MathDB

3

Part of 2013 Saudi Arabia IMO TST

Problems(3)

a Saudi company has 2 offices, no of correspondents from each office

Source: 2013 Saudi Arabia IMO TST I p3

7/23/2020
A Saudi company has two offices. One office is located in Riyadh and the other in Jeddah. To insure the connection between the two offices, the company has designated from each office a number of correspondents so that : (a) each pair of correspondents from the same office share exactly one common correspondent from the other office. (b) there are at least 1010 correspondents from Riyadh. (c) Zayd, one of the correspondents from Jeddah, is in contact with exactly 88 correspondents from Riyadh. What is the minimum number of correspondents from Jeddah who are in contact with the correspondent Amr from Riyadh?
combinatorics
max of p% of all divisors of n with their unit digit equal to 3

Source: 2013 Saudi Arabia IMO TST II p3

7/23/2020
For a positive integer nn, we consider all its divisors (including 11 and itself). Suppose that p%p\% of these divisors have their unit digit equal to 33. (For example n=117n = 117, has six divisors, namely 1,3,9,13,39,1171,3,9,13,39,117. Two of these divisors namely 33 and 1313, have unit digits equal to 33. Hence for n=117n = 117, p=33.33...p =33.33...). Find, when nn is any positive integer, the maximum possible value of pp.
number theoryDigitDivisors
2 circumcircles and a line concurrent

Source: 2013 Saudi Arabia IMO TST III p3

7/23/2020
Let ABCABC be an acute triangle, MM be the midpoint of BCBC and PP be a point on line segment AMAM. Lines BPBP and CPCP meet the circumcircle of ABCABC again at XX and YY , respectively, and sides ACAC at DD and ABAB at EE, respectively. Prove that the circumcircles of AXDAXD and AYEAYE have a common point T≠AT \ne A on line AMAM.
geometrycircumcircle