Editorial of Calibration Round #21

Problems link: Problems | Calibration Round #21 | গণিতযজ্ঞ

Problem A. বহুপদীর বিভাজ্যতা
Divide a^3+a^2+a+1 into two factors. The rest you can understand easily.

Problem B. বৃত্ত-ত্রিভুজ সংঘর্ষ

  • Just to use the Power of point theorem and easy trigonometric approach.
  • Prove that, \triangle NPQ \sim \triangle NOM.

Problem C. বহিরাগত গুণক
i) Use AM-GM inequality.
ii) You might find terms like (x-y), (y-z) and (z-x) while manipulating expressions. And you need to use xy+yz+zx often in expansions. So think about it a bit.

Problem D. ভাগশেষ নির্ণয়

  • Try to express numbers in base 7.
  • You can think of it in terms of expansion of (1+x). Then apply combinatorial logic in terms of the power of x.

Problem E. ভয়কে করো জয়!

  • It has something to do with the power of 2.
  • The maximum value is of the form 1- \frac {1}{2^k}.
    Find k by yourselves.

Problem F. দুই ত্রিভুজ
An easy trigonometric approach using Cosine law. You can use Stewart’s theorem also.