Difference between revisions of "User talk:Bobthesmartypants/Sandbox"
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</asy> | </asy> | ||
<cmath>\text{Find the probability that }b>a \text{.}</cmath> | <cmath>\text{Find the probability that }b>a \text{.}</cmath> | ||
+ | |||
+ | <asy>unitsize(2inch); | ||
+ | import olympiad; | ||
+ | path c2 = dir(90)-dir(120)..dir(90)-dir(150)..dir(90)-dir(180); | ||
+ | path c1 = dir(90)-dir(0)..dir(90)-dir(30)..dir(90)-dir(60); | ||
+ | path c3 = dir(120)..dir(150)..dir(180); | ||
+ | path c4 = dir(0)..dir(30)..dir(60); | ||
+ | |||
+ | draw(dir(0)..dir(30)..dir(60)..dir(90)..dir(120)..dir(150)..dir(180)--dir(0)); | ||
+ | draw(dir(90)-dir(0)..dir(90)-dir(30)..dir(90)-dir(60)..dir(90)-dir(90)..dir(90)-dir(120)..dir(90)-dir(150)..dir(90)-dir(180)--dir(90)-dir(0)); | ||
+ | draw((-1.2,0.8)--(1.2,0.8)); | ||
+ | label("$l_{-n}$", (1.2,0.8),dir(0)); | ||
+ | draw((-1.2,0.5)--(1.2,0.5)); | ||
+ | label("$l_0$", (1.2,0.5),dir(0)); | ||
+ | draw((-1.2,0.2)--(1.2,0.2)); | ||
+ | label("$l_n$", (1.2,0.2),dir(0)); | ||
+ | label("$\vdots$", (1.2, 0.4), dir(0)); | ||
+ | label("$\vdots$", (0, 0.4)); | ||
+ | label("$\vdots$", (1.2, 0.65), dir(0)); | ||
+ | label("$\vdots$", (0, 0.65)); | ||
+ | |||
+ | label("$A_{-n}$", intersectionpoint(c1, (-1.2, 0.8)--(1.2, 0.8)), dir(135)); | ||
+ | label("$C_{-n}$", intersectionpoint(c3, (-1.2, 0.8)--(1.2, 0.8)), dir(135)); | ||
+ | label("$D_{-n}$", intersectionpoint(c4, (-1.2, 0.8)--(1.2, 0.8)), dir(45)); | ||
+ | label("$B_{-n}$", intersectionpoint(c2, (-1.2, 0.8)--(1.2, 0.8)), dir(45)); | ||
+ | |||
+ | label("$X$", intersectionpoint(c1, (-1.2, 0.5)--(1.2, 0.5)), dir(150)); | ||
+ | label("$Y$", intersectionpoint(c2, (-1.2, 0.5)--(1.2, 0.5)), dir(30)); | ||
+ | |||
+ | label("$C_{n}$", intersectionpoint(c3, (-1.2, 0.2)--(1.2, 0.2)), dir(135)); | ||
+ | label("$A_{n}$", intersectionpoint(c1, (-1.2, 0.2)--(1.2, 0.2)), dir(45)); | ||
+ | label("$B_{n}$", intersectionpoint(c2, (-1.2, 0.2)--(1.2, 0.2)), dir(135)); | ||
+ | label("$D_{n}$", intersectionpoint(c4, (-1.2, 0.2)--(1.2, 0.2)), dir(45)); | ||
+ | |||
+ | dot(intersectionpoint(c1, (-1.2, 0.8)--(1.2, 0.8))); | ||
+ | dot(intersectionpoint(c2, (-1.2, 0.8)--(1.2, 0.8))); | ||
+ | dot(intersectionpoint(c3, (-1.2, 0.8)--(1.2, 0.8))); | ||
+ | dot(intersectionpoint(c4, (-1.2, 0.8)--(1.2, 0.8))); | ||
+ | |||
+ | dot(intersectionpoint(c1, (-1.2, 0.5)--(1.2, 0.5))); | ||
+ | dot(intersectionpoint(c2, (-1.2, 0.5)--(1.2, 0.5))); | ||
