MathDB

4

Part of 2017 Hanoi Open Mathematics Competitions

Problems(2)

last digit 2^1+3^5+4^9+5^{13}+ ... +505^{2013}+506^{2017} (HOMC 2017 J Q4)

Source:

8/7/2019
Put S=21+35+49+513+...+5052013+5062017S = 2^1 + 3^5 + 4^9 + 5^{13} + ... + 505^{2013} + 506^{2017}. The last digit of SS is
(A): 11 (B): 33 (C): 55 (D): 77 (E): None of the above.
number theorylastDigit
c(x-a)(x-b)/(c-a)(cb)+a(x-b)(x-c)/(a-b)(a-c)+b(x-c)(x-a)/(b-c)(b-a)+1 HOMC17.4

Source:

9/9/2019
Let a,b,c be three distinct positive numbers. Consider the quadratic polynomial P(x)=c(x−a)(x−b)(c−a)(c−b)+a(x−b)(x−c)(a−b)(a−c)+b(x−c)(x−a)(b−c)(b−a)+1P (x) =\frac{c(x - a)(x - b)}{(c -a)(c -b)}+\frac{a(x - b)(x - c)}{(a - b)(a - c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}+ 1. The value of P(2017)P (2017) is
(A): 20152015 (B): 20162016 (C): 20172017 (D): 20182018 (E): None of the above.
algebrapolynomial