MathDB

3

Part of 2017 Vietnam Team Selection Test

Problems(2)

Vietnam TST 2017 problem 3

Source: Vietnam TST 2017

3/26/2017
Triangle ABCABC with incircle (I)(I) touches the sides AB,BC,ACAB, BC, AC at F,D,EF, D, E, res. Ib,IcI_b, I_c are BB- and C−C- excenters of ABCABC. P,QP, Q are midpoints of IbE,IcFI_bE, I_cF. (PAC)∩AB={A,R}(PAC)\cap AB=\{ A, R\}, (QAB)∩AC={A,S}(QAB)\cap AC=\{ A,S\}. a. Prove that PR,QS,AIPR, QS, AI are concurrent. b. DE,DFDE, DF cut IbIcI_bI_c at K,JK, J, res. EJ∩FK={M}EJ\cap FK=\{ M\}. PE,QFPE, QF cut (PAC),(QAB)(PAC), (QAB) at X,YX, Y res. Prove that BY,CX,AMBY, CX, AM are concurrent.
geometry
Vietnam TST 2017 problem 6

Source:

3/26/2017
For each integer n>0n>0, a permutation a1,a2,…,a2na_1,a_2,\dots ,a_{2n} of 1,2,…2n1,2,\dots 2n is called beautiful if for every 1≤i<j≤2n1\leq i<j \leq 2n, ai+an+i=2n+1a_i+a_{n+i}=2n+1 and ai−ai+1≢aj−aj+1a_i-a_{i+1}\not \equiv a_j-a_{j+1} (mod 2n+12n+1) (suppose that a2n+1=a1a_{2n+1}=a_1). a. For n=6n=6, point out a beautiful permutation. b. Prove that there exists a beautiful permutation for every nn.
combinatorics