# IBC 2021 Math Course

## Monday - Trigonometric equations
### Task 8.387:
------------
Show that the equation:
$$
ctg(2x) + ctg(3x) + \frac{1}{(sin(x)*sin(2x)*sin(3x)}=0
$$

Has no roots

### Task 3.378:
------------
Given:
$$
\frac{\sin(\alpha + \beta)}{\sin(\alpha - \beta)} = \frac{p}{q}
$$

Consider $\alpha$, $p$, $q$ are known, find $ctg(\beta)$

### Task 3.364:
------------

Calculate:
1.$\sin(\frac{\alpha + \beta}{2})$
2.$\cos(\frac{\alpha + \beta}{2})$
if $\sin(\alpha) + \sin(\beta) = -\frac{21}{65}$; $\cos(\alpha) + \cos(\beta) = -\frac{27}{65}$,
$\frac{5\pi}{2}<\alpha<3\pi$ and $-\frac{\pi}{2}<\beta<0$

### Task 3.476:
------------
Find the biggest value of the expression:
$$\sin^2(\frac{15\pi}{8} - 4\alpha) - \sin^2(\frac{17\pi}{8} - 4\alpha)$$
given that $0<\alpha<\frac{pi}{8}$

### Task 3.281:
------------
Simplify an expression:
$$\sin(\frac{5\pi}{2} + 4a) - \sin^6(\frac{5\pi}{2} + 2a) + \cos^6(\frac{7\pi}{2} - 2a)$$

### Task 3.282:
------------
Simplify an expression:
$$\frac{\sin(8\alpha) + \sin(9\alpha) + \sin(10\alpha) + \sin(11\alpha)}{\cos(8\alpha) + \cos(9\alpha) + \cos(10\alpha) + \cos(11\alpha)} * \frac{\cos(8\alpha) - \cos(9\alpha) - \cos(10\alpha) + \cos(11\alpha)}{\sin(8\alpha) - \sin(9\alpha) - \sin(10\alpha) + \sin(11\alpha)}$$

### Task 8.303:
------------
Solve an equation:
$$(\cos^{-6}(z) - tg^6(z) - \frac{7}{3})(\sin(z)+\cos(z)+2)=0$$
### Task 8.313:
------------
Solve an equation:
$$1-\cos(x) = \sqrt{1-\sqrt{4\cos^2(x) - 7 \cos^4(x)}}$$
### Task 8.401:
------------
Solve a system of equations:
$$
\begin{cases}
tg(\frac{x}{2}) + tg(\frac{y}{2})=2
\\ctg(x)+ctg(y)= -1,8 \end{cases}
$$

### Task 3.489:
------------
- Prove that for all numbers $\phi$, that are satisfies inequality: $0 < \phi < \frac{\pi}{4}$, holds equation
$1-tg(\phi) + tg^2(\phi)-tg^3(\phi)+...=\frac{\sqrt{2}\cos(\phi)}{2\sin(\frac{\pi}{4} + \phi)}$
-----------

## Tuesday - Algebraic and logarithmic equations

1. $(x + 3)^4 + (x+5)^4 = 16$
Answer: -5, -3

2. $\sqrt{x-1} + \sqrt{x+3} + 2\sqrt{(x-1)(x+3)}= 4 - 2x$
Answer: 1

3. $\frac{20}{\sqrt(x)} +x\sqrt(x) + x = 22$
Answer: 1, 4

4. \begin{cases}
(\frac{x}{y})^2 + (\frac{x}{y})^3 = 12 \\
(xy)^2 + xy = 6
\end{cases}
Answer: (2;1), (-2;-1)

5. \begin{cases}
(x + y)(x + 2y)(x + 3y) = 60 \\
(y + x)(y + 2x)(y + 3x) = 105
\end{cases}
Answer: $(\frac{19\sqrt[3](4)}{4}; \frac{-17\sqrt[3](4)}{4})$, (2; 1)

6. $x^3 + x + \sqrt[3](x^3 + x - 2) = 12$
Answer: 2

7. $6\sqrt[3](x-3) + \sqrt[3](x-2) = 5\sqrt[6]((x-2)(x-3))$
Answer: $\frac{190}{63}$, $\frac{2185}{728}$

