Problem M111—M115; F118—F122

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TASKS

M111. The square on side 1 contains a figure whose distance between any two points is not equal to 0.001. Prove that the area of ​​this figure does not exceed 0.34. (We can assume that the boundary of the figure referred to in the conditions consists of straight line segments and circular arcs.)
Try to get a more accurate estimate and prove a similar theorem in space.
G. V. Rosenblum

M112. In the table $m \times n$ the numbers are written so that for any two rows and any two columns, the sum of the numbers in the two opposite vertices of the rectangle they form is equal to the sum of the numbers in its two other vertices. Some of the numbers were erased, but let’s say you can restore the erased ones. Prove that there are at least $(n+m-1)$ numbers left.

M113. Prove that for any natural number $n$ there is a number made up of the digits 1 and 2 that is divisible by $2^n$.
B. M. Ivlev

M114. Several numbers are written in a circle. If for some four consecutive numbers $a, b, c, d$ it turns out that $(a-d)(b-c) < 0$, then the numbers $b$ and $c$ can be swapped. Prove that this operation can be performed only a finite number of times.
V. B. Alekseev

M115. A whole number of liters of water are poured into three vessels. It is allowed to pour a column of water into any vessel, as much as it already contains, from any other vessel. Prove,

Fig. 1.
a voltmeter, a current source and connected wires, is it possible to change the value of one of the resistances without breaking a single contact in the circuit?

A. R. Zilberman

F119. Light from source $S$ on its way to the screen passes through a resting glass cube with edge $l$ (Fig. 2). How much faster is light?