Humanities students take mathematics

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HUMANITIES
RENTAL
MATHEMATICS

Given the already tired division into “physicists” and “lyricists,” “physicists” include not only physicists, but also chemists, engineers, biologists, geologists, mathematicians, and “lyricists” include not only lyrical posers, but also philologists, social scientists and, in general, representatives of the humanities.
The complexity of the idea of ​​​​two polar categories of schoolchildren - one fascinating chemistry, physics, mathematics and actively participate in technical modeling classes; other topics of interest include history, art, psychology and are published in school literary magazines. The first should go to colleges, as well as physico-mathematical and chemical-biological faculties; the second - to institutes and faculties of humanitarian professionals.
However, when high school students are thinking about choosing a place for further study, they should not forget that the last decades have seen the deep penetration of mathematics into many branches of knowledge traditionally considered “humanitarian.” Suffice it to say that mathematics is taught today at the faculties of economics and psychology and at the department of structure and applied linguistics of the philological faculty of Moscow University.
This is not the place to talk about the role of mathematics in humanities research, or at least about the role of mathematical disciplines in teaching at these faculties of Moscow State University. In this article we present only the problems in mathematics proposed at the humanities faculties of Moscow State University at the written entrance exams in August 1969. However, the versions of the written tasks partly reflect the requirements for the mathematical preparation of those entering certain faculties and departments.

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TASK OPTIONS FOR THE WRITTEN EXAM*)

Faculty of Psychology

1. The internal point $A$ of a ball of radius $r$ is connected to the surface of the trey ball by straight segments of length $l$ and drawn at an angle $a$ to each other. Find the distance of point $A$ from the center of the ball.

2. Find all positive numbers $a$ for which all distinct non-negative values ​​$x$ satisfying the equation
   $$\cos[(5a - 9)x] = \cos[(9a + 17)x]$$
   and arranged in ascending order form an arithmetic progression.

3. Prove, without using decimals, that
   $$\log_2 5 > \log_5 32.$$

4. The distance between $A$ and $B$ is 7 km. Two pedestrians simultaneously embroidered towards each other and met equals an hour later. If the first one walked twice as fast as he actually walked, and the speed of the second one was 2 km/h more than his actual speed, then by the time they met, the second one would have covered most of the distance. Which pedestrian has the fastest speed?

5. Prove that
   $$8x^2 + y^2 + 11z^2 + 4xy - 12xz - 5yz > 0,$$
   if $x^2 + y^2 + z^2 > 0$.

3 tasks
from other options

a) A diameter $AB$ is drawn into a ball of radius $r$ and three equal chords $AC, AD$ and $AF$ are drawn at an angle $a$ to each other. Find the volume of the body bounded by the planes of triangles $ACD, ADF, ACF, BCD, BDF$ and $BCF$.

6. From $A$ to $B$ a raft floats along the river. At the same time that the raft began its journey from $A$ to $B$, a boat sailed towards it from $B$ to $A$, which met the raft no earlier than 2 hours later, and then arrived at $A$, spending less than three hours and twenty minutes on the entire journey. Will the raft manage to travel from $A$ to $B$ in 5 hours if the distance between $A$ and $B$ is 20 km?

Problem solutions

1. Let $O$ be the center of the ball; $B, D$ and $C$ are the points of intersection with the surface of the ball of the lines drawn from the point $A; O_1$ - the center of the section of the ball by the plane $\Delta BCD$. Considering,

*) The exams were allotted: at the Faculty of Psychology and the Department of Political Economy of the Faculty of Economics - 4 hours, at the Faculty of Philology and the Department of Economics of the Faculty of Economics - 5 hours (one hour is equal to sixty minutes).

that the plane $\Delta BCD$ is perpendicular to the diameter $AO_1$ (except for the case $l = r$), we find
$$OO_1 = \sqrt{r^2 - \frac{4}{3}l^2 \sin^2 \frac{\alpha}{2}}.$$
$$AO_1 = l \sqrt{1 - \frac{4}{3} \sin^2 \frac{\alpha}{2}}.$$

Then
$$AO = OO_1 + AO_1,$$ if points $A$ and $O$ lie on opposite sides of the plane $\Delta BCD$;
$$AO = OO_1 - AO_1,$$ if points $A$ and $O$ lie on the same side of the plane $\Delta BCD$, and points $O$ lie outside the parmid $ABCD$ ($OO_1 > AO_1$);
$$AO = AO_1 - OO_1,$$ if points $A$ and $O$ lie on the same side of the plane $\Delta BCD$, and point $O$ lies inside the parmida $ABCD$.