8. $4^x + 10^x = 25^x$
Answwer: $log_{0.4}(\frac{-1 + \sqrt(5)}{2})$

9. Show that: $log_{3}12 = log_{3}7log_{7}5log_{5}4 + 1$

10. $\sqrt(log_{2}(2x^2)log_{4}(16x)) = log_{4}x^3$
Answer: 16

11. $4^{log_{16}x} - 3^{log_{16}x - 0.5} = 3^{log_{16}x+0.5} - 2^{2{log_{16}x-1}}$
Answer: 64

12. $(2+\sqrt3)^{x^2-2x+1} + (2-\sqrt3)^{x^2-2x-1} = \frac{4}{2-\sqrt3}$
Answer: $1 + \sqrt2$, $1-\sqrt2$, 1

13. $(2^x - 2 * 2^{-x})^{log_{9}(2x+3)-log_{3}x} = 1$
Answer: 1, 3

## Wednesday - Geometry in the plane

1. The center of a circle inscribed in a rectangular trapezoid is at a distance of 3 and 9 cm from the ends of the lateral side. Find the sides of the trapezoid.
Answer: $\frac{36}{\sqrt10}$, $\frac{12}{\sqrt10}$, $\frac{18}{\sqrt10}$, $\frac{30}{\sqrt10}$ cm

2. Inside the right angle, a point M is given, the distances from which to the sides of the angle are 4 and 8 cm. The straight line passing through point M cuts off a triangle with an area of 100 $cm^2$ from the right angle. Find the legs of a triangle.
Answer: 20,10 or 5,40 𝑐𝑚

3. Point 𝑪𝟏 is the midpoint of side 𝑨𝑩 of triangle 𝑨𝑩𝑪; angle 𝑪𝑶𝑪𝟏, where 𝑶 the center of a circle circumscribed about a triangle is right. Prove that |𝒁𝑩 – 𝒁𝑨| = 𝟗𝟎°.

4. The circle touches two adjacent sides of the square and divides each of its two other sides into segments equal to 2 and 23 cm. Find the radius of the circle.
Answer: 17cm.

5. Given a triangle ABC, in which 2hс = AB and ZA = 75°. Find the value of the angle C.
Answer: 75

6. The bisectors of obtuse angles at the base of the trapezoid intersect at its other base. Find the sides of a trapezoid if its height is 12 cm and the lengths of the bisectors are 15 and 13 cm.
Answer: 14, 12.5, 29.4, 16.9

7. Given a triangle ABC such that AB = 15 cm, BC = 12 cm and AC = 18 cm. In what ratio does the center of the circle inscribed in the triangle divide the bisector of angle C?
Answer: $2:1$

8. Two chords 9 and 17 cm long are drawn from one point of the circle. Find the radius of the circle if the distance between the midpoints of these chords is 5 cm.
Answer: $\frac{85}{8}$

9. A circle with a radius of 5 cm is inscribed at a certain angle. The length of the chord connecting the points of tangency is 8 cm. Two tangents are drawn to the circle, parallel to the chord. Find the sides of the resulting trapezoid.
Answer: 5, 20, 12.5, 12.5

10. The vertices of a rectangle inscribed in a circle divide it into four arcs. Find the distance from the middle of one of the larger arcs to the vertices of a rectangle if its sides are 24 and 7 cm.
Answer: 15 and 20 cm

11. The center of a semicircle inscribed in a right-angled triangle so that its diameter lies on the hypotenuse divides the hypotenuse into segments 30 and 40. Find the length of the arc of the semicircle enclosed between the points of its tangency with legs.
Answer: $12\pi$

12. In triangle ABC, the value of angle A is twice the value of angle B, and the lengths of the sides opposite to these angles are 12 and 8 cm, respectively. Find the length of the third side of the triangle.
Answer: 10 cm

13. The largest of the parallel sides of the trapezoid is a, the smaller is b, the non-parallel sides are c and d. Find the area of the trapezoid.
Answer: $\frac{a+b}{4(a-b)}\sqrt{(𝑎−𝑏+𝑐+𝑑)(𝑎−𝑏+𝑑−𝑐)(𝑐+𝑎−𝑏−𝑑)(𝑐−𝑎+𝑏+𝑑)}$

14. The height and median of a triangle, drawn inside it from one of its vertices, are different and form equal angles with the sides extending from the same vertex. Determine the radius of the circumscribed circle if the median is m.
Answer: m

---

## Thursday - More Geometric tasks

### Task 1 (12.004)

In the square ABCD, a straight line is drawn through the middle of AB at point M, intersecting the opposite side of CD at point N. In what ratio does the straight line MN divide the area of the square if the acute angle AMN is equal to $\alpha$?

