Physics questions

Option 2

1. Solve inequality
   $$\log_2(1 + \cos 4x) \leq 1 + \log_{\sqrt{2}} \sin x.$$

2. Find all valid solutions of the system
   $$\begin{cases} |xy - 2| = 6 - x^2, \\ 2 + 3y^2 = 2xy. \end{cases}$$

3. Three racers ($A$, then $B$ and then $C$) start with an interval of 1 minute from one point on the track highway and move in the same direction at constant speeds. Each racer spends more than two minutes on a lap. Having made three laps, the racer $A$ will finish $B$ for the first time at the starting point, and after another 3 minutes he will overtake $C$ for the second time. Racer $B$ agreed for the first time $C$ also at the starting point, having completed 4 laps. How many minutes does a racer $A$ spend on a lap?

4. A convex quadrilateral $ABCD$ contains two circles of the same radius $r$, externally touching each other. The center of the first circle is on the segment connecting vertex $A$ with the midpoint $F$ of side $CD$, and the center of the second circle is on the segment connecting vertex $C$ with the midpoint $E$ of side $AB$. The first circle touches sides $AB$, $AD$ and $CD$; the second circle touches sides $AB$, $BC$ and $CD$. Find $AC$.

5. The length of each edge of the triangular pyramid $SABC$ is equal to 1. $BD$ is the height of triangle $ABC$. The equilateral triangle $BDE$ lies in the plane forming an angle φ with the edge $AC$, and the points $S$ and $E$ lie on the same side of the plane $ABC$. Find the distance between points $S$ and $E$.

Sweeping. In problem 2, the system can be solved differently: from the first equation, express $x$ through $y$ and substitute it into the second. Try this solution method.

Figure 3 is symmetrical, therefore $BM_1 = M_2C$, which are used in the solution of problem 4. One can, however, deduce this equality from the equality of the corresponding triangles.

In problem 5
In the solution to Problem 5, the model signs in $BO'$ are placed because the point $B$ can lie lower than the point $O'$.

Physics questions

1. Is the power of a hydroelectric power plant sufficient to evaporate the water passing through the turbines?

2. What gets cold faster under the same conditions: broth or tea? Why?

3. The terminal rod is suspended on light threads (see Fig. 1). Where is its center of gravity?

4. Two identical boas begin to simultaneously swallow each other from the tail and do this at a constant speed until each of them passes the other’s body completely (but without heads). Draw a diagram of how it all ends.

5. Fill a 2 liter saucepan to the top with water. A thin-walled pan with a capacity of 1.5 liters and a weight of 0.6 kg (Fig. 2.) is placed in it. How much water will come out?

6. Pure water is used for light. Why is fog, which is small drops of water, opaque?