2. Using the difference of cosines formula
   $$2 \sin [(7a + 4)x] \cdot \sin [(2a + 13)x] = 0.$$

Since $a > 0$, the solutions of the equation have the form
$$\frac{\pi k}{7a + 4}, \frac{\pi l}{2a + 13}; k, l - целые.$$

It is clear that the non-negative solutions (corresponding to the non-negative $k$ and $l$) form two arithmetic progressions with the first term 0 and the differences of the corresponding integer factored into the other, i.e. when either $\frac{d_1}{d_2}$ is natural or $\frac{d_1}{d_2}$ the analogous result is: $\frac{d_2}{d_1}$ is natural.

Conversely, if, for example, $\frac{d_2}{d_1}$ is natural, then in connection with the arithmetic progression with the first term 0 and the difference $\frac{d_2}{d_1}$ is also a member of the arithmetic progression with the first term 0 and the difference $\frac{d_1}{d_1}$, therefore the terms of the two progressions together form an arithmetic progression with the first term 0 and the difference $\frac{d_1}{d_1}$.

So, $a > 0$ satisfies the conditions if and only if either $\frac{d_1}{d_2}$ is natural, or $\frac{d_2}{d_1}$ is natural, or $\frac{d_1}{d_2}$ is also a member of an arithmetic progression with the first term 0 and the difference $\frac{d_1}{d_1}$, so the terms of the two progressions together form an arithmetic progression with the first term 0 and the difference $\frac{d_1}{d_1}$.

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Therefore we have to consider two cases:

1. $\frac{d_1}{d_2}$ natural. $\frac{\pi}{7a+4} = n \cdot \frac{\pi}{2a+13}$, $n$ natural. $a = \frac{13-4n}{7n-2}$. $7n-2 > 0$ for natural $n$. Therefore, $a > 0$ if and only if $13-4n > 0$, i.e., when $n=1, 2, 3$, respectively $a = \frac{9}{5}, \frac{5}{12}, \frac{1}{19}$.
2. $\frac{d_2}{d_1}$ natural. Similarly follows 1) we get $a = \frac{9}{5}, \frac{22}{3}, 35$.
   $a = \frac{1}{19}, \frac{5}{12}, \frac{9}{5}, \frac{22}{3}, 35$.

3. Since $\log_2 5 \cdot \log_5 32 = \log_2 32 = 5$, it is enough for us to show that $\log_2 5 > \sqrt{5}$. Since $81 > 80$, i.e. $\frac{81}{16} > 5$, then $\frac{9}{4} > \sqrt{5}$, and to prove the assumed inequality it is enough to verify that $\log_2 5 > \frac{9}{4}$ or $5 > 2^{\frac{9}{4}}$. But $5^4 = 625 > 512 = 2^9$; Taking the fourth root, we obtain the required inequality.
4. The speed of the second pedestrian is higher.
5. $8x^2 + y^2 + 11z^2 + 4xy - 12xz - 5yz = 2\left(2x + \frac{y}{2} - \frac{3}{2}z\right)^2 + \frac{1}{2}(y - 2z)^2 + \frac{9}{2}z^2$.

