Question 15

$$A,M,T,I$$ are positive integers such that $$A+M+T+I=10$$. The maximum possible value of $$A\times M\times T\times I+A\times M\times T+A\times M\times I+A\times T\times I+M\times T\times I+A\times M+A\times T+A\times I+M\times T+M\times I+T\times I$$ is

Solution

The expression equals $$\left(1+A\right)\left(1+M\right)\left(1+T\right)\left(1+I\right)-1-\left(A+M+T+I\right)$$. Under the condition that the four positive integers sum to $$10$$, the maximum occurs at $$A=M=3$$ and $$T=I=2$$. The value is $$4\cdot4\cdot3\cdot3-1-10=133$$.

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$$A,M,T,I$$ are positive integers such that $$A+M+T+I=10$$. The maximum possible value of $$A\times M\times T\times I+A\times M\times T+A\times M\times I+A\times T\times I+M\times T\times I+A\times M+A\times T+A\times I+M\times T+M\times I+T\times I$$ is

Solution

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