MathDB

2017 CMIMC Geometry

Part of 2017 CMIMC

Subcontests

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2017 G8: Parabola-Triangle Interaction

In triangle ABCABC with AB=23AB=23, AC=27AC=27, and BC=20BC=20, let DD be the foot of the AA altitude. Let P\mathcal{P} be the parabola with focus AA passing through BB and CC, and denote by TT the intersection point of ADAD with the directrix of P\mathcal P. Determine the value of DT2−DA2DT^2-DA^2. (Recall that a parabola P\mathcal P is the set of points which are equidistant from a point, called the <spanclass=′latex−italic′>focus</span><span class='latex-italic'>focus</span> of P\mathcal P, and a line, called the <spanclass=′latex−italic′>directrix</span><span class='latex-italic'>directrix</span> of P\mathcal P.)
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2017 G5: Orthogonal Circles on Vertices of Triangle

Two circles ω1\omega_1 and ω2\omega_2 are said to be <spanclass=′latex−italic′>orthogonal</span><span class='latex-italic'>orthogonal</span> if they intersect each other at right angles. In other words, for any point PP lying on both ω1\omega_1 and ω2\omega_2, if ℓ1\ell_1 is the line tangent to ω1\omega_1 at PP and ℓ2\ell_2 is the line tangent to ω2\omega_2 at PP, then ℓ1⊥ℓ2\ell_1\perp \ell_2. (Two circles which do not intersect are not orthogonal.)
Let △ABC\triangle ABC be a triangle with area 2020. Orthogonal circles ωB\omega_B and ωC\omega_C are drawn with ωB\omega_B centered at BB and ωC\omega_C centered at CC. Points TBT_B and TCT_C are placed on ωB\omega_B and ωC\omega_C respectively such that ATBAT_B is tangent to ωB\omega_B and ATCAT_C is tangent to ωC\omega_C. If ATB=7AT_B = 7 and ATC=11AT_C = 11, what is tan⁡∠BAC\tan\angle BAC?
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