Consequently, the original expression is non-negative, as reducible to a sum of squares, and if it vanishes, then $x=y=z=0$, i.e. $x^2+y^2+z^2=0$.
Pozny, if $x^2+y^2+z^2>0$, then
$8x^2+y^2+11z^2+4xy-12xz-5yz>0$.
a) The required volume is $\frac{1}{\sqrt{3}}(2r)^3\left(3-4\sin^2\frac{\alpha}{2}\right)\sin^2\frac{\alpha}{2}$.
6) Let $x$ be the speed of the river, $y$ be the speed of the boat in still water. According to the conditions we have: $\frac{20}{(y-x)+x} \geq 2$, $\frac{20}{y-x} < 3\frac{1}{3}$, whence $y \leq 10$, $x < y-6$; $x<4$, $\frac{20}{x} > 5$. Consequently, the raft will not have time to cover the path from $A$ to $B$ in 5 hours.

Department of Political Economy, Faculty of Economics

1. The Nauchsky single-track railway, 28.5 km long, needs to be laid. There are 30 m and 15 m long rails available for laying. If all 30 m long rails are laid, then 15 m long rails will need to be added to 40% of the total number. If you lay all the rails 15 m long, then 60% of the total number of rails 30 m long must be added. Determine the number of rails of both types.
2. Solve inequality $|x^2-1|-2x<0$.
3. In the acute triangle $ABC$ from the vertex $B$ a medana $BD$ equal to $m$ is drawn. Angle $A$ is equal to $a$, angle $C$ is equal to $y$. Calculate the area of ​​triangle $ABC$.
4. Solve the equation
   $$\sin x \cdot \mathrm{ctg}^2 \frac{x}{2} + 4(1+\cos x) - 5\sin x = 0$$
5. Solve the equation
   $$\sqrt{x} + \sqrt{x(x+2)} - \sqrt{(x+1)^2} = 0$$
   3 tasks from other options
   a) In a bookstore, an antique collection of works costs 350 rubles. discounted twice for the same number of projects. Find this number if it is known that after a double price reduction, the collected works cost 283 rubles. 50 kopecks

6. At what real value $x$ does the expression $\frac{2x-1}{2x-x^2-4}$ take its smallest value?

Problem solutions

1. When drawing up the equation, it should be taken into account that a single-track railway of 28.5 km has a track of 57 km. It is necessary to lay rails: 1500 30 m long and 2000 15 m long.
2. $-1 + \sqrt{2} < x < 1 + \sqrt{2}$.
3. $m^2 \frac{\sin \alpha \cdot \sin \gamma \cdot \sin (\alpha+\gamma)}{4\sin^2 \alpha \cdot \sin^2 \gamma + \sin^2 (\alpha-\gamma)}$.
4. $x_1 = (2k+1)\pi$, $x_2 = \frac{\pi}{2} + 2k\pi$.
5. $x = \frac{\sqrt{5}-1}{2}$.
   a) 10%.

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6. Putting $y = \frac{2x - 1}{2x - x^2 - 4}$, we obtain a quadratic (with respect to $x$) equation $yx^2 + 2(1-y)x + 4y - 1 = 0$, the discriminant of which must be non-negative: $(1-y)^2 + y(4y - 1) \geq 0$, since $x$ is a real number, whence
   $$-\frac{1 + \sqrt{13}}{6} \leq y \leq \frac{-1 + \sqrt{13}}{6}.$$
   Therefore, the smallest value of $y$ is $-\frac{1 + \sqrt{13}}{6}$; the corresponding value of $x$ is, as can be easily calculated, $\frac{1 + \sqrt{13}}{2}$.

