Muscovites are expressing concerns about the difficulty level of advanced mathematics problems featured on the Unified State Exam (EGE). The exam, a standardized test for graduates, has sparked debate regarding the complexity of its content.

The advanced-level math section of the EGE has drawn criticism from students and the public, with many claiming the problems are excessively challenging. The debate centers around whether the exam’s difficulty reflects inadequate student preparation or inherent flaws in the test itself.

The task involves finding all values of the parameter “a” for which the equation has exactly two distinct roots.

To solve, note that x cannot be zero. Substitute x – 9/x = t, resulting in the equation at² – 2t – 49a + 18 = 0. Analyzing the substitution, multiplying x – 9/x = t by x yields x² – tx – 9 = 0. The discriminant D = t² + 36 > 0, indicating two roots for any t. Therefore, for the original equation to have exactly two roots, the new equation must have only one root.

When a = 0, the equation simplifies to -2t + 18 = 0, giving t = 9, meaning a = 0 is a valid solution. When a is not zero, the equation is quadratic, requiring the discriminant D = 0. Therefore: 4 – 4a(18 – 49a) = 0, leading to 1 – a(18 – 49a) = 0 or 49a² – 18a + 1 = 0.

The original equation has two solutions for the following values of a: .

Various opinions are circulating regarding the difficulty of the equation, ranging from considering it simple to expressing disbelief that such problems are given to schoolchildren.