Test for the 4-year program

Question 1

Evaluate:

\[ (6 - 2 \frac{4}{5}) \cdot 3 \frac{1}{8} - 1 \frac{3}{5} \div \frac{1}{4} - 1 \div \frac{1}{3} \]

Question 2

The line \( l_1 \) with equation \( y = k_1x + b_1 \) and \( l_2 \) with equation \( y = k_2x + b_2 \) are shown in the picture. Which of the following statements are true?

• \( k_1 > k_2, b_1 > b_2 \)
• \( k_1 > k_2, b_1 < b_2 \)
• \( k_1 < k_2, b_1 > b_2 \)
• \( k_1 > k_2, b_1 < b_2 \)

Question 3

Which number is larger \( 0.2^{2^{15}} \) or \( 0.05^{2^{10}} \)?

Question 4

Find the value of the expression:

\[ (2-6b)(2a + b) - (5 + 4a)(a-3b) + 2(3b^2 + 2a^2) \]

for \( a = \frac{1}{2}, b = 2 \frac{5}{34} \).

Question 5

An unloaded car traveled from point A to point B at a constant speed and returned along the same road with a load at 60 km/h. What was its speed when traveling unloaded if the average speed for the entire journey was 70 km/h?

Question 6

Solve the equation:

\[ (10 + 410 \div (2x - 31)) \cdot 5 = 60 \]

Question 7

Evaluate:

\[ \frac{-5.13^2 - 0.76^2 + 4.37^2}{15.2 \cdot 1.026} \]

Question 8

Solve the inequality:

\[ \frac{x+1}{4} - \frac{4x+1}{5} \leq \frac{7-3x}{10} \]

Question 10

Solve the system:

\[ \begin{cases} 2x + 3y = 7 \\ 3x + 2y = 3 \end{cases} \]

Question 15

A rectangular bar of chocolate has a mass of 100 g and consists of 20 equal squares. Bob cut the bar into four pieces as shown in the picture. What is the weight of the middle piece?

Test for the 3-year program

Question 1

Find the value of the expression:

\[ (2 - 6b)(2a + b) - (5 + 4a)(a - 3b) + 2(3b^2 + 2a^2) \]

for \( a = \frac{1}{2}, b = 2 \frac{5}{34} \).

Question 2

An unloaded car traveled from point A to point B at a constant speed and returned along the same road with a load at 60 km/h. What was its speed when traveling unloaded if the average speed for the entire journey was 70 km/h?

Question 3

Evaluate:

\[ \frac{-5.13^2 \;-\; 0.76^2 \;+\; 4.37^2}{15.2 \cdot 1.026} \]

Question 4

The parabolas \(p_1\) with equation \(y = a_1 x^2 + b_1 x + c_1\) and \(p_2\) with equation \(y = a_2 x^2 + b_2 x + c_2\) are shown in the picture.

a) Determine the signs of each of the coefficients.

b) Which of the statements are true: \(|a_1| > |a_2|, |b_1| > |b_2|, |c_1| > |c_2|\)?

Question 5

Solve the equations:

a) \(\displaystyle \frac{2x^3 + x^2 + 5x + 7}{x - 1} \;=\; \frac{x^3 + 5x^2 + 2x + 7}{x - 1} \)

b) \(\displaystyle (x^2 - x)\sqrt{x^2 + x - 6} \;=\; 6\sqrt{x^2 + x - 6} \;+\; x \;-\; 6 \)

Question 6

Angle \(\angle ABC\) of an isosceles triangle \(\triangle ABC\) is \(120^\circ\). Find the length of a leg of the triangle if the length of its bisector \(BL\) is 4.

Question 7

Find the minimum value of the expression:

\[ \frac{b^4 + b^2 + 1}{b^2 + 1} \]

Question 8

Find all the values \(a\) such that the polynomial \(\displaystyle p(x) = a x^2 - (2a - 1)x + a + 2\) has two real distinct roots.

Question 9

A circle is inscribed in triangle \(ABC\) where \(AB = 5\), \(BC = 7\), and \(AC = 9\). A tangent to the circle intersects sides \(AB\) and \(BC\) at points \(K\) and \(L\), respectively. Determine the perimeter of triangle \(KBL\).

Question 10

Two objects are moving along a circle: the first completes a lap 2 seconds faster than the second. If they meet every 60 seconds moving in the same direction, what fraction of the circle does each object cover per second?

Question 11

Let \[ p = \frac{\sqrt{2} + \sqrt{\frac{3}{2}}}{\sqrt{2} + \sqrt{2 + \sqrt{3}}} \;\; \div \;\; \frac{\sqrt{2} - \sqrt{\frac{3}{2}}}{\sqrt{2} - \sqrt{2 - \sqrt{3}}}. \] Given \(p\) is rational, find \(p\).

Question 12

20 students are participating in a competition. To win, a student has to answer all 5 questions correctly. It is known that 16 students answered the first question correctly, 14 students answered the second question correctly, 15 students answered the third question correctly, and only one student won the contest by answering all five questions correctly. Determine the maximum number of students who could have answered the last (fifth) question correctly.

