MathDB

2

Part of 2022 Romania Team Selection Test

Problems(2)

Romania TST 2022 Day 2 P2

Source: Romania TST 2022

5/15/2022
Let ABCABC be an acute triangle and let B′B' and C′C' be the feet of the heights BB and CC of triangle ABCABC respectively. Let BA′B_A' and BC′B_C' be reflections of B′B' with respect to the lines BCBC and ABAB, respectively. The circle BBA′BC′BB_A'B_C', centered in OBO_B, intersects the line ABAB in XBX_B for the second time.
The points CA′,CB′,OC,XCC_A', C_B', O_C, X_C are defined analogously, by replacing the pair (B,B′)(B, B') with the pair (C,C′)(C, C'). Show that OBXBO_BX_B and OCXCO_CX_C are parallel.
geometryromaniaRomanian TST
Romania TST 2022 Day 4 P2

Source: Romania TST 2022

6/3/2022
Fix a nonnegative integer a0a_0 to define a sequence of integers a0,a1,…a_0,a_1,\ldots by letting ak,k≥1a_k,k\geq 1 be the smallest integer (strictly) greater than ak−1a_{k-1} making ak−1+aka_{k-1}+a_k{} into a perfect square. Let SS{} be the set of positive integers not expressible as the difference of two terms of the sequence (ak)k≥0.(a_k)_{k\geq 0}. Prove that SS is finite and determine its size in terms of a0.a_0.
number theorySequenceromaniaRomanian TST