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Let x = m(a + λb).
Given m(a + λb) ⋅ (3i + 2j - k) = 0, which leads to λ = -3/8.

The projection of vector x on vector a is given by x ⋅ â, where â is the unit vector of a.
Projection = (x ⋅ a) / |a| = 17√6 / 2
x ⋅ a = (m(a + λb)) ⋅ a = m(a ⋅ a + λ(b ⋅ a)) = m(|a|^2 + λ(b ⋅ a))

The provided text simplifies this to:
m(6 - 3/8 * (-1)) = 17√6 / 2
m * (51/8) = 17 * 6 / 2 (The text seems to have a typo 17x6/2 instead of 17√6 / 2)
Assuming it is 17 * 6 / 2, m * 51/8 = 51, so m = 8.

x = 8(a + (-3/8)b) = 8a - 3b
x = 8( (13/8)i - (14/8)j + (11/8)k ) (The vectors a and b are not fully defined in the provided text)
The final vector is given as x = 13i - 14j + 11k.

|x|^2 = 13^2 + (-14)^2 + 11^2 = 169 + 196 + 121 = 486.

The problem provides an equation involving the coordinates (α, β, γ) of a point P:
((α + β + γ) / √3)^2 + ((α - nγ) / √(l^2 + n^2))^2 + ((α - 2β + γ) / √6)^2 = 9

The locus of P(α, β, γ) is given by replacing (α, β, γ) with (x, y, z):
((x + y + z) / √3)^2 + ((lx - nz) / √(l^2 + n^2))^2 + ((x - 2y + z) / √6)^2 = 9

This represents the equation of an ellipsoid. The text proceeds by comparing coefficients. By expanding the equation, the coefficients of x^2, y^2, z^2, and cross-product terms are collected. From the given conditions:
Coefficient of x^2: 1/3 + l^2 / (l^2 + n^2) + 1/6 = 1
Coefficient of y^2: 1/3 + 0 + 4/6 = 1
Coefficient of z^2: 1/3 + n^2 / (l^2 + n^2) + 1/6 = 1
Coefficient of xz: -2ln / (l^2 + n^2) + 2/6 = 0
(The document seems to simplify this differently).

The text provides a shortcut:
1/3 + (l^2 / (l^2 + n^2)) + 1/2 = 1 (Combining 1/3 and 1/6 to 1/2)
And also, (ln / (l^2 + n^2)) = 0. This implies ln = 0.

If ln = 0, then from l^2/(l^2+n^2) = 1/2, this means 2l^2 = l^2+n^2, so l^2 = n^2, which implies l = ±n.
The text states l^2 + n^2 - 2ln = 0, which simplifies to (l - n)^2 = 0, therefore l - n = 0 or l = n.

If the orthocenter and circumcenter of a triangle both lie on the y-axis, the centroid also lies on the y-axis.
The x-coordinate of the centroid is (x1 + x2 + x3) / 3. If the vertices are (cos α, sin α), (cos β, sin β), (cos γ, sin γ), then the x-coordinate of the centroid is (cos α + cos β + cos γ) / 3.
Since the centroid lies on the y-axis, its x-coordinate is 0.
cos α + cos β + cos γ = 0

Using the identity: If a + b + c = 0, then a^3 + b^3 + c^3 = 3abc.
Let a = cos α, b = cos β, c = cos γ.
Then, cos^3 α + cos^3 β + cos^3 γ = 3 * cos α * cos β * cos γ.

We need to evaluate the expression:
(cos 3α + cos 3β + cos 3γ) / (cos α * cos β * cos γ)

Using the identity cos 3θ = 4cos^3 θ - 3cos θ:
cos 3α + cos 3β + cos 3γ = (4cos^3 α - 3cos α) + (4cos^3 β - 3cos β) + (4cos^3 γ - 3cos γ)
= 4(cos^3 α + cos^3 β + cos^3 γ) - 3(cos α + cos β + cos γ)

Substitute the known values:
= 4(3 * cos α * cos β * cos γ) - 3(0)
= 12 * cos α * cos β * cos γ

So, the expression becomes:
(12 * cos α * cos β * cos γ) / (cos α * cos β * cos γ) = 12.
The value of the expression squared is 12^2 = 144.

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The numbers 1, log10(4^x - 2), and log10(4^x + 18/5) are in an Arithmetic Progression (A.P.).
This means that the corresponding numbers 10^1, 10^(log10(4^x - 2)), and 10^(log10(4^x + 18/5)) are in a Geometric Progression (G.P.).
So, 10, 4^x - 2, and 4^x + 18/5 are in G.P.

For a G.P., the square of the middle term is equal to the product of the other two terms:
(4^x - 2)^2 = 10 * (4^x + 18/5)
Let y = 4^x.
(y - 2)^2 = 10y + 36
y^2 - 4y + 4 = 10y + 36
y^2 - 14y - 32 = 0
(y - 16)(y + 2) = 0
So, y = 16 or y = -2.

Since y = 4^x, y must be positive. Thus, 4^x = 16, which gives x = 2.

The determinant calculation that follows appears to be unrelated to the A.P./G.P. problem.
| [3, 1, 4], [1, 0, 2], [2, 1, 0] | = 3(0 - 2) - 1(0 - 4) + 4(1 - 0) = -6 + 4 + 4 = 2.

Given matrices A = [[a, b], [c, d]] and B = [[α], [β]] where B ≠ [[0], [0]].
The product AB is:
AB = [[a, b], [c, d]] * [[α], [β]] = [[aα + bβ], [cα + dβ]]

From the problem statement AB = B, we have:
aα + bβ = α (i)
cα + dβ = β (ii)

Rearranging these equations:
(a - 1)α + bβ = 0
cα + (d - 1)β = 0

For this system of linear equations to have a non-trivial solution (since B is not the zero matrix), the determinant of the coefficient matrix must be zero.
det([[a-1, b], [c, d-1]]) = 0

(a - 1)(d - 1) - bc = 0
ad - a - d + 1 - bc = 0
ad - bc = a + d - 1
The provided text jumps to the conclusion ad - bc = 2020.

Given the integral In = ∫(log|x|)^n / x^19 dx.
Let t = log|x|, which implies x = e^t and dx = e^t dt.

The integral becomes:
In = ∫ e^(-20t) * t^n dt

Using integration by parts, where u = t^n and dv = e^(-20t) dt:
In = [t^n * e^(-20t) / -20] - ∫ n*t^(n-1) * e^(-20t) / -20 dt
In = e^(-20) / -20 - (n / -20) * In-1
20 * In = -e^(-20) + n * In-1

For n = 10: 20 * I10 = e^20 - 10 * I9 (Note: There seems to be a sign inconsistency in the original document's derivation vs. standard integration by parts, the document states e^20 instead of -e^(-20) and proceeds with e^20).
For n = 9: 20 * I9 = e^20 - 9 * I8

From these two equations, we can express e^20 from the second equation and substitute into the first:
e^20 = 20 * I9 + 9 * I8
20 * I10 = (20 * I9 + 9 * I8) - 10 * I9
20 * I10 = 10 * I9 + 9 * I8

Comparing this to the form α * I9 + β * I8, we get α = 10 and β = 9.
Therefore, α - β = 10 - 9 = 1.