MathDB

2021 Taiwan TST Round 2

Part of TST Round 2

Subcontests

(6)
A
1

Complex inequalities

Prove that if non-zero complex numbers α1,α2,α3\alpha_1,\alpha_2,\alpha_3 are distinct and noncollinear on the plane, and satisfy α1+α2+α3=0\alpha_1+\alpha_2+\alpha_3=0, then there holds ∑i=13(∣αi+1−αi+2∣∣αi∣(1∣αi+1∣+1∣αi+2∣−2∣αi∣))≤0......(∗)\sum_{i=1}^{3}\left(\frac{|\alpha_{i+1}-\alpha_{i+2}|}{\sqrt{|\alpha_i|}}\left(\frac{1}{\sqrt{|\alpha_{i+1}|}}+\frac{1}{\sqrt{|\alpha_{i+2}|}}-\frac{2}{\sqrt{|\alpha_{i}|}}\right)\right)\leq 0......(*) where α4=α1,α5=α2\alpha_4=\alpha_1, \alpha_5=\alpha_2. Verify further the sufficient and necessary condition for the equality holding in (∗)(*).
6
1

Connectedness Testing

Let k≤nk\leq n be two positive integers. IMO-nation has nn villages, some of which are connected by a road. For any two villages, the distance between them is the minimum number of toads that one needs to travel from one of the villages to the other, if the traveling is impossible, then the distance is set as infinite.
Alice, who just arrived IMO-nation, is doing her quarantine in some place, so she does not know the configuration of roads, but she knows nn and kk. She wants to know whether the furthest two villages have finite distance. To do so, for every phone call she dials to the IMO office, she can choose two villages, and ask the office whether the distance between them is larger than, equal to, or smaller than kk. The office answers faithfully (infinite distance is larger than kk). Prove that Alice can know whether the furthest two villages have finite distance between them in at most 2n2/k2n^2/k calls.
Proposed by usjl and Cheng-Ying Chang
5
1
3
1
N
1
G
2

Easy Geo Regarding Euler Line

Let ABCDABCD be a convex quadrilateral with pairwise distinct side lengths such that AC⊥BDAC\perp BD. Let O1,O2O_1,O_2 be the circumcenters of ΔABD,ΔCBD\Delta ABD, \Delta CBD, respectively. Show that AO2,CO1AO_2, CO_1, the Euler line of ΔABC\Delta ABC and the Euler line of ΔADC\Delta ADC are concurrent.
(Remark: The Euler line of a triangle is the line on which its circumcenter, centroid, and orthocenter lie.)
Proposed by usjl

Another easy geometry from Taiwan TST

Let ABCABC be a triangle with circumcircle Γ\Gamma, and points EE and FF are chosen from sides CACA, ABAB, respectively. Let the circumcircle of triangle AEFAEF and Γ\Gamma intersect again at point XX. Let the circumcircles of triangle ABEABE and ACFACF intersect again at point KK. Line AKAK intersect with Γ\Gamma again at point MM other than AA, and NN be the reflection point of MM with respect to line BCBC. Let XNXN intersect with Γ\Gamma again at point SS other that XX.
Prove that SMSM is parallel to BCBC.
Proposed by Ming Hsiao