Question 13

A square \(ABCD\) has an interior point \(P\) such that the areas of triangles \(\triangle APB\), \(\triangle BPC\), and \(\triangle CPD\) are 26, 24, and 6, respectively.

a) Find the area of \(\triangle APD\).
b) Find \(DP\).

Question 14

A six-digit phone number is given. A seven-digit phone number will be called “extended” if removing one of its digits results in the given six-digit number. How many such “extended” numbers are there?

Question 15

Given a cube \(ABCD A_1 B_1 C_1 D_1\) with edge length 4, find the shortest path along the cube’s surface between the midpoint \(M\) of edge \(BC\) and point \(N\) located 1 unit from vertex \(A_1\) on edge \(A_1 D_1\).

Test for the 2-year program

Question 1

An unloaded car traveled from point A to point B at a constant speed and returned along the same road with a load at 60 km/h. What was its speed when traveling unloaded if the average speed for the entire journey was 70 km/h?

Question 2

Solve the equations:

a) \(\displaystyle \frac{2x^3 + x^2 + 5x + 7}{x - 1} \;=\; \frac{x^3 + 5x^2 + 2x + 7}{x - 1} \)

b) \(\displaystyle (x^2 - x)\sqrt{x^2 + x - 6} = 6\sqrt{x^2 + x - 6} + x - 6 \)

Question 3

In triangle \(ABC\), the altitude \(CK\) is drawn, and \(\angle ACB = 90^\circ\). Given that \(CB = 10\) and \(\sin \angle CBA = \frac{3}{5}\), find the lengths of \(BA,\; CA,\) and \(KA\).

Question 4

The parabolas \(p_1\) with equation \(\displaystyle y = a_1 x^2 + b_1 x + c_1\) and \(p_2\) with equation \(\displaystyle y = a_2 x^2 + b_2 x + c_2\) are shown in the picture.

a) Determine the signs of each of the coefficients.

b) Which of the statements are true: \(|a_1| > |a_2|,\; |b_1| > |b_2|,\; |c_1| > |c_2|\)?

Question 5

In trapezium \(ABCD\), the bases are \(BC = 8\) and \(AD = 10\). A point \(E\) is chosen on side \(CD\) such that segment \(BE\) intersects \(AD\) in a \(5:2\) ratio at \(F\).

a) Find the ratio \(CE : ED\).

b) Find the area of triangle \(COE\), if the height of the trapezium is \(7\).

Question 6

There are 20 books randomly arranged on a shelf, including a two-volume set by J. London. Assuming all possible arrangements are equally likely, determine the probability that the two volumes are placed next to each other.

Question 7

Find the minimum and the maximum values of the expression \((0^\circ \le \alpha \le 180^\circ)\):

\[ 7 \cos^2 \alpha \;+\; \bigl|\,7 \sin^2 \alpha \;-\; 3\bigr|. \]

Question 8

Let \[ p \;=\; \frac{\sqrt{2} + \sqrt{\tfrac{3}{2}}}{\sqrt{2} + \sqrt{2 + \sqrt{3}}} \;+\; \frac{\sqrt{2} - \sqrt{\tfrac{3}{2}}}{\sqrt{2} - \sqrt{2 - \sqrt{3}}}. \] Given \(p\) is rational, find \(p\).

Question 9

Two objects are moving along a circle: the first completes a lap 2 seconds faster than the second. If they move in the same direction and meet every 60 seconds, what fraction of the circle does each object cover per second?

Question 10

Find all values of \(a\) such that the polynomial \(\displaystyle p(x) = a x^2 \;-\; (2a - 1)x \;+\; a + 2\) has two real distinct roots.

Question 11

The sum of the first two terms and the difference of the first two terms of an arithmetic progression is 12 more than the sum of its second, third, and fourth terms. Find the sum of the first five terms of this progression.

Question 12

A square \(ABCD\) has an interior point \(P\) such that the areas of triangles \(\triangle APB\), \(\triangle BPC\), and \(\triangle CPD\) are \(26\), \(24\), and \(6\), respectively.

a) Find the area of \(\triangle APD\). b) Find \(DP\).

Question 13

A box contains 3 white balls and 4 black balls. Three players take turns drawing one ball at a time, without replacement. The game continues until someone draws a white ball, at which point that person wins. What is the probability that the second player is the winner?

Question 14

20 students are participating in a competition. To win, a student has to answer all 5 questions correctly. It is known that 16 students answered the first question correctly, 14 students answered the second question correctly, 15 students answered the third question correctly, 18 students answered the fourth question correctly, and only one student won by answering all five questions correctly. Determine the maximum number of students who could have answered the last (fifth) question correctly.

Question 15

Given a cube \(ABCD A_1 B_1 C_1 D_1\) with edge length 4, find the shortest path along the cube’s surface between the midpoint \(M\) of edge \(BC\) and point \(N\) located 1 unit from vertex \(A_1\) on edge \(A_1 D_1\).