MathDB

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Part of 2018 Junior Balkan Team Selection Tests - Romania

Problems(5)

1/a +3/b+5/c >= 4a^2 + 3b^2 + 2c^2 if a,b,c>0 with a^2 + b^2 + c^2 = 3

Source: 2018 Romania JBMO TST 1.2

6/19/2020
Let a,b,ca, b, c be positive real numbers such that a2+b2+c2=3a^2 + b^2 + c^2 = 3. Prove that 1a+3b+5c≥4a2+3b2+2c2\frac{1}{a}+\frac{3}{b}+\frac{5}{c} \ge 4a^2 + 3b^2 + 2c^2 When does the equality hold?
Marius Stanean
algebrainequalities
max of x^3y^2z when 2x^2+3y^2+6z^2+12(x+y+z) =108

Source: 2018 Romania JBMO TST 2.2

6/19/2020
Let x,y,zx, y,z be positive real numbers satisfying 2x2+3y2+6z2+12(x+y+z)=1082x^2+3y^2+6z^2+12(x+y+z) =108. Find the maximum value of x3y2zx^3y^2z.
Alexandru Gırban
inequalitiesmaxalgebra
sum a /\sqrt{(a + 2b)^3 \ge 1 /\sqrt{a + b + c}

Source: 2018 Romania JBMO TST 5.2

6/19/2020
If a,b,ca, b, c are positive real numbers, prove that a(a+2b)3+b(b+2c)3+c(c+2a)3≥1a+b+c\frac{a}{\sqrt{(a + 2b)^3}}+\frac{b}{\sqrt{(b + 2c)^3}} +\frac{c} {\sqrt{(c + 2a)^3}} \ge \frac{1}{\sqrt{a + b + c}} Alexandru Mihalcu
inequalitiesalgebra
DM // AO, circumcircle, midpoint projections on sides, circumcenter

Source: 2018 Romania JBMO TST 4.2

6/1/2020
Let ABCABC be an acute triangle, with AB≠ACAB \ne AC. Let DD be the midpoint of the line segment BCBC, and let EE and FF be the projections of DD onto the sides ABAB and ACAC, respectively. If MM is the midpoint of the line segment EFEF, and OO is the circumcenter of triangle ABCABC, prove that the lines DMDM and AOAO are parallel.
As source was given [url=https://artofproblemsolving.com/community/c629086_caucasus_mathematical_olympiad]Caucasus MO, but I was unable to find this problem in the contest collections
geometrycircumcirclemidpointparallel
\sqrt{x+y }+\sqrt{y+z}+\sqrt{z+x} > 2\sqrt{(x + y)(y + z)(z + x)/(xy + yz + zx)}

Source: 2018 Romania JBMO TST 6.2

6/19/2020
Let k>2k > 2 be a real number. a) Prove that for all positive real numbers x,yx,y and zz the following inequality holds: x+y+y+z+z+x>2(x+y)(y+z)(z+x)xy+yz+zx\sqrt{x + y }+\sqrt{y + z }+\sqrt{z + x} > 2\sqrt{\frac{(x + y)(y + z)(z + x)}{xy + yz + zx}} b) Prove that there exist positive real numbers x,yx, y and zz such that x+y+y+z+z+x<k(x+y)(y+z)(z+x)xy+yz+zx\sqrt{x + y }+\sqrt{y + z}+\sqrt{z + x} <k\sqrt{\frac{(x + y)(y + z)(z + x)}{xy + yz + zx}} Leonard Giugiuc
inequalitiesalgebra