Editorial of Calibration Round #22

Problems link: Problems | Calibration Round #22 | গণিতযজ্ঞ

Problem A. সংখ্যা প্রতিস্থাপন

Let p be a prime dividing a. Clearly, a and p are odd, so a-p is even so that n-p<2022, once if a-p were greater than 2022, a could be written.

Problem B. তিন কন্যা

You have R=\frac{a}{2sinA}=\frac{b}{2sinB}=\frac{c}{2sinC}=\frac{P}{2(sinA+sinB+sinC)} and K=\frac{abc}{4R}

Problem C. ঝঞ্ঝাটময় সমীকরণ

Let log denote log base 3. Then a=log(n)+log(n-1).

Problem D. আতিয়াব সংখ্যা

For any two-digit positive integer \overline {ab}, it is sabroso if 11a+11b is a perfect square, so a+b=11k^2 for some positive integer k.

Problem E. ঝঞ্ঝাটময় ত্রিভুজ

If the feet of perpendiculars from O to the sides of the triangle are X, Y, Z, respectively, then OB will be the radius of the cyclic quadrilateral OZQX. Then, you can use the sine rule.

Problem F. দুই চলকের সমীকরণ

It is a quadratic equation in c with discriminant \Delta: =d^4-2(d+1)^2+72, but for all y>5 you have (d^2-2)^2<\Delta<d^4