MathDB

7

Part of 2008 JBMO Shortlist

Problems(3)

a,b,c>0 , abc=1 => Π (ab + bc +1/ca)>=Π(1+2a)

Source: JBMO 2008 Shortlist A7

10/14/2017
Let a,ba, b and cc be positive real numbers such that abc=1abc = 1. Prove the inequality (ab+bc+1ca)(bc+ca+1ab)(ca+ab+1bc)≥(1+2a)(1+2b)(1+2c)\Big(ab + bc +\frac{1}{ca}\Big)\Big(bc + ca +\frac{1}{ab}\Big)\Big(ca + ab +\frac{1}{bc}\Big)\ge (1 + 2a)(1 + 2b)(1 + 2c).
JBMOinequalitiesalgebra
2008 JBMO Shortlist G7

Source: 2008 JBMO Shortlist G7

10/10/2017
Let ABCABC be an isosceles triangle with AC=BCAC = BC. The point DD lies on the side ABAB such that the semicircle with diameter BDBD and center OO is tangent to the side ACAC in the point PP and intersects the side BCBC at the point QQ. The radius OPOP intersects the chord DQDQ at the point EE such that 5⋅PE=3⋅DE5 \cdot PE = 3 \cdot DE. Find the ratio ABBC\frac{AB}{BC} .
JBMOgeometry
Minimum value

Source: JBMO 2008 Shortlist

10/17/2012
Determine the minimum value of prime p>3p> 3 for which there is no natural number n>0n> 0 such that 2n+3n≡0(modp)2^n+3^n\equiv 0\pmod{p} .
modular arithmeticquadraticsnumber theory unsolvednumber theory