Maths Olympiad Prep

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Number theory Difficulty 5.2 AIME, harder Find the answer

14. (6 points) A group of PLA soldiers formed a three-layer hollow square formation with 9 soldiers left over. If another layer was added inside the hollow part, 7 soldiers would be short. The total number of soldiers in this group is \qquad people.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

【Solution】Solution: Let the number of people on each side of this layer be AA, from the problem we have:
16=(A−4)×4×4,16A=80,A=5, \begin{aligned} 16 & =(A-4) \times 4 \times 4, \\ 16 A & =80, \\ A & =5, \end{aligned}

Then the number of people in the outermost layer is 5+3×2=115+3 \times 2=11,
Total number of people: (11−3)×3×4+9(11-3) \times 3 \times 4+9,
=8×3×4+9,=105 (people),  \begin{array}{l} =8 \times 3 \times 4+9, \\ =105 \text { (people), } \end{array}

Answer: There are a total of 105 soldiers.
The answer is: 105.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.