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3

Part of 2015 Hanoi Open Mathematics Competitions

Problems(2)

sum of even positive <100, nod divis. by 3 (HOMC 2015 J Q3)

Source:

8/6/2019
The sum of all even positive integers less than 100100 those are not divisible by 33 is
(A): 938938, (B): 940940, (C): 16341634, (D): 16381638, (E): None of the above.
number theorydivisiblealgebra
(x-a)(x-b)/(c-a)(c-b)+(x-b)(x-c)/(a-b)(a-c)}+(x-c)(x-a)/(b-c)(b-a)= x (HOMC15-3)

Source:

9/7/2019
Suppose that a>b>c>1a > b > c > 1. One of solutions of the equation (x−a)(x−b)(c−a)(c−b)+(x−b)(x−c)(a−b)(a−c)+(x−c)(x−a)(b−c)(b−a)=x\frac{(x - a)(x - b)}{(c - a)(c - b)}+\frac{(x - b)(x - c)}{(a - b)(a - c)}+\frac{(x - c)(x - a)}{(b - c)(b - a)}= x is
(A): −1-1, (B): −2-2, (C): 00, (D): 11, (E): None of the above.
algebraequation