MathDB

4.4

Part of 2011 Saudi Arabia Pre-TST

Problems(2)

2011 is not a divisor of ab -cd if a+b+c+d = 2011

Source: 2011 Saudi Arabia Pre-TST November 4.4

1/1/2022
Let a,b,c,da, b, c, d be positive integers such that a+b+c+d=2011a+b+c+d = 2011. Prove that 20112011 is not a divisor of ab−cdab - cd.
number theorydivisordivides
MP = MQ wanted, midpoint of AH, circumcenter, orthocenter

Source: 2011 Saudi Arabia Pre-TST February 4.4 https://artofproblemsolving.com/community/c2745403_2011_

1/1/2022
In a triangle ABCABC, let OO be the circumcenter, HH the ortho­center, and MM the midpoint of the segment AHAH. The perpendicular at MM onto OMOM intersects lines ABAB and ACAC at PP and QQ, respectively. Prove that MP=MQMP = MQ.
geometryequal segmentsorthocenter