Incident at the lecture

48

Solution. Let us find using a graphical solution of the system of inequalities

$$x + 2y \leq 3, \quad 2x - y \leq 1, \quad x \geq 0, \quad y \geq 0$$

a set of points whose coordinates satisfy specified conditions. From figure 15 it is clear that this will be a quadrilateral OABC. (The coordinates of its vertices are easily determined from the equations of the boundary lines and are indicated on the same figure.)

Now our task is formulated as follows. Among the points of the quadrilateral AOBC, find a point for which the expression $x^2 + y^2$ is the most 简便. But we know that the expression $x^2 + y^2$ determines the square of the distance from the point with coordinates $(x, y)$ to the origin. Consequently, the task comes down to finding the point in the OABC quadrilateral that is most convenient from the origin. Obviously, such a point is the point $C\left(0; \frac{3}{2}\right)$. Prove it!

Solve two problems of this type for yourself (these problems were presented in the 1968 written entrance exams in mathematics).

Example 12 (physicist at Leningrad University). Among all pairs of numbers that satisfy the inequality $\log_{x^2 + y^2}(x + y) \geq 1$, find those with $y$ the most.

Example 13 (Mechanics and Mathematics of Novosibirsk University). For each $a$ among all points of the plane whose coordinates $(x, y)$ satisfy the system of inequalities

$$\begin{align*} y - 2x &\geq 0, \\ -x^2 + 2ax - a^2 + a + 1 - y &\geq 0, \end{align*}$$

find the point with the most $y$.

At the beginning of the lecture, the lecturer formulated the famous Cauchy theorem, according to which the arithmetic mean of unequal positive numbers is greater than their geometric mean.

At this moment, the student was called to the dean’s office. When he returned to the lecture, conditions were still outside the door as the lecturer finished formulating the problem:

“The clue is that all the roots of the equation are not only real, but also positive.”

When the student entered the classroom, everything on the board had already been erased except for the note $x^{20} - 20x^{19} \dots +1 = 0$.

“Everything is clear,” said the student and named all the roots of the equation that the lecturer wrote down in his absence.

Can you name them?