MathDB

4

Part of 2019 Dutch IMO TST

Problems(3)

Parallel through Random points yield cyclic quadrilateral

Source: Netherlands IMO TST #1 2019 P4

7/16/2019
Let ΔABC\Delta ABC be a scalene triangle. Points D,ED,E lie on side AC‾\overline{AC} in the order, A,E,D,CA,E,D,C. Let the parallel through EE to BCBC intersect ⊙(ABD)\odot (ABD) at FF, such that, EE and FF lie on the same side of ABAB. Let the parallel through EE to ABAB intersect ⊙(BDC)\odot (BDC) at GG, such that, EE and GG lie on the same side of BCBC. Prove, Points D,F,E,GD,F,E,G are concyclic
parallelgeometrycyclic quadrilateralcircumcircle
divisibility functional with primes , f(p) > 0, p| (f(x) + f(p))^{f(p)}- x

Source: Dutch IMO TST2 2019 p4

1/11/2020
Find all functions f:Z→Zf : Z \to Z satisfying ∙\bullet f(p)>0 f(p) > 0 for all prime numbers pp, ∙\bullet p∣(f(x)+f(p))f(p)−xp| (f(x) + f(p))^{f(p)}- x for all x∈Zx \in Z and all prime numbers pp.
number theoryfunctionFind all functionsprime numbersfunctional equation
300 participants, n games of chess, maximum n wanted

Source: Dutch IMO TST3 2019 p4

1/11/2020
There are 300300 participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of nn for which it is possible to satisfy the following conditions at the same time: each contestant plays at most nn games of chess, and for each mm with 1≤m≤n1 \le m \le n, there is a contestant playing exactly mm games of chess.
combinatoricsmaxgraph theory