Problem M86—M90; F98—F102

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TASKS

M86. The bottom of the rectangular box was lined with tiles of sizes $2 \times 2$ and $1 \times 4$. The tiles spilled out of the box and in the process lost one tile $2 \times 2$. Instead, we managed to get tile $1 \times 4$. Prove that it is now impossible to tile up to the box.

L. G. Limanov

M87. Prove that if three circles of the same radius pass through one point, then three other points of pairwise intersection of these circles lie on a circle of the same radius (Fig. 1).

M88. What condition must the coefficients $a, b, c$ of the equation $x^3 + ax^2 + bx + c = 0$ satisfy in order for its three roots to form an arithmetic progression?

M. F. Bezborodnikov

M89. Prove that in any convex polygon except pa-

Fig. 1.

rappelogram, you can select these three sides, when continuing

Fig. 2.

which form a triangle enclosing the given polygon. (For example, at figure 2, where the polygon is outlined with a black line, the three red lines satisfy the required condition, but the three blue lines do not.)

I. M. Yaelom

M90. Prove that if $x_1 < x_2 < x_3 < \dots$ are natural numbers,

$$\frac{\sqrt{x_2-x_1}}{x_2} + \frac{\sqrt{x_3-x_2}}{x_3} + \dots$$

$$\dots + \frac{\sqrt{x_n-x_{n-1}}}{x_n} < 1$$

$$+\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n^2-1} + \frac{1}{n^2}$$

G. I. Natanson