An Exponential Meeting Its Inverse
Difficulty: Hard · Topics: Algebra, Exponents, Inverse Functions
O‘z teskarisi bilan uchrashuvchi ko‘rsatkichli funksiya
For real numbers a<b, the graphs of f(x)=2^x-k and its inverse meet at the two points A≤ft(a,2^a-k ) and B≤ft(b,2^b-k ). Let S 1 be the area of the circle of radius overlineOA and S 2 the area of the circle of radius overlineOB, where O is the origin.
Over all a, b, k with S 2<225S 1, the sum of all values of k for which dfrac2^b-2^a2^k is a natural number equals q/p+log 2 7. Find p+q. (p, q are coprime natural numbers.)
Haqiqiy a<b sonlar uchun f(x)=2^x-k va uning teskarisi grafiklari ikki nuqtada — A≤ft(a,2^a-k ) va B≤ft(b,2^b-k ) da uchrashadi. S 1 — radiusi overlineOA bo‘lgan doira yuzi, S 2 — radiusi overlineOB bo‘lgan doira yuzi, bunda O — koordinatalar boshi.
S 2<225S 1 bo‘lgan barcha a, b, k lar bo‘yicha dfrac2^b-2^a2^k natural son bo‘ladigan barcha k qiymatlar yig‘indisi q/p+log 2 7 ga teng. p+q ni toping. (p, q — o‘zaro tub natural sonlar.)