MathDB

2

Part of 2023 Iran Team Selection Test

Problems(2)

cyclic quadrilateral ABCD

Source: Iran TST 2023 ; Exam 1 Problem 2

3/15/2023
ABCDABCD is cyclic quadrilateral and OO is the center of its circumcircle. Suppose that AD∩BC=EAD \cap BC = E and AC∩BD=FAC \cap BD = F. Circle ω\omega is tanget to line ACAC and BDBD. PQPQ is a diameter of ω\omega that FF is orthocenter of EPQEPQ. Prove that line OEOE is passing through center of ω\omega
Proposed by Mahdi Etesami Fard
geometrycyclic quadrilateralcircumcircle
an insect jumping on [0,1]

Source: Iran TST 2023 ; Exam 2 Problem 2

3/16/2023
Suppose 12<s<1\frac{1}{2} < s < 1 . An insect flying on [0,1][0,1] . If it is on point aa , it jump into point a×s a\times s or (a−1)×s+1(a-1) \times s +1 . For every real number 0≤c≤10 \le c \le 1, Prove that insect can jump that after some jumps , it has a distance less than 11402\frac {1}{1402} from point cc.
Proposed by Navid Safaei
IranTSTIranian TSTcombinatorics