Borys Shumyatskiy
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About
I graduated Kharkiv State University (Ukraine) as a mathematician. My thesis was about one specific Banach space. Then I started (1998) to work as a programmer and I do this till now. My areas in programming are databases, user interface and algorithms.
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Recent Activity

Answered a Question in Math
The AI's solution process is mostly correct. It correctly determined the xcoordinates of the vertices and correctly used the condition "The point P also lies on parabola 2, while the point Q also... 
Answered a Question in Math
AI started correctly from the theorem that states `r_1+r_2+r_3+r_4 = a, ` `r_1r_2+r_1r_3+r_1r_4+r_2r_3+r_2r_4+r_3r_4 = b, ` `r_1r_2r_3+r_2r_3r_4+r_3r_4r_1+r_4r_1r_2 = c, ` `r_1r_2r_3r_4 = d.` But... 
Answered a Question in Math
The main problem here that you incorrectly transformed the problem from the picture to the symbols. Not "(log3^(y4))^3" but "(log_3(y)4)^3". The correct equation is `(log_3(y)4)^3 +... 
Answered a Question in Math
The main AI's error is that w^(log3^4)^2 without additional parentheses denotes `w ^ ( ( log ( 3 ^ 4 ) ) ^ 2 ) , ` not `( w ^ ( log ( 3 ^ 4 ) ) ) ^ 2 . ` Because of this, the value is `( w ^ ( log... 
Answered a Question in Math
AI made many mistakes. First, the total number of passengers is 277, not 276. Second, adding consecutive cars 123 with 234 is absurd because this way we count passengers from the cars 2 and 3... 
Answered a Question in Math
The AI's answer is incorrect. The first incorrect statement is "So, angle STR is 96 degrees." No, the angle STR is the same as the angle RTS, which is `42^@. ` `96^@ ` is the angle SRT. It seems... 
Answered a Question in Math
In the whole, the AI said several stupid things, although it seems it knows enough. First, the figure in question is not "two sectors", but "two equal parts each of which is a difference between... 
Answered a Question in Math
The AI's answer is incorrect even for 3 points because it assumes that two points always determine one specific semicircle, but there are infinitely many. Note that the probability that some two... 
Answered a Question in Math
The answer is almost correct. The formula used is the adequate one, provided `1 0 % ` means formal annual interest rate, and the interest is compounded monthly. In this case, the monthly interest... 
Answered a Question in Math
The solution is rather incorrect. It has some correct parts, it correctly determines the position of the triangle centroid. It was a bad idea to call this centroid G because this letter is already... 
Answered a Question in Math
The answer might be considered correct, partial or incorrect depending on the context. The correct side is that the two nontrivial factors are indeed presented. The incorrect side is that this... 
Answered a Question in Math
Probably you meant `[BPC] / [ABC] * [BMC] / [ABC] .` The AI's response is incorrect. The first statement, `[BPC] = (1/2) * r * BP * PC ` is incorrect because its dimensionality is 3 while it must... 
Answered a Question in Math
First, note that by the definition of logarithm must be `x gt 0 , ` `x != 1 , ` `x != 1 / 16 ` and `x != 1 / 256 .` Then use the formula `log_b a = 1 / ( log_a b ) ` to rewrite the equation the... 
Answered a Question in Math
Note that also `GM = HM = x  ( 3  r ) = x + r  3 .` Recall that the area of a triangle is `1 / 2 r ` multiplied by its perimeter. Here `S ( ABC ) = 1 / 2 * 3 * 4 = S ( ACM ) + S ( ABM ) = 1 / 2... 
Answered a Question in Math
It is clear that `/_ BAC = 20^@ ` and `/_ BPC = 20 + k . ` Use the Sine Law for the triangles `ABC ` and `APC ` to get `( sin 20 ) / ( BC ) = ( sin 80 ) / ( AC ) , ` `( sin k ) / ( AP ) = ( sin ( k... 
Answered a Question in Math
It is simple to compute the area of any "curved corner" (one of them is marked in green on the attached picture). It is `A_1 = 1 / 4 ( 10^2  pi * 5^2 ) .` Also, the area of the triangle ACD is... 
Answered a Question in Math
The equation to solve was `x^2 + 15^2  2 * x * 15 * cos CAP = (15/13)^2 ( x^2 + 13^2  2 * x * 13 * cos BAP ) .` A free term is a summand without `x . ` At the left there is a term `15^2 ` while... 
