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假定有25名短跑选手比赛争夺前三名，赛场上有五条赛道，一次可以有五名选手同时比赛。比赛并不计时，只看相应名次。假定选手的发挥是稳定的，也就是说，张三比李四跑得快...

Jan 19, 2024

假定有25名短跑选手比赛争夺前三名，赛场上有五条赛道，一次可以有五名选手同时比赛。比赛并不计时，只看相应名次。假定选手的发挥是稳定的，也就是说，张三比李四跑得快，李四比王五跑得快，张三一定比王五跑得快。问最少需要几组比赛才能决出前三名？

Solution by Steps

step 1

To determine the minimum number of races needed to find the top three runners out of 25 with 5 lanes available, we start by running 5 initial races with 5 runners each

step 2

After these 5 races, we have 5 winners. We run one more race with these 5 winners to determine the fastest 3

step 3

The fastest runner from step 2 is the overall fastest. The second and third place runners from this race are potential candidates for the overall second and third place

step 4

We need to race the runners who came in second and third in the initial races against the second and third place runners from the winners' race to determine the overall second and third places

step 5

There are 2 runners from the winners' race and 8 runners who came in second and third in the initial races, making a total of 10 runners. We can run 2 more races with 5 runners each

step 6

After these 2 races, we take the top two from each race and run one more race to determine the final order of the second and third places

Answer

The minimum number of races needed to determine the top three runners is 5 (initial) + 1 (winners) + 2 (second and third places) + 1 (final) = 9 races.

Key Concept

Minimizing the number of races to find the top competitors

Explanation

By organizing the races efficiently and using the process of elimination, we can minimize the number of races needed to determine the top three runners out of 25 with only 5 lanes available.

正三角形内部有一个点，到三个顶点的距离分别是3，4，5，求此三角形的面积

Solution by Steps

step 1

Use Viviani's theorem, which states that the sum of the distances from any interior point to the sides of an equilateral triangle is constant and equal to the altitude of the triangle

step 2

The distances from the interior point to the vertices are 3, 4, and 5. The sum of these distances is the altitude

h

of the equilateral triangle

step 3

Calculate the sum of the distances:

3 + 4 + 5 = 12

. This sum is equal to the altitude

h

of the triangle

step 4

The area

A

of an equilateral triangle with side length

a

is given by the formula

A = \frac{\sqrt{3}}{4}a^2

step 5

The altitude

h

also relates to the side length

a

by the formula

h = \frac{\sqrt{3}}{2}a

step 6

Solve for

a

using the altitude

h = 12

a = \frac{2h}{\sqrt{3}} = \frac{2 \cdot 12}{\sqrt{3}} = \frac{24}{\sqrt{3}} = 8\sqrt{3}

step 7

Substitute

a = 8\sqrt{3}

into the area formula to find the area of the triangle:

A = \frac{\sqrt{3}}{4}(8\sqrt{3})^2

step 8

Simplify the area expression:

A = \frac{\sqrt{3}}{4} \cdot 64 \cdot 3

step 9

Calculate the area:

A = \frac{\sqrt{3}}{4} \cdot 192 = 48\sqrt{3}

Answer

A = 48\sqrt{3}

square units

Key Concept

Viviani's theorem and area of an equilateral triangle

Explanation

Viviani's theorem is used to determine the altitude of the equilateral triangle from the given distances of an interior point to the vertices. The altitude is then used to find the side length and subsequently the area of the triangle.