+ | dot(intersectionpoint(c3, (-1.2, 0.5)--(1.2, 0.5))); | ||
+ | dot(intersectionpoint(c4, (-1.2, 0.5)--(1.2, 0.5))); | ||
+ | |||
+ | dot(intersectionpoint(c1, (-1.2, 0.2)--(1.2, 0.2))); | ||
+ | dot(intersectionpoint(c2, (-1.2, 0.2)--(1.2, 0.2))); | ||
+ | dot(intersectionpoint(c3, (-1.2, 0.2)--(1.2, 0.2))); | ||
+ | dot(intersectionpoint(c4, (-1.2, 0.2)--(1.2, 0.2))); | ||
+ | </asy> | ||
+ | |||
+ | Two half-circles are drawn as shown above, with a line <math>l_0</math> throught the two intersections points, <math>X,Y</math> of the half-circles. Lines <math>l_k</math> for <math>k=-n\to n</math> parallel to the bases of the half-circles are drawn such that the distances between <math>l_k</math> and <math>l_0</math> and <math>l_{-k}</math> and <math>l_0</math> are always the same for all <math>k=1\to n</math>. | ||
+ | |||
+ | The intersection points of <math>l_k</math> with one of the half-circles are labeled <math>A_k, B_k</math>, and with the other half-circle at <math>C_k,D_k</math>, as shown in the diagram. | ||
+ | |||
+ | Prove that <cmath>\prod_{k=-n}^n |A_kB_k|+|C_kD_k| \ge \prod_{k=-n}^n |A_kD_k|+|B_kC_k|</cmath> | ||
+ | |||
==Picture 2== | ==Picture 2== | ||
<asy> | <asy> | ||
Line 247: | Line 303: | ||
label("$R$",0.4*dir(120),dir(210)); | label("$R$",0.4*dir(120),dir(210)); | ||
dot((0,0));</asy> | dot((0,0));</asy> | ||
+ | |||
+ | ==moar images== | ||
+ | |||
+ | <asy> | ||
+ | import olympiad; | ||
+ | markscalefactor=0.01; | ||
+ | |||
+ | draw((-1,0)--(1,0)); | ||
+ | draw((-1,0)--dir(30)--(1,0)); | ||
+ | dot(incenter(dir(180),dir(30),dir(0))); | ||
+ | draw(rightanglemark(dir(180),dir(30),dir(0))); | ||
+ | |||
+ | draw((-1,0)--dir(80)--(1,0)); | ||
+ | dot(incenter(dir(180),dir(80),dir(0))); | ||
+ | draw(rightanglemark(dir(180),dir(80),dir(0))); | ||
+ | draw((-1,0)--dir(140)--(1,0)); | ||
+ | dot(incenter(dir(180),dir(140),dir(0))); | ||
+ | draw(rightanglemark(dir(180),dir(140),dir(0))); | ||
+ | |||
+ | draw((-1,0)--dir(200)--(1,0)); | ||
+ | dot(incenter(dir(180),dir(200),dir(0))); | ||
+ | draw(rightanglemark(dir(180),dir(200),dir(0))); | ||
+ | |||
+ | draw((-1,0)--dir(250)--(1,0)); | ||
+ | dot(incenter(dir(180),dir(250),dir(0))); | ||
+ | draw(rightanglemark(dir(180),dir(250),dir(0))); | ||
+ | draw((-1,0)--dir(320)--(1,0)); | ||
+ | dot(incenter(dir(180),dir(320),dir(0))); | ||
+ | draw(rightanglemark(dir(180),dir(320),dir(0))); | ||
+ | |||
+ | label("$1$",(0,0),dir(90)); | ||
