MathDB

2016 Harvard-MIT Mathematics Tournament

Part of Harvard-MIT Mathematics Tournament

Subcontests

(36)
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2016 Guts #36

<spanclass=′latex−bold′>(Self−IsogonalCubics)</span><span class='latex-bold'>(Self-Isogonal Cubics)</span> Let ABCABC be a triangle with AB=2AB = 2, AC=3AC = 3, BC=4BC = 4. The \emph{isogonal conjugate} of a point PP, denoted P∗P^\ast, is the point obtained by intersecting the reflection of lines PAPA, PBPB, PCPC across the angle bisectors of ∠A\angle A, ∠B\angle B, and ∠C\angle C, respectively.
Given a point QQ, let K(Q)\mathfrak K(Q) denote the unique cubic plane curve which passes through all points PP such that line PP∗PP^\ast contains QQ. Consider:
[*] the M'Cay cubic K(O)\mathfrak K(O), where OO is the circumcenter of △ABC\triangle ABC, [*] the Thomson cubic K(G)\mathfrak K(G), where GG is the centroid of △ABC\triangle ABC, [*] the Napoleon-Feurerbach cubic K(N)\mathfrak K(N), where NN is the nine-point center of △ABC\triangle ABC, [*] the Darboux cubic K(L)\mathfrak K(L), where LL is the de Longchamps point (the reflection of the orthocenter across point OO), [*] the Neuberg cubic K(X30)\mathfrak K(X_{30}), where X30X_{30} is the point at infinity along line OGOG, [*] the nine-point circle of △ABC\triangle ABC, [*] the incircle of △ABC\triangle ABC, and [*] the circumcircle of △ABC\triangle ABC.
Estimate NN, the number of points lying on at least two of these eight curves. An estimate of EE earns ⌊20⋅2−∣N−E∣/6⌋\left\lfloor 20 \cdot 2^{-|N-E|/6} \right\rfloor points.
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2016 Guts #24

Let ΔA1B1C\Delta A_1B_1C be a triangle with ∠A1B1C=90∘\angle A_1B_1C = 90^{\circ} and CA1CB1=5+2\frac{CA_1}{CB_1} = \sqrt{5}+2. For any i≥2i \ge 2, define AiA_i to be the point on the line A1CA_1C such that AiBi−1⊥A1CA_iB_{i-1} \perp A_1C and define BiB_i to be the point on the line B1CB_1C such that AiBi⊥B1CA_iB_i \perp B_1C. Let Γ1\Gamma_1 be the incircle of ΔA1B1C\Delta A_1B_1C and for i≥2i \ge 2, Γi\Gamma_i be the circle tangent to Γi−1,A1C,B1C\Gamma_{i-1}, A_1C, B_1C which is smaller than Γi−1\Gamma_{i-1}.
How many integers kk are there such that the line A1B2016A_1B_{2016} intersects Γk\Gamma_{k}?
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