MathDB

2

Part of 2020 JBMO Shortlist

Problems(4)

JBMO Shortlist 2020 A2

Source: JBMO Shortlist 2020

7/4/2021
Consider the sequence a1,a2,a3,...a_1, a_2, a_3, ... defined by a1=9a_1 = 9 and
an+1=(n+5)an+22n+3a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}
for n≥1n \ge 1.
Find all natural numbers nn for which ana_n is a perfect square of an integer.
JuniorBalkanshortlist2020algebraSequence
JBMO Shortlist 2020 C2

Source: JBMO Shortlist 2020

7/4/2021
Viktor and Natalia bought 20202020 buckets of ice-cream and want to organize a degustation schedule with 20202020 rounds such that: - In every round, both of them try 11 ice-cream, and those 22 ice-creams tried in a single round are different from each other. - At the end of the 20202020 rounds, both of them have tried each ice-cream exactly once. We will call a degustation schedule fair if the number of ice-creams that were tried by Viktor before Natalia is equal to the number of ice creams tried by Natalia before Viktor. Prove that the number of fair schedules is strictly larger than 2020!(21010+(1010!)2)2020!(2^{1010} + (1010!)^2).
Proposed by Viktor Simjanoski, Macedonia
JuniorBalkanshortlist2020combinatorics
JBMO Shortlist 2020 G2

Source: JBMO Shortlist 2020

7/4/2021
Let △ABC\triangle ABC be a right-angled triangle with ∠BAC=90∘\angle BAC = 90^{\circ}, and let EE be the foot of the perpendicular from AA to BCBC. Let Z≠AZ \neq A be a point on the line ABAB with AB=BZAB = BZ. Let (c)(c) and (c1)(c_1) be the circumcircles of the triangles △AEZ\triangle AEZ and △BEZ\triangle BEZ, respectively. Let (c2)(c_2) be an arbitrary circle passing through the points AA and EE. Suppose (c1)(c_1) meets the line CZCZ again at the point FF, and meets (c2)(c_2) again at the point NN. If PP is the other point of intersection of (c2)(c_2) with AFAF, prove that the points NN, BB, PP are collinear.
JuniorBalkanshortlist2020geometry
JBMO Shortlist 2020 N2

Source: JBMO Shortlist 2020

7/4/2021
Find all positive integers aa, bb, cc, and pp, where pp is a prime number, such that
73p2+6=9a2+17b2+17c273p^2 + 6 = 9a^2 + 17b^2 + 17c^2.
JuniorBalkanshortlist2020number theoryprime numbers