Editorial of Calibration Round #20

Problems Link: Problems | Calibration Round #20 | গণিতযজ্ঞ

Problem A. সহজ ফাংশন
Let, f(c)=g(c-926)+a

Then, you should get g(P_1(c))+2g(P_2(c))=0 with P_1(c) and P_2(c). After that, you have to prove that, g(c)=0. And you got the function f(c)=a

Problem B. সহজ জ্যামিতি
Ratios of areas and ratios of segments along a line are preserved by an affine transformation. It suffices to find the area of LYC in an equilateral triangle LMN. Easily you can use angle bisector theorem.

Problem D. সহজ সমীকরণ
Just factorization and caseworking makes a perfect solution. For a hint, let \sqrt{p}+2\sqrt{q} =k

Problem E. আরো একটি সহজ জ্যামিতি

  • Join R,X and T,X

  • Look for triangles that have equal bases and altitudes and notice that \triangle PXY= \triangle RXT.

  • Think about breaking \triangle PXT into \triangle PXR and \triangle RXT.

  • Work with \triangle PXT and \triangle TQS and also \triangle PXR and \triangle RQS and then break \triangle TQS into suitable triangles.

  • Try to prove that, \triangle LST = \triangle PXY+\triangle LQR

Problem F. সহজ গণনা

  • Consider the islands as vertices and the bridges as edges connecting the vertices. Then you’ll get a planar graph
  • Using vertices V, edges E, and regions R in that graph you’ll get V-E+R=2 and an inequality 2E \ge 3R. Using these two, eliminate the R and find an inequality containing V and E variables

Problem G. সহজ বিয়ের শর্ত
From f(n-1).f(n)=n, You will get, f(1)×f(2)×f(3)×\dots f(2022)=2×4×6×...×2022. And then working on it, you will reach the answer.

Problem H. আরো একটি সহজ ফাংশন
As the function is true for all values of x and y. Then for x and y be 0, it should be true. You have to prove that, there exists no other function.