MathDB

1

Part of 2022 Romania Team Selection Test

Problems(3)

n-variable inequality with a weird condition

Source: Romanian TST 2022, Test 1, P1

5/14/2022
Given are positive reals x1,x2,...,xnx_1, x_2,..., x_n such that ∑11+xi2=1\sum\frac {1}{1+x_i^2}=1. Find the minimal value of the expression ∑xi∑1xi\frac{\sum x_i}{\sum \frac{1}{x_i}} and find when it is achieved.
inequalities
Romania TST 2022 Day 2 P1

Source: Romania TST 2022

5/15/2022
Let ABCABC be an acute scalene triangle and let ω\omega be its Euler circle. The tangent tAt_A of ω\omega at the foot of the height AA of the triangle ABC, intersects the circle of diameter ABAB at the point KAK_A for the second time. The line determined by the feet of the heights AA and CC of the triangle ABCABC intersects the lines AKAAK_A and BKABK_A at the points LAL_A and MAM_A, respectively, and the lines tAt_A and CMACM_A intersect at the point NAN_A.
Points KB,LB,MB,NBK_B, L_B, M_B, N_B and KC,LC,MC,NCK_C, L_C, M_C, N_C are defined similarly for (B,C,A)(B, C, A) and (C,A,B)(C, A, B) respectively. Show that the lines LANA,LBNB,L_AN_A, L_BN_B, and LCNCL_CN_C are concurrent.
geometryEuler CircleromaniaRomanian TST
Romania TST 2022 Day 4 P1

Source: Romania TST 2022

6/3/2022
A finite set L\mathcal{L} of coplanar lines, no three of which are concurrent, is called odd if, for every line ℓ\ell in L\mathcal{L} the total number of lines in L\mathcal{L} crossed by ℓ\ell is odd.
[*]Prove that every finite set of coplanar lines, no three of which are concurrent, extends to an odd set of coplanar lines. [*]Given a positive integer nn determine the smallest nonnegative integer kk satisfying the following condition: Every set of nn coplanar lines, no three of which are concurrent, extends to an odd set of n+kn+k coplanar lines.
combinatoricsromaniaRomanian TST