Given A, B, and adj(A⁻¹)+adj(B⁻¹) are non-singular matrices of the same order, find the inverse of A[adj(A⁻¹)+adj(B⁻¹)]⁻¹B using the reversal rule and the M·adj(M)=det(M)I identity.
Prove the vector triangle inequality |a+b|≤|a|+|b| using the dot product expansion of |a+b|² and the Cauchy-Schwarz bound a·b≤|a||b|.
Find the term independent of x in the expansion of [(x+1)/(x^(2/3)−x^(1/3)+1) − (x−1)/(x−x^(1/2))]^10, by simplifying both fractions using sum/difference-of-cubes and -squares factoring first.
Find the sum of the squares of all roots of x²+|2x−3|−4=0, by splitting into two cases at the modulus’s sign-change point and checking each candidate root against its case’s domain.
m arithmetic means are inserted between 1 and 31 so the ratio of the 7th and (m−1)th means is 5:9. Find m by expressing both means via the common difference D=30/(m+1).
Given the line ax+y=c is tangent to both the circle x²+y²=1 and the parabola y²=4√2x, find |c| by combining the circle’s tangent-distance condition with the parabola’s tangent-in-slope-form condition.
Find the maximum area of a rectangle inscribed in the ellipse x²/a²+y²/b²=1, using the parametric point (a cosθ, b sinθ) and maximising over θ.
For an infinite GP a, ar, ar², … with Σar^n=57 and Σa³r^3n=9747, find a+18r by dividing the cubed-series sum by the cube of the original sum.
Find the greatest positive integer k such that 49^k+1 divides 1+49+49²+…+49¹²⁵, by writing the sum as a GP and factoring it as a difference of squares.
A girl throws a die: on 5 or 6 she tosses a coin 3 times, otherwise once. Given she got exactly one head, find the probability she threw 1, 2, 3, or 4, using Bayes’ theorem.