Find all arithmetic functions

Find all arithmetic functions

Understand the problem Find all functions satisfying following condition: Source of the problem IMO longlist 1977 Topic Functional equations Difficulty Level Medium Suggested Book Functional Equations by B J Venkatachala Start with hints Hint 0Hint 1Hint 2Hint 3Hint...
Ratio with the sum of digits

Ratio with the sum of digits

Understand the problem Find the 3-digit number whose ratio with the sum of its digits is minimal. Source of the problem Albania TST 2013 Topic Number Theory, Inequalities. Difficulty Level Easy Suggested Book Problem Solving Strategies by Arthur Engel Start with hints...
Does there exist a Magic Rectangle?

Does there exist a Magic Rectangle?

Magic Squares are infamous; so famous that even the number of letters on its Wikipedia Page is more than that of Mathematics itself. People hardly talk about Magic Rectangles. Ya, Magic Rectangles! Have you heard of it? No, right? Not me either! So, I set off to...
Coincident Nine-point Circles

Coincident Nine-point Circles

Understand the problem Let be a triangle, its circumcenter, its centroid, and its orthocenter. Denote by , and the centers of the circles circumscribed about the triangles , and , respectively. Prove that the triangle is congruent to the triangle and that the...
An inequality with unit coefficients

An inequality with unit coefficients

Understand the problem Let be positive real numbers. Show that there exist such that: Source of the problem Iberoamerican olympiad 2011 Topic Inequalities Difficulty Level Easy Suggested Book Inequalities: An Approach Through Problems by B.J. Venkatachala Start with...