Editorial of Calibration Round #23

Problems: Problems | Calibration Round #23 | গণিতযজ্ঞ

Problem A. বৃত্তস্থ ত্রিভুজের ক্ষেত্রফলের অনুপাত
Let R' be the reflection of R about center of the circle. Then work on parallelogram RSR'T.

Problem B. পূর্ণবর্গ রাশি
Clearly d > 2. d2^{d-1} = (x-1)(x+1) hence there exists an odd divisor p of d such that x - 1 = 2p and x+1 = 2^{d-2}\frac{d}{p}.

Problem C. বিচ্ছিরি সমীকরণ
Break the number a into integral part and fractional part. You will obtain a Diophantine equation. Then try out the combinations of factors of 58.

Problem D. ট্রাপিজিয়ামের ভেতর-বাহির
Let E on \overline {PQ} and F on \overline {RS} be collinear with X, s.t \overline {EF} is perpendicular to \overline {PQ} and \overline {RS}. Then a simple length-chase.

Problem E. ৩ চলকের ২ সমীকরণ
Ever heard of Pell’s equation? The solution of the problem requires Pell equation.

Problem F. সাতটি বাস্তব সংখ্যা
Try to apply AM-GM ineqality that involve terms \frac {1}{n{x_n}^n}+x_n. At the end, you will end up with a finite harmonic series.

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