresponds to depression below the horizontal;\n\n• azimuth φ (measured in the horizontal plane from the negative X-axis toward the +Y-axis)   \n with |φ| ≤ φ₀, where the fixed constant φ₀ satisfies 0 < φ₀ < 90°.  \n Thus φ = 0 means “due west’’ and admissible shots are confined to the horizontal wedge −φ₀ ≤ φ ≤ φ₀.\n\nEvery projectile leaves the muzzle with the same speed U (independent of β and φ).\n\nAfter exit the only forces acting are  \n\n1. the practically uniform asteroid gravity   g⃗ = (0,0,−γ) (γ > 0),  \n2. a constant tangential “tidal-stress’’ acceleration p⃗ = (κ,0,0) of known magnitude κ ≥ 0 that points along (and is fixed in) the X-direction.\n\nAir drag is negligible, the rotation of Eros is ignored, and the local surface of the asteroid is approximated by the plane Z = 0.  Only points with Z > 0 are of interest.\n\nFor t>0 introduce  \n\n Q(t;X,Y,Z)= [X−½κt²]² + Y² + [Z+½γt²]²,   \n C(t)=Z+½γt²,   T(t)= Y / [−X+½κt²]  (denominator ≠0).\n\nPut  \n\n s₁ = sin(−70°)=−sin 70°,  s₂ = 1,  τ = tan φ₀.  \n\nLet R be the set of all points P=(X,Y,Z) with Z>0 that can be struck by an ideal projectile under the above restrictions.  Show that\n\n             R = ⋃_{t>0} R(t),\n\nwhere for each fixed flight time t>0\n\n R(t)= { (X,Y,Z) ∈ ℝ³ :\n  (i) Q(t;X,Y,Z) = U²t²,                        \n  (ii) C(t)/(Ut) ∈ [s₁,s₂],                     \n  (iii) either [−X+½κt²>0 and |T(t)|≤τ] or (vertical-shot case) [−X+½κt²=0 and Y=0] }.\n\nFinally, describe geometrically how the simultaneous conditions (i)–(iii) carve the attainable region out of the one-parameter family of “shifted safety spheres’’ of radius Ut whose centres travel along the space parabola (½κt², 0, −½γt²).\n\n————————————————————————————————————",
      "solution": "Step 1.  Equations of motion  \nBecause both g⃗ and p⃗ are constant, the trajectory of any fired projectile is  \n\n  r(t)=v₀t+½(p⃗+g⃗)t²   (t ≥ 0),                             (1)\n\nwith initial velocity  \n\n  v₀ = U(−cos β cos φ, cos β sin φ, sin β).                   (2)\n\nWriting the components of (1) and setting the impact time equal to the flight parameter t gives\n\n X = −U t cos β cos φ + ½κt²,                               (3a)  \n Y =  U t cos β sin φ,                                      (3b)  \n Z =  U t sin β − ½γt².                                     (3c)\n\nConversely, if (X,Y,Z) and (β,φ,t) satisfy (3a)–(3c) with Z>0 and the admissible ranges for β and φ, the projectile indeed reaches P=(X,Y,Z).\n\nStep 2.  A useful change of origins  \nIntroduce the shifted coordinates  \n\n A := X − ½κt²,  B := Y,  C := Z + ½γt².                (4)\n\nWith these, system (3) reads  \n\n A = −U t cos β cos φ,  B = U t cos β sin φ,  C = U t sin β. (5)\n\nSquaring and adding the three equalities yields the identity  \n\n A² + B² + C² = U²t²( cos²β cos²φ + cos²β sin²φ + sin²β ) = U²t².     (6)\n\nThat is, for any feasible shot one necessarily has  \n\n  Q(t;X,Y,Z) = U²t².                                       (7)\n\nStep 3.  Translating the mechanical constraints into inequalities  \n(a) Elevation.  From (5c)  \n\n sin β = C/(Ut).                                           (8)\n\nBecause −70° ≤ β ≤ 90°, condition (8) is equivalent to  \n\n s₁ ≤ C/(Ut) ≤ s₂  with s₁ = −sin 70°, s₂ = 1.          (9)\n\n(b) Azimuth.  Suppose −X+½κt² ≠ 0.  Dividing (5b) by −(5a) gives  \n\n tan φ = B/(−A) = Y/[−X+½κt²].                             (10)\n\nSince |φ| ≤ φ₀, one must have  \n\n |T(t)| = |Y/[−X+½κt²]| ≤ τ  and  −X+½κt² ≥ 0,           (11)\n\nwhere τ = tan φ₀.  \n\nVertical-shot exception.  \nIf −X+½κt² = 0 the first two rows of (5) give cos β cos φ = 0 and Y = 0.  Because |φ|<90°, cos φ ≠ 0; hence cos β = 0 and β = 90°.  Thus the point must lie on the ray  \n\n X = ½κt², Y = 0, Z = Ut − ½γt²,  Z > 0.                 (12)\n\nSuch points satisfy (i) and (ii) automatically; they are included in R(t) by the “vertical-shot’’ clause of (iii).