MathDB

Problem 1

Part of 2006 Federal Math Competition of S&M

Problems(3)

find angle in quadrilateral (Serbia MO 2006 1st Grade P1)

Source:

4/10/2021
In a convex quadrilateral ABCDABCD, ∠BAC=∠DAC=55∘\angle BAC=\angle DAC=55^\circ, ∠DCA=20∘\angle DCA=20^\circ, and ∠BCA=15∘\angle BCA=15^\circ. Find the measure of ∠DBA\angle DBA.
geometryquadrilateral
comparison of discriminant based on quadratic inequality

Source: Serbia MO 2006 2nd Grade P1

4/10/2021
Suppose a,b,c,A,B,Ca,b,c,A,B,C are real numbers with a≠0a\ne0 and A≠0A\ne0 such that for all xx, ∣ax2+bx+c∣≤∣Ax2+Bx+C∣.\left|ax^2+bx+c\right|\le\left|Ax^2+Bx+C\right|.Prove that ∣b2−4ac∣≤∣B2−4AC∣.\left|b^2-4ac\right|\le\left|B^2-4AC\right|.
quadraticsInequalityalgebrainequalities
sum(x/(y^2+z))>=9/4 if x+y+z=1

Source: Serbia MO 2006 3&4th Grades P1

4/10/2021
Let x,y,zx,y,z be positive numbers with the sum 11. Prove that xy2+z+yz2+x+zx2+y≥94.\frac x{y^2+z}+\frac y{z^2+x}+\frac z{x^2+y}\ge\frac94.
Inequalityinequalitiesalgebra