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# JOHN

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Questions62

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• ### ADVICE ON MATHS DRAWING SOFTWARE?

Not having any knowledge of or experience with this type of software, whenever I can’t avoid including a diagram as part of an answer to a question, I hand draw the diagram, scan it in, creating a pdf file in the process, snip the image using the Windows snipping tool and finally upload the picture file image. This is a tedious process and limits my ability to post answers to geometry type questions. I know there are several experts who post regularly on YA Maths who can do beautiful diagramswhen they need to. I wonder whether you can refer me to the software you use, with recommendations as to which might be suitable for a beginner in this area, and whether I may obtain it free on the internet, or if I need to buy it, details of cost and where and how to get it. Many thanks.

• ### ADVICE ON MATHS DRAWING SOFTWARE?

Not having any knowledge of or experience with this type of software, whenever I can’t avoid including a diagram as part of an answer to a question, I hand draw the diagram, scan it in, creating a pdf file in the process, snip the image using the Windows snipping tool and finally upload the picture file image. This is a tedious process and limits my ability to post answers to geometry type questions. I know there are several experts who post regularly on YA Maths who can do beautiful diagramswhen they need to. I wonder whether you can refer me to the software you use, with recommendations as to which might be suitable for a beginner in this area, and whether I may obtain it free on the internet, or if I need to buy it, details of cost and where and how to get it. Many thanks.

Software10 months ago
• • ### a, b and c are positive real numbers. Using only AM-GM (no Jensen, no Holder, no Cauchy-Schwarz, etc) prove that a/√(a² + bc) + b/√(?

a, b and c are positive real numbers. Using only AM-GM (no Jensen, no Holder, no Cauchy-Schwarz, etc) prove that

a/√(a² + bc) + b/√(b² + ca) + c/√(c² + ab) ≥ 1.

• ### ABC is a Δ of which ∠B is obtuse. D is the circumcentre, DF⊥ AC, DG ⊥ AB, DH ⊥BC. Prove that DG + DH = R + r + DF, where r is the?

ABC is a Δ of which ∠B is obtuse. D is the circumcentre, DF⊥ AC, DG ⊥ AB, DH ⊥BC. Prove that DG + DH = R + r + DF, where r is the in-radius and R the circum-radius of ABC.

• ### n is a 4-digit integer not divisible by 10, and Φ(n) is the integer whose digits are those of n in reverse order (eg, Φ(1234) = 4321).?

n is a 4-digit integer not divisible by 10, and Φ(n) is the integer whose digits are those of n in reverse order (eg, Φ(1234) = 4321). Find n such that Φ(n) = 4n + 3.

• ### x, y, z are real numbers for which x + y + z = 1, xa² yb² + zc² = w², a, b, c and w being distinct real numbers.?

w, a, b, c are distinct real numbers and x, y, z are real numbers for which

x + y + z = 1

xa² yb² + zc² = w²

xa³ + yb³ + zc³ = w³

xa⁴ + yb⁴ + zc⁴ =w⁴.

Prove that w = abc/(ab + bc + ca).

• ### A uniform cable AB of line density σ hangs freely under gravity with its two ends A, B fixed at points on a horizontal line. The?

A uniform cable AB of line density σ hangs freely under gravity with its two ends A, B fixed at points on a horizontal line. The tension at the lowest point is T₀. Prove that in equilibrium the cable assumes the shape y = ccosh(x/c), where c = T₀/ σ (the x-axis being below AB and || to it and the y-axis passes through the lowest point.

[This problem can be solved by minimising the potential energy using the methods of the calculus of variations. This is NOT the preferred solution, which is one using entirely elementary methods].