Department of Economic Cybernetics, Faculty of Economics

1. Solve the equation
   $$\left( \sqrt{\cos x} - 4 \sqrt[4]{2} \sqrt[4]{\cos \left( x - \frac{\pi}{4} \right)} + 5 \sqrt[4]{2}\right)^2 - 2 \sqrt{\cos x} \times \left( \sqrt{\cos x} - 4 \sqrt[4]{2} \sqrt[4]{\cos \left( x - \frac{\pi}{4} \right)} + 5 \sqrt[4]{2}\right) - \sin x = 0.$$
2. Four groups of tourists set off on a Sunday approach along four routes of different lengths. The sum of the distances covered by the first and fourth groups is 6 km greater than the sum of the distances covered by the second and third groups. The second group walked 2 km less than the first. The number of tourists of the third group is equal to the distance traveled by the first group, and the number of tourists of the second group is equal to the distance traveled by the fourth group. The sum of the squares of the distance traveled by each group is 494. How much cybernet did each group ask for?
3. In a circle of radius $R$ there is a hexagon ABCDEF. It is known that $\angle A = \angle C = \angle E$, side $AB = a$, CD = b, EF = c. Calculate the area of ​​the hexagon ABCDEF.
4. Solve inequality
   $$\left( 4x - x^2 - 3 \right) \log_2 \left( \cos^2 \pi x + 1 \right) \geq 1.$$
   3 task
   from other options
   a) The segment $AB = a$ is perpendicular to the segment $BC = 3a$. In the plane containing both of these segments, two circles of radius $a$ with centers at points $A$ and $B$ and a circle of radius $3a$ with center at point $C$ are constructed. Calculate the area of ​​part of the three constructed circles.

6. Solve inequality
   $$\left| \sqrt{2} | x | - 1 \right| \cdot \log_2 \left( 2 - 2x^2 \right) \geq 1.$$

Problem solutions

1. Range of acceptable values ​​(OD3):
   $$-\frac{\pi}{4} + 2k\pi \leq x \leq \frac{\pi}{2} + 2k\pi,$$
   $k = 0, \pm 1, \dots$

Let’s put
$$y = \sqrt{\cos x} - 4 \sqrt[4]{2} \sqrt[4]{\cos \left( x - \frac{\pi}{4} \right)} + 5 \sqrt[4]{2},$$
Then
$$y^2 - 2 \sqrt{\cos x} y - \sin x = 0,$$
where
$$y = \sqrt{\cos x} \pm \sqrt{\sqrt{2} \cos \left( x - \frac{\pi}{4} \right)},$$
i.e. we have
$$\sqrt{\cos x} - 4 \sqrt[4]{2} \sqrt[4]{\cos \left( x - \frac{\pi}{4} \right)} + 5 \sqrt[4]{2} = \sqrt{\cos x} \pm \sqrt[4]{2} \sqrt{\cos \left( x - \frac{\pi}{4} \right)}.$$

Putting $z = \sqrt[4]{\cos \left( x - \frac{\pi}{4} \right)}$, we get $-4z + 5 = \pm z^2$. After the equation has roots $z_1 = 1, z_2 = -5, z_3 = 2 \pm i$. Roots $z_2, z_3, z_4$, very/no, construction.
From $z_1 = 1$ follows $\cos \left( x - \frac{\pi}{4} \right) = 1$. Thus, $x = \frac{\pi}{4} + 2k\pi, k = 0, \pm 1$.

2. Let $x_i$ be the distance traveled by the $i$-group, $i = 1, 2, 3, 4$. From the problem condition we have $x_1 + x_4 - 6 = x_2 + x_3, x_1 - x_2 = 2, x_1^2 + x_2^2 + x_3^2 + x_4^2 = 494$, whence
   $$(x_1 - 1)^2 + (x_4 - 2)^2 = 242,$$
   $$x_1 = 1 + \sqrt{242 - (x_4 - 2)^2}.$$

According to the conditions, $x_1$ and $x_4$ are natural numbers ($x_1$ is the number of tourists in the 3rd group, and $x_4$ is the number of light tourists by selection based on the last correspondence (and replacing that by virtue of the same correspondence $x_4 < 18$). Sort through sequentially
$x_4 = 1, 2, 3, \dots, 17,$ we find the only solution:
$x_4 = 13, x_1 = 12.$
In this case $x_2 = 10, x_3 = 9.$

3. $$\frac{3 \sqrt{3}}{4} R^2 + \frac{\sqrt{3}}{8} a \left( \sqrt{12R^2 - 3a^2} - a \right) + \frac{\sqrt{3}}{8} b \left( \sqrt{12R^2 - 3b^2} - b \right) + \frac{\sqrt{3}}{8} C \left( \sqrt{12R^2 - 3C^2} - C \right).$$

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$$\frac{\cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ - 1}{\sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ} = 2 + \sqrt{2} - \sqrt{3} - \sqrt{6}.$$

In conclusion, we suggest solving one option for each table on your own.