+ | draw(Circle((0,0),1),linetype("8 8"));</asy> | ||
+ | |||
+ | |||
+ | <asy> | ||
+ | import olympiad; | ||
+ | markscalefactor=0.01; | ||
+ | draw((-1,0)--(1,0)); | ||
+ | |||
+ | draw((-1,0)--dir(80)--(1,0)); | ||
+ | dot(incenter(dir(180),dir(80),dir(0))); | ||
+ | draw((-1,0)--incenter(dir(180),dir(80),dir(0))--(1,0),linetype("8 8")); | ||
+ | draw(rightanglemark(dir(180),dir(80),dir(0))); | ||
+ | |||
+ | label("$A$",(-1,0),dir(180)); | ||
+ | label("$B$",(1,0),dir(0)); | ||
+ | label("$C$",dir(80),dir(90)); | ||
+ | label("$I$",incenter(dir(180),dir(80),dir(0)),dir(90)); | ||
+ | draw(Circle((0,0),1),linetype("8 8"));</asy> | ||
+ | |||
+ | <asy> | ||
+ | import olympiad; | ||
+ | markscalefactor=0.01; | ||
+ | draw((-1,0)--(1,0)); | ||
+ | |||
+ | draw((-1,0)--dir(80)--(1,0)); | ||
+ | dot(incenter(dir(180),dir(80),dir(0))); | ||
+ | draw(rightanglemark(dir(180),dir(80),dir(0))); | ||
+ | draw(dir(180)..incenter(dir(180),dir(80),dir(0))..dir(0)); | ||
+ | draw(Circle((0,-1),sqrt(2)),linetype("8 8")); | ||
+ | |||
+ | |||
+ | draw(Circle((0,0),1),linetype("8 8"));</asy> | ||
+ | |||
+ | <asy> | ||
+ | import olympiad; | ||
+ | markscalefactor=0.01; | ||
+ | |||
+ | fill(dir(0)..incenter(dir(180),dir(260),dir(0))..dir(180)--dir(180)..incenter(dir(180),dir(80),dir(0))..dir(0)--cycle,grey); | ||
+ | draw((-1,0)--(1,0)); | ||
+ | draw(dir(180)..incenter(dir(180),dir(80),dir(0))..dir(0)); | ||
+ | draw(Circle((0,-1),sqrt(2))); | ||
+ | draw(dir(180)..incenter(dir(180),dir(260),dir(0))..dir(0)); | ||
+ | draw(Circle((0,1),sqrt(2))); | ||
+ | draw(dir(0)--dir(90)--dir(180)--dir(-90)--cycle);</asy> | ||
+ | |||
+ | ==yay== | ||
+ | |||
+ | <asy> | ||
+ | import olympiad; | ||
+ | |||
+ | draw(Circle((0,0),1)); | ||
+ | draw((0,0)--dir(75)); | ||
+ | draw((1.5,0)--(-1.5,0),grey); | ||
+ | draw((0,1.5)--(0,-1.5),grey); | ||
+ | markscalefactor=0.01; | ||
+ | draw(anglemark(dir(0),(0,0),dir(75))); | ||
+ | label("$\theta$",0.07*dir(37.5),dir(37.5)); | ||
+ | draw(dir(180)--dir(75)--dir(0)); | ||
+ | label("$P$",dir(75),dir(75)); | ||
+ | label("$A$",dir(0),dir(0)); | ||
+ | label("$B$",dir(180),dir(180));</asy> | ||
+ | |||
+ | ==solution reflection== | ||
+ | |||
+ | <asy>draw(dir(0)--(0,0)--dir(15)--(0,0)--dir(30)); | ||
+ | draw(dir(0)--dir(15)--dir(30),linetype("8 8")); | ||
+ | dot(0.75*dir(7)); | ||
+ | draw(0.75*dir(7.5)--0.574*dir(15)--0.4*dir(0)); | ||
+ | draw(0.574*dir(15)--0.4062*dir(30),linetype("8 8")); | ||
+ | label("$A$",(0,0),dir(180)); | ||
