1971 Mathematics Entrance Exam Options

65

Moscow State University named after M.V. Lomonosov

Department of Economic Cybernetics, Faculty of Economics

1. Two types of parts were produced. The first part was made from an alloy of metals $M_1$ and $M_2$, and the second part was made from an alloy of metals $M_3$ and $M_4$. If parts of the first type were made only of metal $M_1$, then their total weight would be 1.6 kg, and if only of metal $M_2$, then 5.2 kg. If the parts of the second type were made only of metal $M_3$, then their total weight would be 3 kg, and if only of metal $M_4$, then 4.5 kg. The cost of one part of the first type is 3 rubles, its weight is 0.31 kg. The cost of one part of the second type is 5 rubles, its weight is 0.34 kg. How many parts of each type were produced if their total cost is 68 rubles?

2. Solve the equation:

$$tg(\pi ctg x) = ctg(\pi tg x).$$

3. Find the minimum term of a sequence

$$a_n = n^2 - 8n + 15 - \frac{9}{(3n - 16)^2 + 6},$$

$$n = 1, 2, 3, \dots$$

4. On the plane lie a straight circular cylinder of radius $R$ (that is, the cylinder touches the plane along some image) and, without intersecting with it, radius $r$. The distance from the cylinder axis to the ball center is $p$. Find the minimum possible radius of a ball that would simultaneously touch the cylinder, the plane and the given ball.

5. How many six-digit numbers are there in which each place has a digit one greater than the one to the left of it (the digits are numbered from left to right)?

Department of Political Economy, Faculty of Economics

1. Product output during the year of operation of the enterprise increased by $P\%$, and the next year it increased by 10% more than in the first year. Determine by what percentage the output increased in the first year, if it is extracted that in two years it increased by 48.59%.

2. Solve inequality:

$$x > \frac{1}{x - 1}.$$

3. Solve inequality:

$$11 \sin x + \cos 2x - 6 \leq 0.$$

4. Calculate the sum of all natural numbers that do not exceed 1000 and do not divide by 13.

5. A ball of radius $R$ is inscribed into the dihedral angle $60^\circ$. Find the radius of a ball inscribed at the same angle and tangent to the given ball without intersecting it, a ball of radius $r$. The distance from the cylinder axis to the center of the ball is $p$. Find the minimum possible radius of a ball that would simultaneously touch the cylinder, the plane and the given ball.

Department of biological biology faculty

1. Solve the equation:

$$\log \frac{1}{8} \left[ 2^x + 1 + tg^2 \frac{\pi}{4} \right] + \log \frac{1}{8} \left[ 2^x + ctg^2 \frac{\pi}{6} \right] = \frac{5}{3} + 3 \log \frac{1}{8} 5.$$

2. A truncated cone and a regular hexagonal prism are arranged so that the upper base of the truncated cone is circumscribed about the lower base of the prism. It is known that the height of a truncated cone is equal to the sum of the radii of its bases. Find the ratio of the surface sizes of these bodies.

3. By mixing 2 $cm^3$ of three substances, we obtained 16 $e$ mixtures. It is known that 4 $e$ of the second of these substances occupy a volume $\frac{1}{2} cm^3$ larger than 4 $e$ of the third substance. Find the density of the third substance if it is known that the mass of the second substance in the mixture is twice that of the first substance.