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Home » Want a Birthday Cake That Looks Like a Stitch? Here’s How You Can Get It! - Crankk.io

Want a Birthday Cake That Looks Like a Stitch? Here’s How You Can Get It! - Crankk.io

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 This is known as the **quadratic functional equation**. The general solution over $ \mathbb{R} $, assuming regularity (e.g., continuity or boundedness), is $ f(x) = cx^2 $. We verify this form.

📌 Let $ f(x) = cx^2 $. Then:

📌 f(x+y) + f(x-y) = c(x+y)^2 + c(x-y)^2 = c(x^2 + 2xy + y^2 + x^2 - 2xy + y^2) = c(2x^2 + 2y^2) = 2cx^2 + 2cy^2 = 2f(x) + 2f(y)

📌 So it satisfies the equation.

📌 Given $ f(1) = c(1)^2 = 3 \Rightarrow c = 3 $, so $ f(x) = 3x^2 $.

📌 Then $ f(5) = 3 \cdot 25 = \boxed{75} $

📌 Question: A cartographer uses a polynomial $ p(x) $ of degree 3 such that $ p(0) = 2 $, $ p(1) = 5 $, $ p(2) = 12 $, and $ p(3) = 31 $. Find the remainder when $ p(x) $ is divided by $ x^4 - 1 $.

📌 Solution: Since $ p(x) $ is a cubic polynomial, when dividing by $ x^4 - 1 $, the remainder $ r(x) $ must be a polynomial of degree less than 4. However, because $ x^4 - 1 $ has roots $ \pm 1, \pm i $, and $ p(x) $ is real, the remainder can be expressed as any cubic or lower, but here we are to find the **actual remainder**, which is unique and of degree $ < 4 $. But since $ p(x) $ is already degree 3, and division by degree 4 polynomial yields a unique remainder of degree $ < 4 $, and since $ p(x) $ agrees with any degree ≤3 polynomial at these 4 points, we conclude:

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\( S_5 = 300 \times \frac{1.18^5 - 1}{0.18} \)
Calculate \( 1.18^5 \approx 2.2877 \)
\( S_5 = 300 \times \frac{2.2877 - 1}{0.18} = 300 \times \frac{1.2877}...
Total distance ≈ 2146 meters
An AI seismic network monitors 8 fault zones. Each zone has a 12% prob...
Use binomial: \( n = 8 \), \( p = 0.12 \), find \( P(X \geq 2) = 1 - P...
\( P(X=0) = (1 - 0.12)^8 = 0.88^8 \approx 0.3596 \)
\( P(X=1) = \binom{8}{1} (0.12)^1 (0.88)^7 = 8 \times 0.12 \times 0.ли...
More accurately:
\( 0.88^7 \approx 0.3900 \), so \( P(X=1) = 8 \times 0.12 \times 0.390...

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