+ | label("$B$",dir(15),dir(15)); | ||
+ | label("$C$",dir(0),dir(0)); | ||
+ | label("$C'$",dir(30),dir(30));</asy> | ||
+ | |||
+ | |||
+ | <asy> | ||
+ | for(int i = 0; i < 60; ++i){ | ||
+ | draw((0,0)--dir(6*i)); | ||
+ | draw(dir(6*i)--dir(6*i+6),linetype("8 8")); | ||
+ | } | ||
+ | draw(1.2*dir(3)--1.2*dir(177)); | ||
+ | label("Diagram not to Scale",dir(-90),dir(-90));</asy> | ||
+ | |||
+ | ==origami== | ||
+ | |||
+ | <asy>draw((1,1)--(-1,1)--(-1,-1)--(1,-1)--cycle); | ||
+ | dot((0,0)); | ||
+ | label("$O$",(0,0),dir(0)); | ||
+ | dot((-0.5,0.75)); | ||
+ | label("$P$",(-0.5,0.75),dir(0));</asy> | ||
+ | |||
+ | <asy>draw((11/16,1)--(1,1)--(1,-1)--(-1,-1)--(-1,-1/8)); | ||
+ | draw((-1,-1/8)--(-1,1)--(11/16,1),linetype("8 8")); | ||
+ | dot((0,0)); | ||
+ | label("$O$",(0,0),dir(0)); | ||
+ | dot((-0.5,0.75)); | ||
+ | label("$P$",(-0.5,0.75),dir(0)); | ||
+ | draw((-1,-1/8)--(11/16,1)--(1/26,-29/52)--cycle); | ||
+ | draw((-0.5,0.7)..(-0.3,0.3)..(-0.05,0.05),Arrow());</asy> | ||
+ | |||
+ | ==combos== | ||
+ | |||
+ | <asy>label("$C$",(0,0)); | ||
+ | label("$C$",(1,-1)); | ||
+ | label("$O$",(1,0)); | ||
+ | label("$O$",(2,-1)); | ||
+ | label("$O$",(0,1)); | ||
+ | label("$M$",(2,0)); | ||
+ | label("$M$",(1,1)); | ||
+ | label("$M$",(0,2)); | ||
+ | label("$M$",(3,-1)); | ||
+ | label("$B$",(3,0)); | ||
+ | label("$B$",(2,1)); | ||
+ | label("$B$",(1,2)); | ||
+ | label("$B$",(0,3)); | ||
+ | label("$B$",(4,-1)); | ||
+ | label("$O$",(4,0)); | ||
+ | label("$O$",(3,1)); | ||
+ | label("$O$",(2,2)); | ||
+ | label("$O$",(1,3)); | ||
+ | label("$O$",(0,4)); | ||
+ | label("$O$",(5,-1)); | ||
+ | label("$*$",(0,-1)); | ||
+ | </asy> | ||
+ | |||
+ | ==circles== | ||
+ | |||
+ | <asy> | ||
+ | draw(Circle((0,0),3.5)); | ||
+ | draw((-3.5,0)--(3.5,0)); | ||
+ | label("7", (0,0), dir(90)); | ||
+ | dot((0,0)); | ||
+ | draw(Circle((-2,1.4),1)); | ||
+ | draw((-2,1.4)--(-1,1.4)); | ||
+ | label("1", (-1.5,1.4),dir(90)); | ||
+ | </asy> | ||
+ | |||
+ | ==more circles== | ||
+ | |||
+ | <asy> | ||
+ | draw(Circle((0,0),20)); | ||
+ | draw(Circle((0,0),14)); | ||
+ | dot((0,0)); | ||
+ | dot(20*dir(60)); | ||
+ | dot(14*dir(180)); | ||
+ | dot((-17,29.445)); | ||
+ | draw(20*dir(60)--14*dir(180)--(-17,29.445)--cycle); | ||
+ | label("O",(0,0),dir(0)); | ||
+ | label("A",20*dir(60),dir(60)); | ||
+ | label("B",(-14,0),dir(180)); | ||
+ | label("P",(-17,29.445),dir(180)); | ||
+ | draw((-17,29.445)--(0,0),red); | ||
+ | </asy> | ||
+ | |||
