Question 1. Evaluate $ \left(6-2\dfrac{4}{5} \right)\cdot 3\dfrac{1}{8}-1\dfrac{3}{5}:\dfrac{1}{4}-1:\dfrac{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 on the picture. Which of the following statements are true:
a) $k_1 > k_2$, $b_1 > b_2$;
b) $k_1 > k_2$, $b_1 < b_2$;
c) $k_1 < k_2$, $b_1 > b_2$;
d) $k_1 > k_2$, $b_1 < b_2$?

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 $a=\dfrac{1}{2}$, $b=2\dfrac{5}{34}$.

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 $(10+410:(2x-31))\cdot 5=60$.

Question 7. Evaluate $\dfrac{-5.13^2-0.76^2+4.37^2}{15.2\cdot 1.026}$.

Question 8. Solve the inequality $\dfrac{x+1}{4}-\dfrac{4x+1}{5}\le \dfrac{7-3x}{10}$.

Question 9. 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 10. Solve the system: $\begin{cases} 2x+3y=7\\ 3x+2y=3 \end{cases}$.

Question 11. Alice has a sheet of paper measuring $51 \text{\,cm} \times 21 \text{\,cm}$ and a pair of scissors. She cuts a square from the sheet, with one side equal to the shorter side of the sheet. She continues cutting squares of the same size until it’s no longer possible to do so. Then, she repeats the process with the remaining (non-square) portion of the sheet, and so on. How many squares will Alice have in total? What is the side length of the smallest square?

Question 12. The angle bisectors drawn from vertices $A$ and $B$ of triangle $ABC$ intersect at point $D$. Find the angle $\angle ADB$ if $\angle BAC = 50^\circ$, $\angle ABC = 100^\circ$.

Question 13. The monthly gas production volumes at the first, second, and third fields are in the ratio 3:8:13. It is planned to reduce the monthly gas production at the first field by $13\%$ and at the second field by $13\%$ as well. By what percentage should the monthly gas production at the third field be increased so that the total gas production remains unchanged?

Question 14. The line $l_1$ with equation $y=k_1x+b_1$ passes through points $(2,4)$ and $(3,6)$. The line $l_2$ with equation $y=k_2x+b_2$ is perpendicular to $l_1$ and passes through $(0,5)$. Find the value of coefficients $k_1$, $b_1$, $k_2$, and $b_2$.

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?

Question 1. Find the value of the expression $(2-6b)(2a+b)-(5+4a)(a-3b)+2(3b^2+2a^2)$ for $a=\dfrac{1}{2}$, $b=2\dfrac{5}{34}$.

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 $\dfrac{-5.13^2-0.76^2+4.37^2}{15.2\cdot 1.026}$.

Question 4. The parabolas $p_1$ with equation $y=a_1x^2+b_1x+c_1$ and $p_2$ with equation $y=a_2x^2+b_2x+c_2$ are shown on the picture.
a) Determine 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) $\dfrac{2x^3+x^2+5x+7}{x-1}=\dfrac{x^3+5x^2+2x+7}{x-1}$;
b) $(x^2-x)\sqrt{x^2+x-6}=6\sqrt{x^2+x-6}$.

Question 6. Angle $\angle ABC$ of an isosceles triangle $\Delta ABC$ is equal to $120^\circ$. 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: $\dfrac{b^4+b^2+1}{b^2+1}$.

Question 8. Find all the values $a$ such that polynomial $p(x)=ax^2-(2a-1)x+a+2$ 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 $BC$ in 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 $p=\dfrac{\sqrt{2}+\sqrt{\dfrac{3}{2}}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\dfrac{\sqrt{2}-\sqrt{\dfrac{3}{2}}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}$. 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, 18 students answered the fourth 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 $\Delta APB$, $\Delta BPC$, and $\Delta CPD$ are 26, 24, and 6 respectively.
a) Find the area of $\Delta APD$.
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 $ABCDA_1B_1C_1D_1$ with edge length 4, find the shortest path along the cube's surface between the midpoint $M$ of edge $BC$ and point $P$ located 1 unit from vertex $A_1$ on edge $A_1D_1$.

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) $\dfrac{2x^3+x^2+5x+7}{x-1}=\dfrac{x^3+5x^2+2x+7}{x-1}$;
b) $(x^2-x)\sqrt{x^2+x-6}=6\sqrt{x^2+x-6}$.

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

Question 4. The parabolas $p_1$ with equation $y=a_1x^2+b_1x+c_1$ and $p_2$ with equation $y=a_2x^2+b_2x+c_2$ are shown on the picture.
a) Determine 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 diagonal $AC$ and divides it in a $5:2$ ratio, counted from vertex $A$. A point $O$ is the intersection of $AC$ and $BE$.
a) Find the ratio $CE:ED$.
b) Find the area of triangle $COE$, if the height of the trapezium is equal to 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 ($0^\circ\le \alpha\le 180^\circ$): $7\cos^2 \alpha+\left|7\sin^2\alpha-3\right|.$

Question 8. Let $p=\dfrac{\sqrt{2}+\sqrt{\dfrac{3}{2}}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\dfrac{\sqrt{2}-\sqrt{\dfrac{3}{2}}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}$. 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 the objects move in the same direction, they meet every 60 seconds. What fraction of the circle does each object cover per second?

Question 10. Find all the values $a$ such that polynomial $p(x)=ax^2-(2a-1)x+a+2$ has two real distinct roots.

Question 11. The sum of twice the first term and three times the fifth term of an arithmetic progression is 12 more than the sum of its second, third, and fourth terms. Find the sum of the first seven terms of this arithmetic progression.

Question 12. A square $ABCD$ has an interior point $P$ such that the areas of triangles $\Delta APB$, $\Delta BPC$, and $\Delta CPD$ are 26, 24, and 6 respectively.
a) Find the area of $\Delta APD$.
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 the contest 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 $ABCDA_1B_1C_1D_1$ with edge length 4, find the shortest path along the cube's surface between the midpoint $M$ of edge $BC$ and point $P$ located 1 unit from vertex $A_1$ on edge $A_1D_1$.