HMMT 二月 2016 · 冲刺赛 · 第 36 题

英文解析

36. [ 20 ] (Self-Isogonal Cubics) Let ABC be a triangle with AB = 2, AC = 3, BC = 4. The isogonal
    ∗
    conjugate of a point P , denoted P , is the point obtained by intersecting the reflection of lines P A ,
    P B , P C across the angle bisectors of ∠ A , ∠ B , and ∠ C , respectively.
    Given a point Q , let K ( Q ) denote the unique cubic plane curve which passes through all points P such
    ∗
    that line P P contains Q . Consider:
    (a) the M’Cay cubic K ( O ), where O is the circumcenter of 4 ABC ,
    (b) the Thomson cubic K ( G ), where G is the centroid of 4 ABC ,
    (c) the Napoleon-Feurerbach cubic K ( N ), where N is the nine-point center of 4 ABC ,
    (d) the Darboux cubic K ( L ), where L is the de Longchamps point (the reflection of the orthocenteracross point O ),
    (e) the Neuberg cubic K ( X ), where X is the point at infinity along line OG ,
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    (f) the nine-point circle of 4 ABC ,
    (g) the incircle of 4 ABC , and
    (h) the circumcircle of 4 ABC .
    Estimate N , the number of points lying on at least two of these eight curves. An estimate of E earns
    ⌊ ⌋
    −| N − E | / 6
    20 · 2 points.
    Proposed by: Evan Chen
    Answer: 49
    The first main insight is that all the cubics pass through the points A , B , C , H (orthocenter), O ,
    and the incenter and three excenters. Since two cubics intersect in at most nine points, this is all theintersections of a cubic with a cubic.
    On the other hand, it is easy to see that among intersections of circles with circles, there are exactly
    3 points; the incircle is tangent to the nine-point circle at the Feurerbach point while being containedcompletely in the circumcircle; on the other hand for this obtuse triangle the nine-point circle and thecircumcircle intersect exactly twice.
    All computations up until now are exact, so it remains to estimate:
    • Intersection of the circumcircle with cubics. Each cubic intersects the circumcircle at an evennumber of points, and moreover we already know that A , B , C are among these, so the numberof additional intersections contributed is either 1 or 3; it is the former only for the Neuberg cubicwhich has a “loop”. Hence the actual answer in this case is 1 + 3 + 3 + 3 + 3 = 13 (but an estimateof 3 · 5 = 15 is very reasonable).
    • Intersection of the incircle with cubics. Since ∠ A is large the incircle is small, but on the otherhand we know I lies on each cubic. Hence it’s very likely that each cubic intersects the incircletwice (once “coming in” and once “coming out”). This is the case, giving 2 · 5 = 10 new points.
    • Intersection of the nine-point with cubics. We guess this is close to the 10 points of the incircle,
    as we know the nine-point circle and the incircle are tangent to each other. In fact, the exactcount is 14 points; just two additional branches appear.
    In total, N = 9 + 3 + 13 + 10 + 14 = 49.
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