Problem M121—M125; F133—F137

PROBLEM WORKER
QUANTA

This section of the magazine is maintained from issue to issue. In it, along with relatively easy problems, problems consisting of several, gradually developing questions are published. The most difficult tasks are marked with asterisks. They are often difficult even for specialists in mathematics and physics. Therefore, do not check in if you cannot cope with this or that task. Try to get back to it again over time.

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3GOALS

M121. Prove that for any $n$ real numbers $a_1, a_2, \dots, a_n$ there is a natural number $k \leqslant n$ such that each of the $k$ numbers

$$a_k, \frac{a_{k-1}+a_k}{2}, \frac{a_{k-2}+a_{k-1}+a_k}{3}, \dots, \frac{a_1+a_2+\dots+a_k}{k}$$ does not exceed

$$\frac{a_1+a_2+\dots+a_n}{n}.$$

M122. The pentagon $ABCDE$ is inscribed in a circle. It is known that the distances from the point $E$ to the lines $AB, BC$ and $CD$ are equal to the correspondence $p, q, r$. Find the distance from point $E$ to line $AD$.

M123. Find all natural numbers $m$ for which

$$1! \cdot 3! \cdot 5! \cdot \dots \cdot (2m-1)! = \left( \frac{m(m+1)}{2} \right)!$$

(the product $1 \cdot 2 \cdot \dots \cdot n$ is denoted by $n!$).

V. P. Beshkarev

M124. Given a triangle $ABC$. Find a point $O$ inside it that has the following property: for any line passing through the point $O$ and intersecting the sides of the triangle $AB$ at the point $K$ and $BC$ at the point $L$, the equality

$$\frac{AK}{KB} + \frac{CL}{LB} = 1.$$

In general, prove that if $p$ and $q$ are arbitrary given positive numbers, then inside the triangle $ABC$ one can specify a point $O$ such that for any line $KL$ passing through this point ($K$ lies on $AB$, $L$ on $BC$),

$$p \frac{AK}{KB} + q \frac{CL}{LB} = 1.$$

G. Nomen (10th grade student)

M125. a) Is there an infinite sequence of natural numbers with the following property:

None of these numbers is divisible by the other, but among each three numbers you can choose two whose sum is divisible by the third?

6. If not, how many numbers can there be in a set that have this property?

b) Solve the same problem with the additional condition: only odd numbers are allowed to be included in the set.

Here is one example of such a set of four numbers: 3, 5, 7, 107. Here, among the three numbers 3, 5, 7, the sum 5+7 is divisible by 3; in the triple 5, 7, 107, the sum 107+5 is divided by 7; in the triple 3, 7, 107, the sum 7+107 is divided by 3; finally, in the triple 3, 5, 107, the sum 3+107 is divided by 5.

Yu. I. Nonin

F133*. A closed metal chain is connected by a thread to the centering axis of the machine and rotates with an angular velocity $\omega$ (Fig. 1). In this case, the thread makes an angle $\alpha$ with the vertical.

Fig. 1.