Circle Touches x-axis at (a,0), y-axis Intercept b — Find (2a, b²)
A circle x²+y²−αx+βy+γ=0 touches the x-axis at (a,0) and cuts a y-axis intercept of length b, lying below the x-axis. Find the ordered pair (2a, b²).
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AP: First Term 3, Sum of First 4 Terms = 1/5 of Next 4 — Sum of 20 Terms
An AP has first term 3, and the sum of its first 4 terms equals 1/5 of the sum of the next 4 terms. Find the sum of the first 20 terms.
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2×2 Matrix With Entries 0 or 1 — Variance of the Determinant
A 2×2 matrix has entries from {0,1}. Let X be its determinant. Find the variance of X by listing all 16 matrices and their determinant probabilities.
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16-Square Board: Probability Two Random Squares Share No Side
A 4×4 board has 16 squares. Two are chosen at random. Find the probability they share no side, by counting adjacent pairs directly.
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sinx+sin²x=1: Find (cos¹²x+tan¹²x)+3(cos¹⁰x+tan¹⁰x+cos⁸x+tan⁸x)+(cos⁶x+tan⁶x)
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.
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sec²(tan⁻¹α)+cosec²(cot⁻¹β)=36, α+β=8, α<β — Find α²+β
Given sec²(tan⁻¹α)+cosec²(cot⁻¹β)=36 and α+β=8 with α<β, find α²+β by converting to a system of equations in α+β and αβ.
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Product of Rational Roots of (x²−9x+11)²−(x−4)(x−5)=3
Find the product of all rational roots of (x²−9x+11)²−(x−4)(x−5)=3 by substituting y=x²−9x+11 to reduce it to a simple quadratic in y.
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Limit of Σ(k³+6k²+11k+5)/(k+3)! From k=1 to n as n→∞
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.
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Ellipse E and Hyperbola H Share Focal Distance 2√3, a−A=2 — Sum of Latus Rectums
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.
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αx+βy=109 Is Chord of x²/9+y²/4=1 With Midpoint (5/2,1/2), Find α+β
Given αx+βy=109 is the equation of a chord of x²/9+y²/4=1 with midpoint (5/2,1/2), find α+β using the T=S₁ chord-of-midpoint formula.