|z|≤1: Minimum Value of |z+1/2(3+4i)|
If z is a complex number with |z|≤1, find the minimum value of |z+1/2(3+4i)| by interpreting it as the distance from a fixed point to the unit disk.
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Triangle ABC: y=x Bisects ∠B, AC is 2x−y=2, 2AB=BC, A=(4,6) — Find α+2β
In triangle ABC, y=x bisects angle B, AC is 2x−y=2, and 2AB=BC. Given A=(4,6) and B=(α,β), find α+2β using the angle bisector’s side-ratio property.
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Sum of Prime Factors of |P⁻¹AP−2I| for Given 3×3 Matrices A, P
Given 3×3 matrices A and P, find the sum of the prime factors of |P⁻¹AP−2I| using the property that similar matrices share the same determinant after shifting by 2I.
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Integers Between 100 and 1000 With Digit Sum Equal to 14
Count the 3-digit integers between 100 and 1000 whose digits sum to 14, using stars-and-bars with inclusion-exclusion for the digit-range constraints.
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f(x)=2x/√(1+9x²) Composed 10 Times, Find √(3α+1)
Consider f:R→R defined by f(x)=2x/√(1+9x²). If (f∘f∘⋯∘f, 10 times)(x)=2^10x/√(1+9αx²), find the value of √(3α+1).
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Roots of x²−√2x−√3=0, Pn=αⁿ−βⁿ, Find (11√3−10√2)P10+(11√2+10)P11−11P12
Let α, β (α>β) be roots of x²−√2x−√3=0 and Pn=αⁿ−βⁿ. Find (11√3−10√2)P10+(11√2+10)P11−11P12 using the Newton recurrence for power sums.