Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# English Issue, December 2002

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## New exercises for practice - competition C (690-694.)

C. 690. Is it possible for the sum of the volumes of three cubes of integer edges to be 2002 units?

C. 691. Express in mm2 the area of Hungary on a globe of radius 25 cm.

C. 692. For the real numbers x, y, z,

x+2y+4z$\displaystyle \ge$3 and y-3x+2z $\displaystyle \ge$5.

Prove that y-x+2z$\displaystyle \ge$3.

C. 693. In what interval may the apex angle of an isosceles triangle vary if a triangle can be constructed out of its altitudes?

C. 694. Evaluate the sum [log21]+[log22]+[log23]+...+[log22002].

Suggested by Ádám Besenyei, Budapest