MathDB

1

Part of 2019 Dutch IMO TST

Problems(3)

|a|,|b| >= 2017 , P(a^2+b^2) >= P(2ab) , P trinomial

Source: Dutch IMO TST1 2019 p1

1/10/2020
Let P(x)P(x) be a quadratic polynomial with two distinct real roots. For all real numbers aa and bb satisfying ∣a∣,∣b∣≥2017|a|,|b| \ge 2017, we have P(a2+b2)≥P(2ab)P(a^2+b^2) \ge P(2ab). Show that at least one of the roots of PP is negative.
trinomialquadratic trinomialalgebrapolynomial
pupils and best friends go to either Rome or Paris

Source: Dutch IMO TST2 2019 p1

1/11/2020
In each of the different grades of a high school there are an odd number of pupils. Each pupil has a best friend (who possibly is in a different grade). Everyone is the best friend of their best friend. In the upcoming school trip, every pupil goes to either Rome or Paris. Show that the pupils can be distributed over the two destinations in such a way that (i) every student goes to the same destination as their best friend; (ii) for each grade the absolute difference between the number of pupils that are going to Rome and that of those who are going to Paris is equal to 11.
combinatorics
Tangent circles to random line yielding cyclic quadrilateral

Source: Netherlands IMO TST #3 2019 P1

7/16/2019
Let ABCDABCD be a cyclic quadrilateral (In the same order) inscribed into the circle ⊙(O)\odot (O). Let AC‾\overline{AC} ∩\cap BD‾\overline{BD} == EE. A randome line ℓ\ell through EE intersects AB‾\overline{AB} at PP and BCBC at QQ. A circle ω\omega touches ℓ\ell at EE and passes through DD. Given, ω\omega ∩\cap ⊙(O)\odot (O) == RR. Prove, Points B,Q,R,PB,Q,R,P are concyclic.
geometrycyclic quadrilateral