MathDB

3

Part of 2011 JBMO Shortlist

Problems(4)

Simple inequality

Source: JBMO 2011 Shortlist A3

5/15/2016
A3\boxed{\text{A3}}If a,ba,b be positive real numbers, show that:a2+ab+b23+ab≤a+b \displaystyle{\sqrt{\dfrac{a^2+ab+b^2}{3}}+\sqrt{ab}\leq a+b}
inequalitiesalgebraJBMO ShortlistHigh school olympiad
transform 21062011 into 1012011 by 3 operations

Source: JBMO 2011 Shortlist C3

10/14/2017
We can change a natural number nn in three ways: a) If the number nn has at least two digits, we erase the last digit and we subtract that digit from the remaining number (for example, from 123123 we get 12−3=912 - 3 = 9); b) If the last digit is different from 00, we can change the order of the digits in the opposite one (for example, from 123123 we get 321321); c) We can multiply the number nn by a number from the set {1,2,3,...,2010} \{1, 2, 3,..., 2010\}. Can we get the number 2106201121062011 from the number 10120111012011?
JBMOcombinatorics
2011 JBMO Shortlist G3

Source: 2011 JBMO Shortlist G3

10/8/2017
Let ABCABC be a triangle in which (BL{BL}is the angle bisector of ∠ABC{\angle{ABC}} (L∈AC)\left( L\in AC \right), AH{AH} is an altitude of△ABC\vartriangle ABC (H∈BC)\left( H\in BC \right) and M{M}is the midpoint of the side AB{AB}. It is known that the midpoints of the segments BL{BL} and MH{MH} coincides. Determine the internal angles of triangle △ABC\vartriangle ABC.
geometryJBMO
y^2 + xy + 3x = n(x^2 + xy + 3y)

Source: JBMO 2011 Shortlist N3

10/14/2017
Find all positive integers nn such that the equation y2+xy+3x=n(x2+xy+3y)y^2 + xy + 3x = n(x^2 + xy + 3y) has at least a solution (x,y)(x, y) in positive integers.
JBMOnumber theory