\n\nStep 4.  Definition of R(t)  \nFor each fixed flight time t>0 set  \n\n R(t)= { (X,Y,Z) with Z>0 :                                (13)  \n  (i) Q(t;X,Y,Z)=U²t²,  \n  (ii) C(t)/(Ut)∈[s₁,s₂],  \n  (iii) either (−X+½κt²>0 and |T(t)|≤τ) or (vertical case 12) }.\n\nStep 5.  Necessity and sufficiency of the three conditions  \nNecessity.  \nFor any admissible shot the derivations of (7), (9), (11) and (12) show that its impact point belongs to R(t).\n\nSufficiency.  \nConversely, suppose (X,Y,Z)∈R(t).  Then  \n\n • From (i) we have A²+B²+C²=U²t².  \n • Condition (ii) puts C/(Ut) inside [s₁,s₂], so we can choose  \n\n  β = arcsin[C/(Ut)]  ∈ [−70°,90°].                       (14)\n\n • If −X+½κt²>0, requirement (iii) supplies |T(t)|≤τ, whence a legal azimuth  \n\n  φ = arctan[T(t)]  ∈ [−φ₀,φ₀].                           (15)\n\nWith these angles the three equalities (5) are fulfilled, so (3) holds and the projectile indeed hits (X,Y,Z) after flight time t.  \n • If instead −X+½κt²=0, (iii) forces Y=0 and (12) gives β=90°, fulfilling (3) with any φ (take φ=0).  \n\nThus the conditions (i)–(iii) are jointly necessary and sufficient; hence  \n\n         R = ⋃_{t>0} R(t).                        (16)\n\nStep 6.  Geometry of the attainable region  \nFor each t the equality (i) represents the sphere of radius Ut centred at  \n\n Cₜ = (½κt², 0, −½γt²).                                   (17)\n\nAs t varies, these “safety spheres’’ slide along the space parabola Cₜ.  \nCondition (ii) clips each sphere to a horizontal angular slab corresponding to the admissible elevations, while (iii) further removes the parts lying outside the double wedge of azimuthal aperture 2φ₀.  The union over all positive t of the thus-trimmed spherical caps yields a three-dimensional solid whose boundary is the envelope sometimes called a “shifted safety paraboloid’’—the three-dimensional analogue, in the presence of a steady horizontal acceleration, of the classical planar parabola of safety.\n\n————————————————————————————————————",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.352965",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimensionality.  \n   The original problem is planar; the current kernel variant stays planar though angles may be negative.  The enhanced version is fully three-dimensional, forcing the solver to manage vectors, azimuth and elevation simultaneously.\n\n2. Multiple constant accelerations.  \n   Besides uniform gravity a second, independent, horizontal acceleration κ complicates the kinematics, destroying axial symmetry and producing a sheared envelope rather than a simple paraboloid.\n\n3. Dual angular constraints.  \n   Both elevation and azimuth are bounded, so one must translate two independent angular windows into nonlinear inequalities (15) and (17) after eliminating β and φ.\n\n4. Elimination of two internal variables.  \n   The solver must remove β and φ from (7)–(9) by an algebraic trick (squaring-and-adding) and then recover them to verify feasibility, rather than dealing with one angle as in the original.\n\n5. Nontrivial geometry of the boundary.  \n   The final region is not a surface of revolution; its boundary is the union of a 1-parameter family of tilted paraboloids intersected with two moving vertical half-planes and two moving horizontal slabs—considerably more intricate than a single 2-D parabola.\n\n6. More steps and deeper insight.  \n   One must recognise the need for the translation (10), work out three coupled inequalities, and interpret the geometry in 3-D—steps absent from the original planar case.\n\nAltogether these extra dimensions, forces, and constraints demand linear-algebraic manipulation, multivariable inequality handling and spatial geometric reasoning well beyond the original problem’s scope."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}