MathDB

2020 Ukrainian Geometry Olympiad - April

Part of Ukrainian Geometry Olympiad

Subcontests

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2<BCQ =<AQP in a square, <QPN = <NCB, NC=NP (Ukr. Geom. Olympiad '20 VIII p4)

On the sides ABAB and ADAD of the square ABCDABCD, the points NN and PP are selected respectively such that NC=NPNC=NP. The point QQ is chosen on the segment ANAN so that ∠QPN=∠NCB\angle QPN = \angle NCB. Prove that 2∠BCQ=∠AQP2\angle BCQ = \angle AQP.

angle bisector wanted, <PAB = <PCB =1/4 (<A+ < C)

Inside triangle ABCABC, the point PP is chosen such that ∠PAB=∠PCB=14(∠A+∠C)\angle PAB = \angle PCB =\frac14 (\angle A+ \angle C). Let BLBL be the bisector of △ABC\vartriangle ABC. Line PLPL intersects the circumcircle of △APC\vartriangle APC at point QQ. Prove that the line QBQB is the bisector of ∠AQC\angle AQC.
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P interior of ABC, BP > AP and BP > CP => <ABC< 90^o

Inside the triangle ABCABC is point PP, such that BP>APBP > AP and BP>CPBP > CP. Prove that ∠ABC\angle ABC is acute.

equal angles wanted, circles, tangents, symmetric point given

Let Γ\Gamma be a circle and PP be a point outside, PAPA and PBPB be tangents to Γ\Gamma , A,B∈ΓA, B \in \Gamma . Point KK is an arbitrary point on the segment ABAB. The circumscirbed circle of △PKB\vartriangle PKB intersects Γ\Gamma for the second time at point TT, point P′P' is symmetric to point PP wrt point AA. Prove that ∠PBT=∠P′KA\angle PBT = \angle P'KA.

tangent wanted, isosceles, 3 circles, tangents given

Let ABCABC be an isosceles triangle with AB=ACAB=AC. Circle Γ\Gamma lies outside ABCABC and touches line ACAC at point CC. The point DD is chosen on circle Γ\Gamma such that the circumscribed circle of the triangle ABDABD touches externally circle Γ\Gamma. The segment ADAD intersects circle Γ\Gamma at a point EE other than DD. Prove that BEBE is tangent to circle Γ\Gamma .
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