Problems

The Newton method (see Problem 61328) does not always allow us to approach the root of the equation \(f(x) = 0\). Find the initial condition \(x_0\) for the polynomial \(f(x) = x (x - 1)(x + 1)\) such that \(f(x_0) \neq x_0\) and \(x_2 = x_0\).

A pharmacist has three weights, with which he measured out and gave 100 g of iodine to one buyer, 101 g of honey to another, and 102 g of hydrogen peroxide to the third. He always placed the weights on one side of the scales, and the goods on the other. Could it be that each weight used is lighter than 90 grams?

Does there exist a number \(h\) such that for any natural number \(n\) the number \(\lfloor h \times 2021^n\rfloor\) is not divisible by \(\lfloor h \times 2021^{n-1}\rfloor\)?

Decipher the following puzzle. All the numbers indicated by the letter E, are even (not necessarily equal); all the numbers indicated by the letter O are odd (also not necessarily equal).

In a certain kingdom there were 32 knights. Some of them were vassals of others (a vassal can have only one suzerain, and the suzerain is always richer than his vassal). A knight with at least four vassals is given the title of Baron. What is the largest number of barons that can exist under these conditions?

(In the kingdom the following law is enacted: “the vassal of my vassal is not my vassal”).

a) Could an additional \(6\) digits be added to any \(6\)-digit number starting with a \(5\), so that the \(12\)-digit number obtained is a complete square?

b) The same question but for a number starting with a \(1\).

c) Find for each \(n\) the smallest \(k = k (n)\) such that to each \(n\)-digit number you can assign \(k\) more digits so that the resulting \((n + k)\)-digit number is a complete square.

Hannah placed 101 counters in a row which had values of 1, 2 and 3 points. It turned out that there was at least one counter between every two one point counters, at least two counters lie between every two two point counters, and at least three counters lie between every two three point counters. How many three point counters could Hannah have?

The numbers \(1, 2, 3,\dots , 10\) are written around a circle in a particular order. Peter calculated the sum of each of the 10 possible groups of three adjacent numbers around the circle and wrote down the smallest value he had calculated. What is the largest possible value he could have written down?

There are two symmetrical cubes. Is it possible to write some numbers on their faces so that the sum of the points when throwing these cubes on the upwards facing face on landing takes the values 1, 2, ..., 36 with equal probabilities?

A sailor can only serve on a submarine if their height does not exceed 168 cm. There are four teams \(A\), \(B\), \(C\) and \(D\). All sailors in these teams want to serve on a submarine and have been rigorously selected. There remains the last selection round – for height.

In team \(A\), the average height of sailors is 166 cm.

In team \(B\), the median height of the sailors is 167 cm.

In team \(C\), the tallest sailor has a height of 169 cm.

In team \(D\), the mode of the height of the sailors is 167 cm.

In which team, can at least half of the sailors definitely serve on the submarine?