MathDB

3

Part of 2019 Dutch IMO TST

Problems(3)

max of gcd(a, b) + gcd(b, c) + gcd(c, a) when a + b + c = 5n.

Source: Dutch IMO TST1 2019 p3

1/10/2020
Let nn be a positive integer. Determine the maximum value of gcd(a,b)+gcd(b,c)+gcd(c,a)gcd(a, b) + gcd(b, c) + gcd(c, a) for positive integers a,b,ca, b, c such that a+b+c=5na + b + c = 5n.
maximumGCDgreatest common divisornumber theory
2 lines and a circle concurrent, 2 circumcircles and parallelogram related

Source: Dutch 2019 IMO TST2 p3

8/2/2019
Let ABCABC be an acute angles triangle with OO the center of the circumscribed circle. Point QQ lies on the circumscribed circle of △BOC\vartriangle BOC so that OQOQ is a diameter. Point MM lies on CQCQ and point NN lies internally on line segment BCBC so that ANCMANCM is a parallelogram. Prove that the circumscribed circle of △BOC\vartriangle BOC and the lines AQAQ and NMNM pass through the same point.
geometryconcurrencyconcurrentparallelogramcircumcircle
functional in Z, f(f(x)) = x, if x + y is odd then f(x) + f(y) \ge x + y

Source: Dutch IMO TST3 2019 p3

1/11/2020
Find all functions f:Z→Zf : Z \to Z satisfying the following two conditions: (i) for all integers xx we have f(f(x))=xf(f(x)) = x, (ii) for all integers xx and y such that x+yx + y is odd, we have f(x)+f(y)≥x+yf(x) + f(y) \ge x + y.
Functional inequalityFind all functionsalgebra