MathDB

3

Part of 2022 Romania Team Selection Test

Problems(3)

Romania TST 2022 Day 2 P3

Source: Romania TST 2022

5/15/2022
Let n≥2n\geq 2 be an integer. Let aij, i,j=1,2,…,na_{ij}, \ i,j=1,2,\ldots,n be n2n^2 positive real numbers satisfying the following conditions:
[*]For all i=1,…,ni=1,\ldots,n we have aii=1a_{ii}=1 and, [*]For all j=2,…,nj=2,\ldots,n the numbers aij, i=1,…,j−1a_{ij}, \ i=1,\ldots, j-1 form a permutation of 1/aji, i=1,…,j−1.1/a_{ji}, \ i=1,\ldots, j-1.
Given that Si=ai1+⋯+ainS_i=a_{i1}+\cdots+a_{in}, determine the maximum value of the sum 1/S1+⋯+1/Sn.1/S_1+\cdots+1/S_n.
inequalitiesalgebraromaniaRomanian TST
Congruence NT

Source: Romanian TST 2022, Day 3 P3

5/14/2023
Consider a prime number p⩾11p\geqslant 11. We call a triple a,b,ca,b,c of natural numbers suitable if they give non-zero, pairwise distinct residues modulo pp{}. Further, for any natural numbers a,b,c,ka,b,c,k we define fk(a,b,c)=a(b−c)p−k+b(c−a)p−k+c(a−b)p−k.f_k(a,b,c)=a(b-c)^{p-k}+b(c-a)^{p-k}+c(a-b)^{p-k}.Prove that there exist suitable a,b,ca,b,c for which p∣f2(a,b,c)p\mid f_2(a,b,c). Furthermore, for each such triple, prove that there exists k⩾3k\geqslant 3 for which p∤fk(a,b,c)p\nmid f_k(a,b,c) and determine the minimal kk{} with this property.
Călin Popescu and Marian Andronache
number theoryprime numberscongruence
Romania TST 2022 Day 4 P3

Source: Romania TST 2022

6/3/2022
Let ABCABC be a triangle and let its incircle γ\gamma touch the sides BC,CA,ABBC,CA,AB at D,E,FD,E,F respectively. Let PP be a point strictly in the interior of γ.\gamma. The segments PA,PB,PCPA,PB,PC cross γ\gamma at A0,B0,C0A_0,B_0,C_0 respectively. Let SA,SB,SCS_A,S_B,S_C be the centres of the circles PEF,PFD,PDEPEF,PFD,PDE respectively and let TA,TB,TCT_A,T_B,T_C be the centres of the circles PB0C0,PC0A0,PA0B0PB_0C_0,PC_0A_0,PA_0B_0 respectively. Prove that SATA,SBTBS_AT_A, S_BT_B and SCTCS_CT_C are concurrent.
geometryromaniaRomanian TST