falzoon

On entering Table[log(n), {n, 3, 5}], WolframAlpha gives

only the exact values of log(3), log(4), log(5).

How do I get it to give the approximate values instead?

... and find n by simpler means than trial and error on testing with n = 1, 2, 3, etc.?

Draw a triangle inside a square such that the vertices of the triangle touch three

different sides (not the corners) of the square. If all line segments are different

integers, then what is the value of the side of the square if the triangle is of

minimum area?

Is this related to my previous problem? -

http://answers.yahoo.com/question/index?qid=201402...

Draw a convex quadrilateral inside a square such that each vertex of the quadrilateral

touches a different side of the square. If all line segments are different integers, then

what is the value of the side of the square if the quadrilateral is of minimum area?

Please see diagram at: http://s205.photobucket.com/user/falzoon/media/Lin...

This problem that I haven't solved yet, and not likely to, has plagued me for decades.

Starting at the origin, straight lines, a_1, a_2, a_3, ..., a_n, of equal length (1 unit),

are placed end to end, so that a_1 makes a 1º angle with the X-axis, a_2 makes a

2º angle with a_1 produced, a_3 makes a 3º angle with a_2 produced, etc.

Angles are always made to the same side so that the lines turn in the same

'circular' direction.

Questions:

1) Is there a line, a_n, that eventually meets a_1 at any point on it? If so, what is n?

2) Does the meeting with a_1 happen more than once, or infinitely often, or does the

figure continually spiral away from a_1?

Hoping someone has some software to attack this problem and maybe provide a

diagram of proceedings.

Addenda for research:

(i) What if a_1 = 1 unit, a_2 = 2 units, a_3 = 3 units, etc., with the above angles?

(ii) What if, given the same lengths in (i), but all the angles are 1º?

Two circles with a common centre have radii, r = 1 and R = 2. The annulus contains

an ellipse of maximum area, with a = semi-major axis and b = semi-minor axis.

Can you find the value of (a^2 + b) ?

Q.1: What is the smallest perimeter of an integer-sided scalene triangle,

such that the quotient (circumradius / inradius) is an integer?

Q.2: Same as question 1, but the quotient is a different integer.

Q.3: (For bonus points) Same again, but quotient is yet another different integer.

Q.4: (More bonus points) Can you explain what is going on?

This is a spin-off from Rita the dog's recent question. See diagram at -

http://au.answers.yahoo.com/question/index;_ylt=Ah...

Place the yellow triangle, as it appears in the diagram, along the X-axis with the

left-hand vertex on the origin, (0, 0). Then position the triangles in the order, blue,

bright green, red and finally, dull green. There will be a gap between the dull green

and yellow triangles. Let point A be the 90º vertex of the dull green triangle.

Question: what is the exact distance between A and the origin?

When you have the answer, consider it in relation to the sides of the triangles, as if

they were measured in millimetres, i.e., imagine the triangles to be approximately

106 metres x 101 metres x 33 metres. Do you think the gap would be visible at the

scale of the diagram, and thus, can we regard Maple's effort as adequate?

I'll be surprised if this question hasn't been asked before, but I couldn't find one.

On one side of a tall rectangular prism, made from a stack of 1cm cubes, a straight

line (call it L), is drawn from the lower left corner of the bottom cube to the upper right

corner of the top cube. Somewhere along this prism side, L crosses the horizontal

junction line between two cubes, such that a/b = π, where a = length of junction line

to the left of L, while b = length of junction line to the right of L.

Questions: If the bottom cube is No.1 and π is measured to within 10^(-13), then -

(a) What is the least height of the prism?

(b) Between which numbered cubes lies the particular junction?

Pluto Planar, a resident of Flatland, slid a square, ABCD (side = 2 units),

down the right-hand arm of the parabolic well, y = x^2. It came to rest

with B and C tangent to the right arm, and D tangent to the left arm.

Question: Why did his calculation of the area below line CD, that

is bounded by the parabola, cause him to feel mildly amazed?

Diagram: http://s205.beta.photobucket.com/user/falzoon/medi...

A semicircle, BCD, sits atop an isosceles triangle, ABD, with EA = 125 and

ED = 50 units, looking like a flat, tasteless version of one of my favourite foods.

Question:

Is ABD the largest isosceles triangle that can fit inside the whole figure, ABCD?

If not, then for proof of the largest one, I'll need at least one parameter, e.g.,

area, side or angle, as a decimal value, or preferably, an exact expression.

Are there points A on y = x^3 and points B on y = x^3 + |x| + 1,

such that the distance between A and B does not exceed 0.01?

Diagram: http://s205.beta.photobucket.com/user/falzoon/medi...

Divide a unit square (ADEH) into three equal rectangles, then fold point A onto

segment FG, such that point B lies on segment CF. Find ratio a/b (= EA / AH).