Can anyone help me with this math olympiad task?
(2 votes)
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Math?
Can someone please calculate these written tasks for me? It would be very nice because they will be graded and I really didn't have time to study today.
Math global extreme point?
Solution: Why did you use the left limit for f1(x)? Could one have also shown that there is no global maximum/minimum for f1(x) using the normal limit (x tends to infinity)? Since the same argument was used for f2(x)? Thanks!
(a) was helpful to see what this is all about 😉
So on with b)
D, E and F are the points of contact of the circle AB, BC or AC.
Angle DAM = angle MAF.
Angle MDA = angle AFM (= 90°).
Since the triangles DMA and MFA have the distance AM and also the incirculation radius r together, they are congruent.
This is followed by AF = AD. (I also know this as a tangent set.)
Also BE = BD.
AF + ME = AF + r = AD + r = a.
(AF and ME are parallel.)
And it is BE + MF = BE + r = BD + r = b.
The sum is AD + BD + 2r = a + b.
Because of AD + BD = c (hypotenuse)
is c + 2r = a + b.
c is at the same time the diameter of the circle, i.e.
2R + 2r = a + b
R + r = (a + b)/2, which was to be proven.
Again very elegant. I am not sure whether you have confused a and b in the meantime (AF + ME = b, BD + r = a), but has no consequences.
That can happen. But if I present the solution of Matheolympiaden tasks, the questioner should urgent check again. 😉