Question

Три сталкера подошли к волшебной тропе длиной 100 м. Известно, что первый пошедший по тропе окаменеет в начале тропы, второй окаменеет в произвольной точке тропы. Оба оживут, если третий окажется в такой точке тропы, что сумма расстояний от него до двоих спутников будет равна 100 м. Докажите, что сталкеры пройдут тропу.

Answer 1

Пусть а - точка, в которой окаменел второй, x - координата третьего.
Тогда сумма расстояний от него до двоих спутников описывается функцией
f(x) = x +|x-a|.
При х = 0 ее значение f(0) = 0 +|0-a| = а <100.
А при х = 100 ее значение f(100) = 100 +|100-a| = 200-а >100
Так как функция непрерывна, то по теореме Больцано-Коши она принимает промежуточное значение, равное 100, между точками х=0 и 100, то есть между началом и концом тропы.

Answer 2

Если тропа 100 метров, значит они пройдут, потому что максимум расстояние 100 метров!

Answer 3

Problem Analysis

We are given that three stalkers are approaching a magical path that is 100 meters long. The first stalker will turn into stone at the beginning of the path, the second stalker will turn into stone at a random point on the path, and both will come back to life if the third stalker reaches a point on the path such that the sum of the distances from him to the other two stalkers is equal to 100 meters. We need to prove that the stalkers will be able to traverse the entire path.

Proof

To prove that the stalkers will be able to traverse the entire path, we will consider two cases: when the second stalker turns into stone before the first stalker, and when the first stalker turns into stone before the second stalker.

Case 1: Second stalker turns into stone before the first stalker In this case, the second stalker will turn into stone at a point on the path before the first stalker. Let's assume that the second stalker turns into stone at a point x meters from the beginning of the path. Since the first stalker turns into stone at the beginning of the path, the distance between the first and second stalker at this point is x meters.

Now, for the third stalker to revive the first and second stalker, the sum of the distances from the third stalker to the other two stalkers should be equal to 100 meters. Let's assume that the third stalker reaches a point y meters from the beginning of the path. The distance between the third stalker and the first stalker is y meters, and the distance between the third stalker and the second stalker is (100 - y) meters.

According to the problem statement, the sum of these distances should be equal to 100 meters: y + (100 - y) = 100 Simplifying the equation, we get: y + 100 - y = 100 100 = 100

Since the equation holds true, it means that the third stalker can revive the first and second stalker when the second stalker turns into stone before the first stalker. Therefore, in this case, the stalkers will be able to traverse the entire path.

Case 2: First stalker turns into stone before the second stalker In this case, the first stalker will turn into stone at the beginning of the path before the second stalker. Let's assume that the first stalker turns into stone at a point x meters from the beginning of the path. Since the second stalker turns into stone at a random point on the path, the distance between the first and second stalker at this point is (100 - x) meters.

Now, for the third stalker to revive the first and second stalker, the sum of the distances from the third stalker to the other two stalkers should be equal to 100 meters. Let's assume that the third stalker reaches a point y meters from the beginning of the path. The distance between the third stalker and the first stalker is (100 - y) meters, and the distance between the third stalker and the second stalker is y meters.

According to the problem statement, the sum of these distances should be equal to 100 meters: (100 - y) + y = 100 Simplifying the equation, we get: 100 - y + y = 100 100 = 100

Since the equation holds true, it means that the third stalker can revive the first and second stalker when the first stalker turns into stone before the second stalker. Therefore, in this case, the stalkers will be able to traverse the entire path.

Since we have considered both cases and in both cases, the stalkers are able to traverse the entire path, we can conclude that the stalkers will be able to traverse the entire 100-meter path.

Note: The proof assumes that the third stalker can choose any point on the path. If there are any additional constraints or limitations on the third stalker's choice of point, please provide them for a more accurate analysis.

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