Problem M1141—M1145; F1148—F1152

21

Quests
M1141—M1145, F1148—F1152

M1141. A trapezoid is circumscribed about a circle. Prove that at least one of its diagonals forms an angle of no more than 45° with the base.

H. M. Sedrakin

M1142. The table $m \times n$ is filled with $mn$ numbers so that in each row and in each column these numbers form an arithmetic progression. The sum of the four numbers in the corners of the table is $s$. What is the sum of all the numbers in the table?

M1143. The mass of each of the 101 weights arranged around a circle is a natural number, and their total mass is 300 g. Prove that from this set one or more weights located in a row can be selected with a total mass of 200 g.

B. B. Prozovolyov

M1144. Let $a_1, a_2, \dots, a_n$ be non-negative numbers. Which number is greater:

$$\sqrt{a_1^{1988} + a_2^{1988} + \dots + a_n^{1988}}$$ or $$\sqrt{a_1^{1989} + a_2^{1989} + \dots + a_n^{1989}}$$

A. I. Shekhorsky

M1145*. From the point $P$ two tangents PB and PC are drawn to the circle, and $\angle BPC > 90^\circ$. Point A is taken on the smaller arc BC. Prove that the area of ​​the triangle cut off from angle BPC by the tangent to the circle at point A does not exceed the area of ​​triangle ABC.

V. Yu. Progasov

F1148. A cylinder with a thread wound around it, the second end of which is fixed, is placed on a smooth inclined plane making an angle $\alpha$ with the horizon, as shown in figure 1. At the moment when the thread was vertical, the angular velocity of rotation of the cylinder was equal to $\omega$. Determine what is equal at this moment: a) the speed of the cylinder axis; b) the speed of a point on the cylinder touching the inclined plane. The radius of the cylinder is $R$.

S. S. Krotov

F1149. Thin long bare copper conductors are randomly thrown onto the rails of a toy railroad, closing the rails. Estimate the resistance between the rails if the distance between them is $l = 5$ cm, the diameter of the conductor is $d = 0,2$ mm, its length is $h = 30$ cm, $N = 100$ conductors are thrown.

A. P. Zilberman

F1150. A body is supplied to an ideal monatomic gas enclosed inside an oil bubble. Find the heat capacity of the gas (per 1 mole) in this process, if the pressure of the bubble can be avoided.

A. A. Sheronov

F1151*. An electrostatic voltmeter is a flat capacitor, one of the plates of which