MathDB

3

Part of 2012 Pan African

Problems(2)

Equation with Floor Functions

Source: Pan African MO 2012 Day 1 Q3

10/14/2012
Find all real solutions xx to the equation ⌊x2−2x⌋+2⌊x⌋=⌊x⌋2\lfloor x^2 - 2x \rfloor + 2\lfloor x \rfloor = \lfloor x \rfloor^2.
functionfloor functionalgebra proposedalgebra
Median, altitude relationships

Source: Pan African MO 2012 Day 2 Q3

10/14/2012
(i) Find the angles of △ABC\triangle ABC if the length of the altitude through BB is equal to the length of the median through CC and the length of the altitude through CC is equal to the length of the median through BB.
(ii) Find all possible values of ∠ABC\angle ABC of △ABC\triangle ABC if the length of the altitude through AA is equal to the length of the median through CC and the length of the altitude through CC is equal to the length of the median through BB.
trigonometrygeometry proposedgeometry