### Task 2 (12.017)

The length of the diagonal of the rectangle is equal to $d$​​. It divides the angle of the rectangle with ratio $m:n$​​​. Find the perimeter of the rectangle.

### Task 3 (12.040)

The height of the isosceles trapezoid is equal to $h$. The upper base of the trapezoid from the middle of the lower base is visible at an angle of $2\alpha$, and the lower base from the middle of the upper one is visible at an angle of $2\beta$. Find the area of the trapezoid in this general case and calculate it numerically if $h=2, \alpha=15°, \beta=75°$.

### Task 4 (12.088)

All the lateral edges of a triangular pyramid form the same angle with the base plane, equal to one of the acute angles of the right triangle lying at the base of the pyramid. Find this angle if the hypotenuse of the triangle is equal to $c$, and the volume of the pyramid is equal to $V$.

### Task 5 (12.091)

The side of the rhombus is equal to $a$, its acute angle is equal to $\alpha$. The rhombus rotates around a straight line passing through its vertex parallel to the larger diagonal. Find the volume of the solid of revolution.

### Task 6 (12.131)

In the acute-angled triangle ABC, the height AD = $a$, the height CE = $b$, the acute angle between AD and CE is equal to $\alpha$. Find AC.

### Task 7 (12.167)

Circle sector O has a radius R, the central angle AOB equals $\alpha$. A straight line parallel to the radius OB is drawn from point C bisecting the radius OA and intersecting the arc AB at point D. Find the area of the triangle OCD.

### Task 8 (12.198)

A circle is inscribed in the rhombus. Another circle is inscribed in one of the resulting curved triangles (having an acute angle). Find its radius if the height of the rhombus is equal to $h$, and the acute angle is equal to $\alpha$.

### Task 9 (12.121)

The base of the pyramid is a regular triangle with the side a. The two side faces of the pyramid are perpendicular to the plane of the base, and the equal side edges form an angle $\alpha$ between them. Find the height of a straight triangular prism that is equal to this pyramid and
has a common base with it.

### Task 10 (12.128)

A regular six-angle prism with equal edges is inscribed in the cone, the generatrix of which is equal to l. Find the area of the side surface of the prism if the angle between the generatrix and the height of the cone is $\alpha$.

### Task 11 (12.205)

At the base of the straight prism $ABCA_1B_1C_1 (AA_1||BB_1||CC_1)$ lies a right triangle ABC, in which the larger side (not a hypotenuse) AC is equal to a, and the opposite angle B is equal to $\alpha$. The hypotenuse AB is the diameter of the base of the cone, the vertex of which lies on the edge of $A_1C_1$. Find the height of the cone if $A_A1 = 0.5a$

### Task 12 (12.221)

The bases of the truncated pyramid are regular triangles. A straight line passing through the middle of the side of the upper base and the side of the lower base parallel to it is perpendicular to the planes of the bases. The greater lateral edge is equal to l and forms an angle $\alpha$ with the plane of the base. Find the length of the segment connecting the centers of the upper and lower bases.

## Answers:

1. $\frac{1-\cot(\alpha)}{1+\cot(\alpha)}=\tan(\alpha-\pi/4)$
2. $2\sqrt2*d*\cos(\frac{\pi(m-n)}{4(m+n)})$
3. $h^2(\tan(\alpha)+\tan(\beta)), 16$ units
4. $\arcsin(\frac{\sqrt{12Vc}}{c^2})$
5. $2\pi a^3\sin(\alpha)\sin(\alpha/2)$
6. $\frac{\sqrt{a^2+b^2-2ab\cos\alpha}}{\sin\alpha}$
7. $\frac{R^2}{8}\sin(\alpha)[\sqrt{4-\sin^2\alpha}-\cos\alpha]$
8. $\frac{h}{2}\tan^2(\frac{\pi-\alpha}{4})$
9. $\frac{a}{3sin{\frac{\alpha}{2}}}\sqrt{\sin{(\frac{\pi}{6}+\frac{\alpha}{2}})\sin{(\frac{\pi}{6}-\frac{\alpha}{2})}}$
10. $\frac{3l^2\sin^2{(2\alpha)}}{4\sin^2{(\frac{\pi}{4}+\alpha)}}$
11. $\frac{a\sqrt{\sin^2\alpha+\cot^2{\alpha}}}{2\sin{\alpha}}$
12. $\frac{l\sqrt{5-4\cos{(2\alpha)}}}{3}$

