MathDB

Problem 3

Part of 2017 Vietnamese Southern Summer School contest

Problems(3)

Kissing circles

Source: Southern Summer School, gr. 10

7/9/2017
Let ABCABC be a triangle with right angle ACBACB. Denote by FF the projection of CC on ABAB. A circle ω\omega touches FBFB at point PP, touches CFCF at point QQ, and the circumcircle of ABCABC at point RR. Prove that the points A,Q,RA, Q, R all lie on the same line and AP=ACAP=AC.
geometrytangent circlescircumcircle
3-addict valuation

Source: Southern Summer School, gr. 11

7/9/2017
Prove that, for any integer n≥2n\geq 2, there exists an integer xx such that 3n∣x3+20173^n|x^3+2017, but 3n+1∤x3+20173^{n+1}\not | x^3+2017.
number theoryp-adic
A lot of symmetry

Source: Southern Summer School, gr. 12

7/9/2017
Let ω\omega be a circle with center OO and a non-diameter chord BCBC of ω\omega. A point AA varies on ω\omega such that ∠BAC<90∘\angle BAC<90^{\circ}. Let SS be the reflection of OO through BCBC. Let TT be a point on OSOS such that the bisector of ∠BAC\angle BAC also bisects ∠TAS\angle TAS. 1. Prove that TB=TC=TOTB=TC=TO. 2. TB,TCTB, TC cut ω\omega the second times at points E,FE, F, respectively. AE,AFAE, AF cut BCBC at M,NM, N, respectively. Let SMSM intersects the tangent line at CC of ω\omega at XX, SNSN intersects the tangent line at BB of ω\omega at YY. Prove that the bisector of ∠BAC\angle BAC also bisects ∠XAY\angle XAY.
geometrygeometric transformationreflectionsymmetry