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...
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...
An integer with perfect square digits

An integer with perfect square digits

Understand the problem Find all positive integers that have 4 digits, all of them perfect squares, and such that is divisible by 2, 3, 5 and 7. Source of the problem Centroamerican olympiad 2016 Topic Number theory Difficulty Level Easy Suggested Book An Excursion...
Euler’s theorem and an inequality

Euler’s theorem and an inequality

Understand the problem Let be the circumcenter and be the centroid of a triangle . If and are the circumcenter and incenter of the triangle, respectively,prove that Source of the problem Balkan MO 1996 Topic Geometry Difficulty Level Easy Comments Let be the...