MathDB

2

Part of 2021 Iran RMM TST

Problems(3)

easy function on Tst

Source: Iranian RMM TST 2021 Day1 P2

4/16/2021
Let f:R+→Rf : \mathbb{R}^+\to\mathbb{R} satisfying f(x)=f(x+2)+2f(x2+2x)f(x)=f(x+2)+2f(x^2+2x). Prove that if for all x>14002021x>1400^{2021}, xf(x)≤2021xf(x) \le 2021, then xf(x)≤2021xf(x) \le 2021 for all x∈R+x \in \mathbb {R}^+
Proposed by Navid Safaei
functionalgebra
GGG4 geometry in a Tst

Source: Iranian RMM TST 2021 Day2 P2

4/16/2021
Let ABCABC be a triangle with AB≠ACAB \neq AC and with incenter II. Let MM be the midpoint of BCBC, and let LL be the midpoint of the circular arc BACBAC. Lines through MM parallel to BI,CIBI,CI meet AB,ACAB,AC at EE and FF, respectively, and meet LBLB and LCLC at PP and QQ, respectively. Show that II lies on the radical axis of the circumcircles of triangles EMFEMF and PMQPMQ.
Proposed by Andrew Wu
geometryincenterpower of a pointradical axiscircumcircle
putting queen's on the chess board.

Source: Iranian RMM TST 2021 Day3 P2

4/16/2021
In a chess board we call a group of queens independant if no two are threatening each other. In an nn by nn grid, we put exaxctly one queen in each cell ofa greed. Let us denote by MnM_n the minimum number of independant groups that hteir union contains all the queens. Let kk be a positive integer, prove that M3k+1≤3k+2M_{3k+1} \le 3k+2
Proposed by Alireza Haghi
combinatoricschess