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 asked on 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_99971 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 ???? TIME CODES: Start – 00:00 Problem 1 – 02:20 In triangle ABC, angle A is 56°, angles B and C are acute, heights BD and CE intersect at point O. Find angle DOE. Give your answer in degrees. Problem 2 – 04:52 Vectors a ⃗ (0;3), b ⃗ (-2;4), and c ⃗ (4;-1) are given. Find the length of the vector a ⃗+2b ⃗+c ⃗. Problem 3 – 07:40 A sphere with a volume of 35π is inscribed in a cube. Find the volume of the cube. Problem 4 – 11:12 70 athletes are participating in a gymnastics championship: 25 from the USA, 17 from Mexico, and the rest from Canada. The order in which the gymnasts perform is determined by lot. Find the probability that the athlete performing first is from Canada. Problem 5 – 13:55 There are 11 blue, 6 red, and 8 green marker pens in a box. Two marker pens are selected randomly. Find the probability that one blue and one red marker will be selected. Problem 6 – 20:23 Find the root of the equation ∛(x+3)=5. Problem 7 – 22:49 Find the value of the expression (5^(3/5)∙7^(2/3) )^15/35^9 . Problem 8 – 25:20 The figure shows the graph of the function y=f(x), defined on the interval (-4;13). Determine the number of points at which the tangent to the graph of the function y=f(x) is parallel to the line y=14. Problem 9 – 28:41 The dependence of the demand volume q (units per month) for the products of a monopolist on the price p (thousand rubles) is given by the formula q=190-10p. The enterprise's revenue for the month r (in thousands of rubles) is calculated using the formula r(p)=q∙p. Determine the highest price p at which the monthly revenue r(p) will be at least 700 thousand rubles. Give your answer in thousands of rubles. Problem 10 – 32:06 There are two alloys. The first contains 10% nickel, the second – 35% nickel. From these two alloys, a third alloy was obtained weighing 150 kg, containing 25% nickel. By how many kilograms was the mass of the first alloy less than the mass of the second? Problem 11 – 37:20 The figure shows the graph of a function of the form f(x)=log_a⁡x. Find the value of f(8). Problem 12 – 41:04 Find the minimum point of the function y=(x^2-9x+9)∙e^(x+27). Problem 13 – 46:15 a) Solve the equation 49^(cos^2 x)=7^(√2 cos⁡x ). b) Indicate the roots of this equation that belong to the segment [2π;3π]. Analysis of errors 13 – 57:20 Problem 15 – 01:01:22 Solve the inequality (2^(x+1)-17∙2^(2-x))/(2^x-2^(6-x) )≥1. Analysis of errors 15 – 01:11:40 Problem 16 – 01:21:34 In July 2025, it is planned to take out a ten-year loan in the amount of 800 thousand rubles. The terms of its repayment are as follows: - each January the debt will increase by r% compared to the end of the previous year; - from February to June of each year, it is necessary to repay part of the debt in one payment; – in July 2026, 2027, 2028, 2029 and 2030 the debt must be some equal amount less than the debt as of July of the previous year; – at the end of 2030 the debt will be 200 thousand rubles; – in July 2031, 2032, 2033, 2034 and 2035 the debt must be another equal amount less than the debt as of July of the previous year; – the debt must be repaid in full by July 2035. Find r if the total amount of payments after full repayment of the loan is equal to 1,480 thousand rubles. Problem 18 – 01:48:53 Find all values ​​of a for each of which the equation x^2-2x-6a+a^2=6x-2a has exactly two different roots. Problem 19 – 02:22:38 In each cell of a 5×5 square table there is a natural number less than 6. Vasya finds the sum of the numbers in each column and chooses the smallest sum from the resulting sums. Petya finds the sum of the numbers in each row and chooses the smallest sum from the resulting sums. a) Can Petya’s number be twice as big as Vasya’s? b) Can Petya’s number be five times as big as Vasya’s? c) What is the greatest number of times Petya’s number can be greater than Vasya’s? Problem 17 – 02:34:50 In a right triangle ABC, points M and N are the midpoints of the hypotenuse AB and the leg BC, respectively. The bisector of angle BAC intersects line MN at point L. a) Prove that triangles AML and BLC are similar. b) Find the ratio of the areas of these triangles if cos⁡〖∠BAC〗=7/25. Problem 14 – 02:53:27 On edge AA_1 of rectangular parallelepiped ABCDA_1 B_1 C_1 D_1 point E is taken such that A_1 E:EA=3:1, on edge BB_1- point F such