MathDB

1

Part of 2014 China Team Selection Test

Problems(3)

Perpendicular diagonals in cyclic quadrilateral

Source: 2014 China TST 1 Day 1 Q1

3/18/2014
ABCDABCD is a cyclic quadrilateral, with diagonals AC,BDAC,BD perpendicular to each other. Let point FF be on side BCBC, the parallel line EFEF to ACAC intersect ABAB at point EE, line FGFG parallel to BDBD intersect CDCD at GG. Let the projection of EE onto CDCD be PP, projection of FF onto DADA be QQ, projection of GG onto ABAB be RR. Prove that QFQF bisects ∠PQR\angle PQR.
geometrycyclic quadrilateralperpendicular bisector
Number of prime factors vs prime powers

Source: 2014 China TST 2 Day 1 Q1

3/20/2014
Prove that for any positive integers kk and NN, (1N∑n=1N(ω(n))k)1k≤k+∑q≤N1q,\left(\frac{1}{N}\sum\limits_{n=1}^{N}(\omega (n))^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q}, where ∑q≤N1q\sum\limits_{q\leq N}\frac{1}{q} is the summation over of prime powers q≤Nq\leq N (including q=1q=1). Note: For integer n>1n>1, ω(n)\omega (n) denotes number of distinct prime factors of nn, and ω(1)=0\omega (1)=0.
floor functioninequalitiesnumber theory proposednumber theory
circumcentres and orthocentres

Source: 2014 China Tst 3 Day1 Q1

3/26/2014
Let the circumcenter of triangle ABCABC be OO. HAH_A is the projection of AA onto BCBC. The extension of AOAO intersects the circumcircle of BOCBOC at A′A'. The projections of A′A' onto AB,ACAB, AC are D,ED,E, and OAO_A is the circumcentre of triangle DHAEDH_AE. Define HB,OB,HC,OCH_B, O_B, H_C, O_C similarly. Prove: HAOA,HBOB,HCOCH_AO_A, H_BO_B, H_CO_C are concurrent
geometrycircumcirclegeometric transformationhomothetytrigonometrycyclic quadrilateralgeometry proposed