Question

大于任意一个有一条边长为 $5$ 、另一条边长为 $19$ 的三角形周长的最小整数是多少？

What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19?$

$24$

$29$

$43$

$48$

$57$

使用三角不等式限制第三边

Use the triangle inequality to bound the third side.

周长可以任意接近 $5+19$ 的两倍，但不能等于它

The perimeter can get arbitrarily close to twice $5+19,$ but not equal it.

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设第三边长为 $s$ 。三角不等式给出 $s<5+19=24$ ，所以周长满足 $5+19+s<48\text{。}$

周长可以从下方任意接近 $48$ ，因此大于所有这类三角形周长的最小整数是 $48$ 。

所以正确答案是 D。

Let the third side length be $s.$ The triangle inequality gives $s<5+19=24,$ so the perimeter satisfies $5+19+s<48.$

Perimeters can be made arbitrarily close to $48$ from below, so the smallest whole number larger than the perimeter of any such triangle is $48.$

Thus, D is the correct answer.