Hi, my name is Evgeniy, and I have been preparing for the Unified State Exam and Basic State Exam in mathematics for 13 years. In this video, we will analyze the Unified State Exam 2025 version for 100 points. The version is made up of problems that have already been included in the Unified State Exam and from FIPI, so the versions are of the same difficulty level as the real Unified State Exam 👍 LINKS: Download the version: https://vk.com/wall-40691695_100964 VK group: https://vk.com/shkolapifagora Video courses: https://vk.com/market-40691695 How I passed the Unified State Exam: https://vk.com/wall-40691695_66680 Reviews: https://vk.com/wall-40691695_98328 Insta: / shkola_pifagora 🔥 TIMECODES: Start – 00:00 Problem 1 – 02:21 In triangle ABC AC=BC, AB=15, AH is the height, BH=6. Find the cosine of angle BAC. Problem 2 – 05:33 Vectors a ⃗ (-13;4) and b ⃗ (-6;1) are given. Find the scalar product a ⃗∙b ⃗. Problem 3 – 07:38 In a cube ABCDA_1 B_1 C_1 D_1, find the angle between the lines CD_1 and AD. Give your answer in degrees. Problem 4 – 09:23 The performance competition is held over 3 days. A total of 70 performances have been announced – one from each country participating in the competition. A performer from Russia is participating in the competition. 28 performances are scheduled for the first day, the rest are distributed equally between the remaining days. The order of performances is determined by drawing lots. What is the probability that a performer from Russia will perform on the third day of the competition? Problem 5 – 14:26 There are two identical vending machines in a shopping center selling coffee. The probability that the first machine will run out of coffee by the end of the day is 0.1. The probability that the second machine will run out of coffee is the same. The probability that both machines will run out of coffee is 0.05. Find the probability that both machines will have coffee left by the end of the day. Problem 6 – 20:58 Find the root of the equation log_27⁡〖3^(5x+5) 〗=2. Problem 7 – 25:51 Find the value of the expression log_8⁡14/log_64⁡14 . Problem 8 – 28:19 The figure shows the graph of y=f^' (x), the derivative of the function f(x), defined on the interval (-4;6). Find the abscissa of the point at which the tangent to the graph of the function y=f(x) is parallel to the line y=3x or coincides with it. Problem 9 – 31:21 An observer is at a height h (in km). The distance l (in km) from the observer to the horizon line observed by him is calculated using the formula l = √2Rh, where R = 6400 km is the radius of the Earth. At what height is the observer if he sees the horizon line at a distance of 96 km? Give your answer in km. Problem 10 – 35:37 Two pipes fill a pool in 1 hour 55 minutes, and the first pipe alone fills the pool in 46 hours. How many hours does it take for the second pipe to fill the pool? Problem 11 – 41:58 The figure shows the graph of a function of the form f(x) = k/x. Find the value of f(10). Problem 12 – 44:24 Find the smallest value of the function y = e ^ 2x - 2e ^ x + 8 on the interval [-2; 1]. Problem 13 – 47:44 a) Solve the equation log_13⁡(cos⁡2x-9√2 cos⁡x-8)=0. b) Indicate the roots of this equation that belong to the segment [-2π;-π/2]. Analysis of errors 13 – 01:00:35 Problem 15 – 01:05:45 Solve the inequality (2^x+8)/(2^x-8)+(2^x-8)/(2^x+8)≥(2^(x+4)+96)/(4^x-64). Analysis of errors 15 – 01:17:40 Problem 16 – 01:23:55 Vladimir is the owner of two factories in different cities. The factories produce absolutely identical goods, but the factory located in the second city uses more advanced equipment. As a result, if workers at the plant located in the first city work a total of t^2 hours per week, then they produce 4t units of goods this week; if workers at the plant located in the second city work a total of t^2 hours per week, then they produce 5t units of goods this week. For each hour of work (at each of the plants), Vladimir pays the worker 500 rubles. Vladimir needs to produce 410 units of goods every week. What is the smallest amount that will have to be spent weekly on paying workers? Problem 18 – 01:43:17 Find all values ​​of a 0 for each of which the equation 1-6√x=3(x+a) has exactly two roots. Problem 19 – 01:59:06 Several different natural numbers are written on the board, the product of any two of which is greater than 45 and less than 120. a) Can there be 5 numbers on the board? b) Can there be 6 numbers on the board? c) What is the smallest value that the sum of the numbers on the board can take if there are four of them? Problem 17 – 02:11:19 In trapezoid ABCD, point E is the midpoint of base AD, point M is the midpoint of side AB. a) Prove that the areas of quadrilateral AMOE and triangle COD are equal if O is the intersection point of segments CE and DM. b) Find what part of the area of ​​the trapezoid is the area of ​​quadrilateral AMOE if BC=5, AD=7. Problem 14 – 02:30:22 In triangular pyramid SABC, the lateral edges are known: SA=SB=13, SC=3√17. The base of the height of this pyramid is the midpoint of median CM of triangle ABC. This height is 12. a) Prove that triangle ABC is isosceles. b) Find the volume of pyramid SABC. #OptionsUSEprofilePythagorasSchool