---

## Friday - Math application to other sciences' problems

**Problem 13.212**

Innopolis Bicycle Club has a budget of n rubles. But because the \cost of one bicycle decreased by c rubles, the club bought b additional bicycles (more than they expected). How many bicycles are now in the Innopolis Bicycle Club?

**Answer:** $\frac{-bc +\sqrt{b^2c^2+4bcn}}{2c}$

**Problem 13.227**

Two developers received a task. Second one started coding one hour later than the first one did. Three hours have passed after the first developer started and developers found out that 45% of the task is not completed. When the task was finished, they found out that each of them did exactly 50% of the task. Help our manager to find out how many hours it would take for each developer to complete that task separately.

**Answer**: x = 10, y = 8

**Problem 13.236**

InnoChess Club has invited a chess master to perform a session of simul aneous chess games. By the end of the first two hours he had won 10% of number of total games in session, 8 of the players got a draw as a result of their games with a master. In the next two hours master had won 10% of players that were still in the game after 2 first hours, lost in two and all other 7 games that left were finished with a draw. How many chess boards were there in the session?

**Answer**: x = 20, y = 10

**Problem 13.247**

Tatarstan farmers collect milk with fat percent of 5% and produce cheese with fat percent of 15,5% and they are left with subs ance with fat percent of 0.5%. How many cheese Tatarstan farmers can produce from 1000 kg of milk?

**Answer**: 300 kg

**Problem 13.278**

Passenger of the train knows that the speed of the train on this part of railroad is 40 km/h. As soon as the passenger noticed another train from his window going in the opposite direction, he started counting and found out that the opposite train passed in 3 seconds. length of the opposite train was 75m. Find out the speed of a train.

**Answer**: 14 m/s

**Problem 13.298**

Two students are running at the circle stadium near the football field. Speed of each student is cons ant. The first student spends 10 seconds less than the second one to run one lap.If both students start from one position, they will meet again after 720 seconds. What fraction of one lap each student runs in one second?

**Answer**: $v_{1} = \frac{l}{80}$,$v_{2} = \frac{l}{90}$

**Problem 13.308**

We need to find a two-digit number that is less by 11 than the sum of squares of digits and greater than the doubled multiplication of these digits by 5.

**Answer**: 95 or 15

**Problem 13.310**

There are two types of steel: one with concentration of nickel of 5% and other with concentration of 40%. How many steel do we need to take of each type to get 140 tons of steel with concentration of 30%?

**Answer**: x = 40, y = 100

**Problem 13.358**

We need to find a 6-digit number that starts from 1 such that if we put the starting digit to the end (move 1 to the end of the number) we will get a number that is three times bigger than a given one.

**Answer**: 142857

**Problem 13.386**

Points A1, B1 and C1 are located on the sides of the equilateral triangle ABC such that AA1 = BB1 = CC1 = x. Side of the triangle equals to a. We need to find such x that the fraction of areas of triangles A1B1C1 and ABC is equal to a given positive number m. In which range m can be so that the condition of the problem holds?

**Answer**: $x = \frac{a}{6}(3 +\sqrt{(12m-3)}), m \in [\frac{1}{4},1)$

**Problem 13.434**

Prove that if we have three consecutive natural numbers then the cube of the largest one cannot be the sum of cubes of two other numbers from the sequence.

**Problem 13.439**

We have a two-digit number x. If we add 46 to it, we will get a number $$46*x$$ in which product of it’s digits equals to 6. Sum of x’s digits is 14. Find x.

**Answer**: 77 or 86