MathDB

3

Part of 2019 May Olympiad

Problems(2)

BQ = 2QC, CR = 2RA, < PRQ = 90^o , equal angles wanted

Source: Olimpíada de Mayo / Maio (May Olympiad) 2019 L2

7/11/2019
On the sides AB,BCAB, BC and CACA of a triangle ABCABC are located the points P,QP, Q and RR respectively, such that BQ=2QC,CR=2RABQ = 2QC, CR = 2RA and ∠PRQ=90o\angle PRQ = 90^o. Show that ∠APR=∠RPQ\angle APR =\angle RPQ.
geometryequal angles
Determine the maximum number of distinct numbers the Gus list can contain.

Source: May Olimpiad 2019 Level 1 P3

3/9/2021
Gus has to make a list of 250250 positive integers, not necessarily distinct, such that each number is equal to the number of numbers in the list that are different from it. For example, if 1515 is a number from the list so the list contains 1515 numbers other than 1515. Determine the maximum number of distinct numbers the Gus list can contain.
number theory