Faculty of Philology

1. There are three pipes installed into the pool through which water flows into it. The first and second pipes together fill the pool 35/18 hours faster than the first and third pipes together fill it in 10 hours. What part of the pool does the second pipe fill per hour if it alone takes $\frac{7}{5}$ times longer to fill than the first pipe alone?
2. Solve the equation:
   $$\log_{\sin x} \left( \sin x - \frac{1}{4} \cos x \right) = 3.$$
3. Solve inequality:
   $$2\log_3 x - \log_1/5 (4 - x) \leq \log_3 (x - 1)^2 + 2\log_2 (10 - x).$$
4. N people exchanged handshakes. How many handshakes were there?
5. Indicate at least one pair of positive integers $k_1$ and $k_2$ such that $36 \cdot k_1 - 25 \cdot k_2 = 1$.
6. Prove, without using tables, that $\log_4 9 > \log_5 11$.
7. Does a triangle have an obtuse angle whose sides are 10, 14 and 17 respectively?

Faculty of Psychology

1. A sphere contains a diameter $AB$ and two equal chords $AM$ and $AN$, each at an angle $a$ to the diameter. Find the angle between the chords if the segment $MN$ is visible from the center of the ball at an angle $\beta$.
2. Find all positive numbers $a$ for which all distinct non-negative values ​​$x$ satisfying the equation $\cos [(8a-3)x] = \cos [(14a+5)x]$ and arranged in ascending order form an arithmetic progression.
3. Prove, without using decimals, that $\log_3 16 > \log_16 729$.
4. The distance between stations $A$ and $B$ is 360 km. At the same time, two trains leave from $A$ and from $B$ towards each other. Trains left $A$ and arrive at station $B$ no earlier than 5 hours later. If his speed had been 1.5 times higher than it actually was, then he would have met the second train more than 2 hours after he left $A$. Which train is faster?
5. Prove that
   $$2x^2 + 5y^2 + 3z^2 - 6xy - 2xz + 5yz > 0,$$
   if $x^2 + y^2 + z^2 > 0$.

Department of Political Economy

1. Five camping tents and twenty stopley sleeping bags 490 RUR. After a ten percent discount on one tent and a five percent discount on one bag, for the same number of tents and bags you need to pay 457 rubles. How much is one stop for one tent and how much is one stop for one bag?
2. Solve inequality: $x^2 + x - 10 < 2/x - 2$.
3. In an acute triangle $ABC$, angle $A$ is equal to $a$, side $AC$ is equal to $b$. From the vertex $B$ the median $m$ is drawn. Calculate the area of ​​triangle $ABC$.
4. Solve the equation: $(3 - \text{ctg}^2 x)\sin 2x = 2(1 + \cos 2x)$.
5. Find the maximum value of the fraction $\frac{3x - 2}{4x - x^2 - 6}$ if $x$ can take any real value.

Department of Economic Library

1. Solve the equation
   $$\left( 4\sqrt{\cos \frac{x}{2}} - 5 - \frac{\sqrt{2}}{2} \right)^2 + \sqrt{2} \left( 4\sqrt{\cos \frac{x}{2}} - \frac{\sqrt{2}}{2} - 5 \right) - \frac{\cos x}{2} = 0.$$
2. There are four art clubs at the pioneer camp. Give each member of the circle the same set of colors. In the first and fourth circles together, and in the second and fourth circles together there are even more pioneers than in the first and third circles together. There are no more than ten pioneers in the third circle. How many pioneers are in each circle, if the price in rubles of one paint is equal to the number of members of the first circle divided by ten, and 52 rubles were paid for paints? 80 kopecks?
3. A circle is inscribed in the trapezoid $ABCD$. Find the difference in the lengths of the diagonal segments $AC$ located in a circle if the height of the trapezoid is $h$, and the angles at the base $AD$ are $a$ and $\delta$.
4. Solve inequality: $2^{-1}x - 2^1\log_2(4x - x^2 - 2) \geq 1$.

A. S. KUZICHEV
V. S. USPENSKY