7

Part of 2002 AIME Problems

Problems(2)

Source:

The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers

x, y,

r

|x| > |y|,

(x \plus{} y)^r \equal{} x^r \plus{} rx^{r \minus{} 1}y \plus{} \frac {r(r \minus{} 1)}2x^{r \minus{} 2}y^2 \plus{} \frac {r(r \minus{} 1)(r \minus{} 2)}{3!}x^{r \minus{} 3}y^3 \plus{} \cdots What are the first three digits to the right of the decimal point in the decimal representation of \left(10^{2002} \plus{} 1\right)^{10/7}?

modular arithmeticAMCAIMEalgebrabinomial theorem

Sum of First k Squares

Source:

It is known that, for all positive integers

k,

1^{2}+2^{2}+3^{2}+\cdots+k^{2}=\frac{k(k+1)(2k+1)}{6}.

Find the smallest positive integer

k

1^{2}+2^{2}+3^{2}+\cdots+k^{2}

is a multiple of

200.