MathDB

4

Part of 2014 JBMO Shortlist

Problems(4)

2015 Azerbaijan JBMO TST

Source: 2015 Azerbaijan JBMO TST

5/2/2015
With the conditions a,b,c∈R+a,b,c\in\mathbb{R^+} and a+b+c=1a+b+c=1, prove that 7+2b1+a+7+2c1+b+7+2a1+c≥694\frac{7+2b}{1+a}+\frac{7+2c}{1+b}+\frac{7+2a}{1+c}\geq\frac{69}{4}
algebrainequalitiesAzerbaijan
2015 Azerbaijan JBMO TST

Source: 2015 Azerbaijan JBMO TST

5/2/2015
A=1⋅4⋅7⋯2014A=1\cdot4\cdot7\cdots2014.Find the last non-zero digit of AA if it is known that A≡1mod  3A\equiv 1\mod3.
combinatorics
Very simple,however beautiful geometry problem

Source: JBMO Shortlist 2014,G4

11/6/2016
Let ABCABC be an acute triangle such that AB≠AC.AB\not=AC.Let MM be the midpoint BC,HBC,H the orthocenter of △ABC\triangle ABC,O1,O_1 the midpoint of AHAH and O2O_2 the circumcenter of △BCH\triangle BCH.. Prove that O1AMO2O_1AMO_2 is a parallelogram.
geometrycircumcircle
AZE JBMO TST

Source: AZE JBMO TST

5/2/2015
Prove that there are not intgers aa and bb with conditions, i) 16a−9b16a-9b is a prime number. ii) abab is a perfect square. iii) a+ba+b is also perfect square.
number theoryprime numbersAZE JBMO TST