Answered a Question in Math
There are some results in which I am sure. First, there is only one such (acute) angle. Second, `a_0 = 2 ` and `a_4 = 9 ` and `a_(4n+8) = 9 a_(4n+4)  16 a_(4n) , ` `n gt= 0 , ` so all `a_(4n) `... 
Answered a Question in Math
It is easy to find `cos^2 A: ` `sinA = 2 cos^2 A , ` `1  cos^2 A = 4 (cos^2 A)^2, ` so `cos^2 A = ( sqrt ( 17 )  1 ) / 8 . ` Also, it shows that `1 / ( cos^2 A ) = 4 cos^2 A + 1 = ( sqrt (17) + 1... 
Answered a Question in Math
If we denote the height of XABY as `h, ` the area will be `A = h * (AB + XY) / 2 . ` It is clear that `AB = XY/2 = 12, ` so `A = 18h ` and it is sufficient to find `h.` Denote by M the point of... 
Answered a Question in Math
Denote the initial set as A and the set of collections in question as S. For each `s ` in S consider `Q ( s ) = min {a: a in s} . ` Here `a ` is a subset of A and `a ` is the number of elements... 
Answered a Question in Math
The picture is attached. Indeed, it may be some misunderstanding in conditions. Yes, if we finish the computations, we obtain `m^2 = 13^2 + 7^2  2 * 13 * 7 * ( 13 ^ 2 + 14 ^ 2  15 ^ 2 ) / ( 2 *... 
Answered a Question in Math
The angles ABP and ACP are based on the same chord AP, so their sines are equal. Because of this, the Sine Law gives `13 / ( sin ( APB ) ) = (AP) / ( sin ( ABP ) ) = 15 / ( sin ( APC ) ) . ` The... 
Answered a Question in Math
First, note that 1, 2, 3 is NOT an example of such subset. Then, each such subset contains exactly one pair of consecutive integers, i.e. (1, 2), (2, 3), ..., (9, 10). For each such pair, there are... 
Answered a Question in Math
Let `r = a / b , ` where a and b are positive and relatively prime. Then the fraction `55r = ( 55a ) / b ` is either reducible or not. If not, we obtain the equation a + b = 55a + b, which has no... 
Answered a Question in Math
For any polynomial p, p(x)  p(2) is divisible by x  2. In this specific case, `p ( x )  p ( 2 ) = x^3 + ax^2 + bx + c  8 4a  2b  c = ( x^3  8 ) + a( x^2  4 ) + b ( x  2 ) =` `( x  2 ) (... 
Answered a Question in Math
It is clear that `13! = 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 = 2^10 * 3^5 * 5^2 * 7 * 11 * 13 .` For `( 13! ) / m ` to be a perfect square, `m ` must be divisible by all prime factors... 
Answered a Question in Math
Let's find `tan^2 alpha . ` In the previous question we found that `sin^2 alpha = ( 3 * 55^2 ) / ( 4 * 2299 ) . ` Actually, it is possible to reduce this by 11 twice: `2299 = 11 * 209 = 11^2 * 19 ,... 
Answered a Question in Math
It is clear that angles FPE, FPD and DPE are all equal to 120 degrees. Denote PD = d, PE = e, PF = f. Then by the Cosine law `ED^2 = e^2+d^2+de = 48^2+30^248*30 = 42^2 ,` `EF^2 = e^2+f^2+ef =... 
Answered a Question in Math
Note that `f ( 3 ) / f ( 2 ) = 8 / 3 lt 3 . ` The idea is to prove that `g ( n ) = ( f ( n + 1 ) ) / ( f ( n ) ) gt= 3 ` for `n gt= N , ` where `N ` should be small enough. Then we can test all `n... 
Answered a Question in Math
I suppose it is "incircle" because a nonsquare rhombus has no encircle. Please look on the attached picture. Denote a half of the angle A as `alpha .` It is clear that `r = ( 9 + 16 ) / 2 = 25 / 2... 
Answered a Question in Math
First, note that if M is the least common multiple of 2, 3, 4, 5 and 6, and k is an extradistinct number, then all numbers of the form k + nM are also extradistinct. Because of this, it is... 
Answered a Question in Math
Suppose that P is a point on the circumscribing circle, and that it is between points B and C. Denote the square side as x, then we need `x^2 . ` Denote PA as a, PB as b, PC as c and PD as d. The... 
Answered a Question in Math
I think you are speaking about the function `( sin^2 x ) / ( sin ( 3 / 2 x ) * sin x ) * ( cos x * cos ( x / 2) ) / ( sqrt ( 1 + sin^2 x ) + 1 ) ` from the previous question. First, we can reduce... 
