Kvant
Information

Correspondence Physics and Technology School at MISiS

Kvant 1989, No. 1p. 69Tag PMRX

Rough automatic draft, low quality: expect mistakes and awkward wording. It will be replaced by a better translation. Original →

69

  1. A vessel with a volume of 20 liters is divided into two parts by a thin movable partition. On the left side there is 1 mole of water, on the right there is 0.5 mole of nitrogen. The temperature is kept at 100 °C. Determine the volume of the right side.

Mathematics

  1. Two tourists simultaneously walked from point A to point B. The first tourist walked at a speed of 5 km/h, and the other half at a speed of 4 km/h. The second tourist walked 4/9 of the way at a speed of 6 km/h, and the rest of the way at a speed of 4 km/h. Which tourist came closer to point B?

  2. The length of the midline of the trapezoid is 5 cm, and the length of the segment connecting the midpoints of the bases is 3 cm. Find the bases of the trapezoid if the angles at the larger base are 30° and 60°.

  3. Find the most possible value of the quotient of a three-digit number divided by the sum of its digits.

  4. What two digits must be added to the right of the number 1989 so that the resulting number is divisible by 8 and 9?

  5. It is known that a+1aa + \frac{1}{a} is an integer.
    Prove that a3+1a3a^3 + \frac{1}{a^3} is also an integer.

  6. On the coordinate plane, consider triangles with vertices at points A (0, 0), B (1, 1) and C (x, y). Find and draw on figure the set of points C such that the triangle ABC is acute.

  7. Point K is the midpoint of side AD of rhombus ABCD, and point L lies on the diagonal AC, AL=3LC. In what ratio does the intersection point of segments BD and KL divide segment BD?

  8. For what values ​​of the parameter a does the equation x+x=ax + \sqrt{x} = a have a unique solution?

  9. What values ​​can the parameter c apply if it is known that ∣x2−x+c∣≤1|x^2 - x + c| \leq 1 at 0≤x≤10 \leq x \leq 1?

  10. In a convex quadrilateral of area S, segments are drawn connecting opposite sides. Find the smallest possible value of the product of the lengths of the segments.

  11. The average of natural numbers from 1 to 99 select those that are not divisible by either 2 or 3. Find the sum of these numbers.

  12. A particle flies out from a point on the circumference of the base of the cylinder, which is reflected from the base and side surface of the cylinder according to the law: the angle of incidence is equal to the angle of reflection, and no reflection occurs at points on the circles of the bases. It is known that the first reflection will occur at a point at the base of the cylinder, and after a series of reflections the particle returned to the starting point. Find the smallest possible number of reflections.

Correspondence School of Physics and Technology at MISiS

The branch of the Western Union Correspondence Mathematical School - the Correspondence Physics and Technology School at the Moscow Institute of Steel and Alloys - announces enrollment of eighth, ninth and tenth grade students for the 1989/90 academic year.
The goal of the school is to assist students in in-depth study of the secondary school curriculum in physics and mathematics, as well as in career guidance in the MISiCa profile.
The mathematics curriculum fully complies with the VZMS course. The physics course was developed by teachers from the Department of General Physics and the Department of Theoretical Physics at MISiCa.
The work of the school is organized as follows. Four to five times a year, students will be given test assignments in physics and mathematics, along with brief information on theory and examples of problem solving, as well as career guidance materials. The work submitted by students is checked along with grades A (0, 0), B (1, 1) and C (x, y). Find and draw on figure the set of points C such that the triangle ABC is acute.

  1. Point K is the midpoint of side AD of rhombus ABCD, and point L lies on the diagonal AC, AL=3LC. In what ratio does the intersection point of segments BD and KL divide segment BD?
  2. For what values ​​of the parameter a does the equation x+x=ax + \sqrt{x} = a have a unique solution?
  3. What values ​​can the parameter c apply if it is known that ∣x2−x+c∣≤1|x^2 - x + c| \leq 1 at 0≤x≤10 \leq x \leq 1?
  4. In a convex quadrilateral of area S, segments are drawn connecting opposite sides. Find the smallest possible value of the product of the lengths of the segments.
  5. Average of numbers from 1 to 99, select those that are not divisible by either 2 or 3. Find the sum of these numbers.
  6. A particle flies out from a point on the circumference of the base of the cylinder, which is reflected from the base and side surface of the cylinder according to the law: the angle of incidence is equal to the angle of reflection, and reflection does not occur at points on the circles. It is known that the first reflection will occur at a point at the base of the cylinder, and after a series of reflections the particle returned to the starting point. Find the smallest possible number of reflections.

Classes at ZFTS begin on October 1. To enroll in the school, you must complete the first (introductory) assignment in mathematics, which is offered to all applicants to the VZMSH (Task 1 itself and the rules for completing the solution are given in the information “New admission to the Vesosoyuznaya Correspondence Mathematics School” in this issue of the magazine). Attach a statement to the decision (indicate in the statement -