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. This version is made up of problems that have already appeared 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 version: https://vk.com/wall-40691695_100710 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 Instagram: / shkola_pifagora 🔥 TIMECODES: Start – 00:00 Problem 1 – 01:21 Angle ACO is 28°. Its side CA touches the circle centered at point O. Side CO intersects the circle at points B and D (see fig.). Find the measure of the arc AD of the circle enclosed by this angle. Give your answer in degrees. Problem 2 – 04:26 Vectors a ⃗ (41;0) and b ⃗ (1;-1) are given. Find the length of the vector a ⃗-20b ⃗. Problem 3 – 07:54 The base of a right prism is a right triangle with legs 10 and 9. The lateral edges of the prism are equal to 2/π. Find the volume of the cylinder described about this prism. Problem 4 – 11:21 A taxi company has 60 cars; 27 of them are black with yellow lettering on the sides, the rest are yellow with black lettering. Find the probability that a yellow car with black lettering will arrive in response to a random call. Problem 5 – 13:37 A die was thrown twice. It is known that six points never came up. Find the probability of the event "the sum of the points is 9" under these conditions. Problem 6 – 19:36 Find the root of the equation (6x-13)^2=(6x-11)^2. Problem 7 – 22:47 Find the value of the expression (2^3.2∙6^6.2)/12^5.2 . Problem 8 – 25:30 The figure shows the graph y=F(x) of one of the antiderivatives of a function f(x) and eight points on the abscissa axis are marked: x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8. At how many of these points is the function f(x) negative? Problem 9 – 28:49 A load with a resistance of R Ohms is connected to a source with an EMF of ε=115 V and an internal resistance of r=0.6 Ohm. The voltage across this load, expressed in volts, is given by the formula U=εR/(R+r). At what minimum value of the load resistance will the voltage across it be at least 100 V? Express your answer in ohms. Problem 10 – 33:02 There are two vessels. The first contains 60 kg and the second 20 kg of acid solutions of different concentrations. If these solutions are mixed, the result will be a solution containing 30% acid. If equal masses of these solutions are mixed, the result will be a solution containing 45% acid. What percentage of acid does the first vessel contain? Problem 11 – 40:42 The figure shows the graph of a function of the form f(x)=ax^2+bx+c. Find the value of f(-3). Problem 12 – 46:57 Find the maximum point of the function y=-x/(x^2+225). Problem 13 – 52:24 a) Solve the equation √2 sin⁡(x+π/4)+2sin^2 x=sin⁡x+2. b) Indicate the roots of this equation that belong to the segment [2π;7π/2]. Analysis of errors 13 – 01:03:23 Problem 15 – 01:05:44 Solve the inequality (log_0,2^2 (x+2)-log_5⁡(x^2+4x+4)+1)∙log_5⁡(x+1)≤0. Analysis of errors 15 – 01:21:05 Problem 16 – 01:35:31 In July 2025, it is planned to take out a bank loan for 8 years. The terms of its repayment are as follows: - in January 2026, 2027, 2028 and 2029, the debt increases by 15% compared to the end of the previous year; - in January 2030, 2031, 2032 and 2033, the debt increases by 11% compared to the end of the previous year; - part of the debt must be repaid from February to June of each year; - in July of each year, the debt must be the same amount less than the debt as of July of the previous year; - the loan must be fully repaid by July 2033. What amount is planned to be taken out in loan if the total amount of payments after its full repayment will be 650 thousand rubles? Error Analysis 16 – 01:51:25 Problem 18 – 01:55:50 Find all values ​​of a for each of which the equation (4 cos⁡x-3-a)∙cos⁡x-2.5 cos⁡2x+1.5=0 has at least one root. Problem 19 – 02:12:42 There are three boxes: the first box contains 64 stones, the second contains 77, and the third is empty. In one move, you are allowed to take a stone from two boxes and put it in the remaining one. a) Could the first box contain 64 stones, the second - 59, and the third - 18? b) Could the third box contain 141 stones? c) What is the greatest number of stones that could be in the third box? Problem 17 – 02:31:34 In trapezoid ABCD, base AD is twice as large as base BC. Inside the trapezoid, a point M is taken so that angles ABM and DCM are right angles. a) Prove that AM=DM. b) Find the angle BAD if angle ADC is 70° and the distance from point M to line AD is equal to side BC. Problem 14 – 02:48:43 An isosceles triangle ABC with base AC lies at the base of a right triangular prism ABCA_1 B_1 C_1. Point K is the midpoint of edge A_1 B_1, and point M divides edge AC in the ratio AM:MC=1:3. a) Prove that KM is perpendicular to AC. b) Find the angle between line KM and plane ABB_1 if AB=8, AC=12 and AA_1=5. #OptionsUSEprofilePythagorasSchool