Answered a Question in Math
It is obvious that `EF = (BE) / ( tan x ) . ` It is also obvious that `BC = sin x , ` so `BD = sqrt ( 1 + sin^2 x )  1 .` The angle ADC is equal to `pi / 2  x / 2 , ` thus the angle EDB is equal... 
Answered a Question in Math
Let's factor both sides: `3^x ( 3^x  1 ) = y ( y + 2 ) ( y^2 + 1 ) .` For negative x's the left side is not integer while the right side is always integer. For `x = 0 ` there are two solutions, `y... 
Answered a Question in Math
It is easy to rewrite the given condition as `x^2 ( y^2  1 )  14 x y + 49  y^2 = 0 , ` which is a quadratic equation for `x . ` The solution is ` x = ( 7 y + sqrt ( 49 y^2  ( y^2  1 ) ( 49 ... 
Answered a Question in Math
Let's express `y ` as a function of `x : ` `x + y = x y  7 ` is equivalent to `y ( x  1 ) = x + 7 ` and to `y = ( x + 7 ) / ( x  1 ) .` This function is monotone decreasing for `x gt= 2 `... 
Answered a Question in Math
The triangles APF and BPF have a common height (the one that starts at the point P). Because of this, the ratio between areas is the same as the ratio between bases AF and BF, so `(A ( BPF )) / (A... 
Answered a Question in Math
Let's use the following formula: area of the triangle = 1/2 * side * side * sinus of angle between them. By this formula, `A ( A X Z ) = 1 / 2 * A X * A Z * sin A = 1 / 2 * 1 / 3 * 2 / 3 * A B * A... 
Answered a Question in Math
Let's raise the second equation to the power of `z ` and the third equation to the power of `x + y ` and obtain: `x^( z ( x + y ) ) = y^( 9z^2 ) ,` `x^( z ( x + y ) ) = y^( ( x + y ) ^2 ) .` From... 
Answered a Question in Math
There are several divisors of 24, but only 1, 2, 3, 4, and 6 are suitable because of the first equation. Moreover, from the third equation, we see that x, y, and z cannot be 6. Also, from the third... 
Answered a Question in Math
Let's factor out the left side using `a^3 + b^3 = ( a + b ) ( a^2 + b^2  ab ) :` `sin^6 x + cos^6 x = ( sin^2 x + cos^2 x ) ( sin^4 x + cos^4 x  sin^2 x cos^2 x ) = ` `sin^4 x + cos^4 x  sin^2 x... 
Answered a Question in Math
Note that all conditions on a, b, and c are invariant under their permutations but the value in question is not. It changes sign when, say, a and b are interchanged. In other words, only the... 
Answered a Question in Math
Denote the angle `B A C ` as `alpha .` Then the angles `A B C ` and `A C B ` are equal to `pi / 2  alpha / 2 . ` Because of this, `P X = P B * sin (pi / 2  alpha / 2 ) = P B * cos (alpha / 2 ) `,... 
Answered a Question in Math
Let's express `x ^ 2 =  m  x ` from the second equation and substitute it to the first one: ` m  x + m x =  1 , ` or `x ( m  1 ) = m  1 .` There are two different cases: one for `m = 1 ` and... 
Answered a Question in Math
Denote AB = x, AD = y. Then by the Pythagorean Theorem, ` AC^2 = x^2 + 4 = y^2 + 1 , ` and by the Cosine Theorem, ` BD^2 = x^2 + y^2 xy = 4 + 1  2 .` Express `y^2 = x^2 + 3 ` from the first... 
Answered a Question in Math
It is clear that the value is somewhat near `1 0 0 ^ 2 , ` or rather `1 0 1 . 5 ^ 2 . ` Perform the following transformations of the expression under the root: `1 0 0 * 1 0 1 * 1 0 2 * 1 0 3 + 1... 
Answered a Question in Math
Denote `/_ AOB = alpha, ` then `/_ BOC = alpha ` and `/_ COD = 180^@  alpha . ` Denote also `AO = r .` From the isosceles triangle AOB `sin ( alpha / 2 ) = ( 4 sqrt ( 3 ) ) / r , ` from the right... 
Answered a Question in Math
Because ABC is equilateral, AM is perpendicular to BC. This means that `/_ C = 90^@  /_ MAC ` and `/_ CMD = 90^@  /_ C = /_ MAC , ` so `cos ( /_ MAC ) = cos ( /_ MAD ) = (AD) / (AM) , ` `cos ( /_...
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