5

Part of 2010 AIME Problems

Problems(2)

Counting Integers with Constraints

Source: AIME 2010I Problem 5

Positive integers

a

b

c

d

a > b > c > d

, a \plus{} b \plus{} c \plus{} d \equal{} 2010, and a^2 \minus{} b^2 \plus{} c^2 \minus{} d^2 \equal{} 2010. Find the number of possible values of

a

Logarithmic Equations

Source: 2010 AIMEII Problem 5

Positive numbers

x

y

z

satisfy xyz \equal{} 10^{81} and (\log_{10}x)(\log_{10} yz) \plus{} (\log_{10}y) (\log_{10}z) \equal{} 468. Find \sqrt {(\log_{10}x)^2 \plus{} (\log_{10}y)^2 \plus{} (\log_{10}z)^2}.

logarithmsSFFTspecial factorizationsAMC