MathDB

1

Part of 2014 Korea National Olympiad

Problems(2)

square number

Source: Korea National 2014 #1

1/21/2015
For x,yx, y positive integers, x2−4y+1x^2-4y+1 is a multiple of (x−2y)(1−2y)(x-2y)(1-2y). Prove that ∣x−2y∣|x-2y| is a square number.
number theorygreatest common divisornumber theory proposed
Nice Geometry

Source: 2014 Korean Mathematical Olympiad #5.

1/22/2015
There is a convex quadrilateral ABCD ABCD which satisfies ∠A=∠D \angle A=\angle D . Let the midpoints of AB,AD,CD AB, AD, CD be L,M,N L,M,N . Let's say the intersection point of AC,BD AC, BD be E E . Let's say point F F which lies on ME→ \overrightarrow{ME} satisfies ME‾×MF‾=MA‾2 \overline{ME}\times \overline{MF}=\overline{MA}^{2} .
Prove that ∠LFM=∠MFN \angle LFM=\angle MFN . :)
geometrygeometry proposed