Given sinx+sin²x=1 for x in (0,π/2), evaluate a large trigonometric expression by first proving tanx=cosx, then spotting a binomial cube pattern.
Evaluate the infinite series limit of (k³+6k²+11k+5)/(k+3)! by rewriting the numerator in terms of j=k+3 to reveal a telescoping factorial pattern.
Ellipse E and hyperbola H have the same distance between their foci (2√3), with a−A=2 and eccentricity ratio 1/3. Find the sum of their latus rectum lengths.
A and B alternately throw a pair of dice; A wins by throwing sum 5 before B throws sum 8. Full solution using independent events and an infinite GP.
The common tangents to 4(x²+y²)=9 and y²=4x meet at Q. An ellipse has semi-minor axis OQ and semi-major axis 6. Find l/e² for this ellipse.
Circle C passes through A(2,−1) and B(3,4), AB not a diameter, and its centre lies on (x−5)²+(y−1)²=13/2. Find r² using the perpendicular-bisector property.
A and B are intersection points of a circle and hyperbola; P moves on 2x−3y+4=0. Find the line on which the centroid of triangle PAB always lies.
For some n≠10, the 5th, 6th and 7th coefficients of (1+x)^(n+4) are in AP. Find n, then the largest coefficient of the expansion.
For an AP a1,…,a2024 with a1+(a5+a10+…+a2020)+a2024=2233, find a1+a2+…+a2024 by summing the selected terms as their own smaller AP.
Let A: |(z+1)/(z−1)|<1 and B: arg((z−1)/(z+1))=2π/3 in the complex plane. Find A∩B by converting both conditions into geometric loci.