Problem 4. Positive numbers $a, b, c, d$ are greater than 1. Find the smallest possible value of the expression

$$
\log _{a}\left(a b^{2}\right)+\log _{b}\left(b^{2} c^{3}\right)+\log _{c}\left(c^{5} d^{6}\right)+\log _{d}\left(d^{35} a^{36}\right)
$$
Answer: 67.

Solution. From the properties of logarithms, it follows that $\log _{a} b \cdot \log _{b} c \cdot \log _{c} d \cdot \log _{d} a=1$. Also, all these four factors are positive, since all numbers $a, b, c, d$ are greater than 1.

Transform and estimate the given expression

$$
\begin{gathered}
S=\log _{a}\left(a b^{2}\right)+\log _{b}\left(b^{2} c^{3}\right)+\log _{c}\left(c^{5} d^{6}\right)+\log _{d}\left(d^{35} a^{36}\right)= \\
=\left(\log _{a} a+\log _{a} b^{2}\right)+\left(\log _{b} b^{2}+\log _{b} c^{3}\right)+\left(\log _{c} c^{5}+\log _{c} d^{6}\right)+\left(\log _{d} d^{35}+\log _{d} a^{36}\right)= \\
=\left(1+2 \log _{a} b\right)+\left(2+3 \log _{b} c\right)+\left(5+6 \log _{c} d\right)+\left(35+36 \log _{d} a\right) \geqslant \\
\geqslant(1+2+5+35)+4 \sqrt[4]{2 \log _{a} b \cdot 3 \log _{b} c \cdot 6 \log _{c} d \cdot 36 \log _{d} a}=43+4 \cdot 6=67,
\end{gathered}
$$

here in the last transition, the inequality between the arithmetic and geometric means for four positive numbers $2 \log _{a} b, 3 \log _{b} c, 6 \log _{c} d, 36 \log _{d} a$ was used.

Also note that the value $S=67$ is achieved, for example, when $a=2, b=8, c=d=64$, since all four numbers $2 \log _{a} b, 3 \log _{b} c, 6 \log _{c} d, 36 \log _{d} a$ will be equal to 6.

## Criteria

Points for the estimate and example are summed.

Estimate. The largest suitable criterion is used:

5 p. The estimate $S \geqslant 67$ is proven.

1 p. The problem is reduced to finding the minimum of the sum $2 \log _{a} b+3 \log _{b} c+6 \log _{c} d+36 \log _{d} a$.

0 p. The positivity of the numbers $\log _{a} b, \log _{b} c, \log _{c} d, \log _{d} a$ is mentioned.

Example. The largest suitable criterion is used:

2 p. It is proven that the value $S=67$ is achieved for some $a, b, c, d$, greater than 1.

0 p. Only the correct answer is given.