Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation

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Problem B. 4048. (December 2007)

B. 4048. The vertices of a regular polygon of 2n sides are $A_1, A_2, \ldots, A_{2n}$ in this order. Prove that

$\frac{1}{A_1A_2^2} + \frac{1}{A_1A_n^2} = \frac{4}{A_1A_3^2}.$

(3 pont)

Deadline expired on January 15, 2008.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás: Legyen $x=\frac{\pi}{2n}$ . A sokszög köré írható kör sugarát tekintve egységnek, A₁A₂=2sin x, A₁A₃=2sin 2x és A₁A_n=2sin (n-1)x=2cos x, hiszen x+(n-1)x= $\pi$ /2. Ezért a bizonyítandó állítás ekvivalens a

sin²2x(sin²x+cos²x)=4sin²xcos²x

egyenlőséggel, ami nyilván teljesül, hiszen sin²x+cos²x=1 és sin 2x=2sin xcos x.

Statistics:

88 students sent a solution.

3 points: 80 students.

2 points: 4 students.

1 point: 3 students.

0 point: 1 student.

Problems in Mathematics of KöMaL, December 2007

88 students sent a solution.
3 points:	80 students.
2 points:	4 students.
1 point:	3 students.
0 point:	1 student.