2016 AIME II Problem 3

Question

Let

x,

y,

and

z

be real numbers satisfying the system

\log_2(xyz - 3 + \log_5 x) = 5

\log_3(xyz - 3 + \log_5 y) = 4

\log_4(xyz - 3 + \log_5 z) = 4.

Find the value of

|\log_5 x| + |\log_5 y| + |\log_5 z|.

Answer

Undo each outer logarithm: the equations become

xyz + \log_5 x = 35,

xyz + \log_5 y = 84,

xyz + \log_5 z = 259

Set

x = 5^a,

etc., and add all three equations:

3 \cdot 5^{a+b+c} + (a + b + c) = 378

has exactly one solution

Exponentiating each equation gives

xyz - 3 + \log_5 x = 2^5

and similarly for the others, so

xyz + \log_5 x = 35,

xyz + \log_5 y = 84,

xyz + \log_5 z = 259.

Write

x = 5^a,

y = 5^b,

z = 5^c,

so that

xyz = 5^{a+b+c}.

Adding the three equations gives

3 \cdot 5^s + s = 378,

where

s = a + b + c.

The left side is strictly increasing in

s,

and

s = 3

works since

375 + 3 = 378,

so

s = 3

and

5^s = 125.

Then

a = 35 - 125 = -90,

b = 84 - 125 = -41,

and

c = 259 - 125 = 134,

so

|a| + |b| + |c|

= 90 + 41 + 134

= 265.

Problem 3 in other years