+ | ==checkerboasrd== | ||
+ | |||
+ | <asy> | ||
+ | for(int i = 0; i < 8; ++i){ | ||
+ | for(int j = 0; j < 8; ++j){ | ||
+ | filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle,gray((i%3+3*(j%3))/8)); | ||
+ | } | ||
+ | } | ||
+ | |||
+ | </asy> | ||
+ | |||
+ | ==Fermat point== | ||
+ | |||
+ | <asy> | ||
+ | import math; | ||
+ | |||
+ | draw((0,0)--(4,0)--(0,3)--cycle); | ||
+ | draw((0,0)--3*dir(30)--4*dir(-60)--cycle); | ||
+ | draw((4,0)--4*dir(60)--(4*dir(60)+3*dir(30))--cycle); | ||
+ | draw((0,3)--3*dir(150)--(3*dir(150)+4*dir(-60))--cycle); | ||
+ | |||
+ | draw((0,0)--(4*dir(60)+3*dir(30)),blue+linetype("8 8")); | ||
+ | draw((0,3)--4*dir(-60),blue+linetype("8 8")); | ||
+ | draw((4,0)--3*dir(150),blue+linetype("8 8")); | ||
+ | |||
+ | draw((0,0)--(3*dir(150)+4*dir(-60)),red+linetype("8 8 0 8")); | ||
+ | draw((0,3)--4*dir(60),red+linetype("8 8 0 8")); | ||
+ | draw((4,0)--3*dir(30),red+linetype("8 8 0 8"));</asy> | ||
+ | |||
+ | ==cenn driagrma== | ||
+ | |||
+ | <asy> | ||
+ | draw(Circle((1,0),2)); | ||
+ | draw(Circle((-1,0),2)); | ||
+ | label("3",(0,0)); | ||
+ | label("2", (2,0)); | ||
+ | label("2", (-2,0)); | ||
+ | label("Spotted",(-2,2),dir(90)); | ||
+ | label("5 Legs",(2,2),dir(90)); | ||
+ | </asy> | ||
+ | |||
+ | ==cyclic square== | ||
+ | |||
+ | <asy> | ||
+ | draw(Circle((0,0),0.5)); | ||
+ | draw(0.5*dir(15)--0.5*dir(105)--0.5*dir(195)--0.5*dir(285)--cycle); | ||
+ | label(scale(6)*"CS",(0,0)); | ||
+ | </asy> | ||
+ | |||
+ | <asy> | ||
+ | import olympiad; | ||
+ | |||
+ | size(10cm); | ||
+ | draw(Circle((-5,0),4)); | ||
+ | fill((0,0)--(-10,-1)--(-10,-5)--(0,-5)--cycle,white); | ||
+ | dot(4*dir(60)-5); | ||
+ | dot(4*dir(30)-5); | ||
+ | dot(4*dir(100)-5); | ||
+ | dot(4*dir(150)-5); | ||
+ | label(scale(5)*"Cyclic Squares",(0,0)); | ||
+ | draw((-0.75,2)--(-0.25,-2)--(9.25,-0.9)--(9,1.1)); | ||
+ | draw(rightanglemark((-0.75,2),(-0.25,-2),(9.25,-0.9))); | ||
+ | draw(rightanglemark((-0.25,-2),(9.25,-0.9),(9,1.1))); | ||
+ | </asy> | ||
+ | |||
+ | |||
+ | ==diagram == | ||
+ | |||
+ | |||
+ | <cmath>\text{Given that }\theta\le 90^{\circ}\text{, prove }a^2+b^2\le D^2\text{, where }D\text{ is the diameter of the circle.}</cmath> | ||
+ | <asy> | ||
+ | draw(Circle((0,0),1)); | ||
+ | draw(dir(0)--dir(40)--dir(170)--dir(260)--dir(0)--dir(170)--dir(260)--dir(40)); | ||
+ | |||
+ | label("$\theta$", extension(dir(0),dir(170),dir(40),dir(260))-0.05*dir(30),-dir(30)); | ||
+ | label("a",(dir(170)+dir(260))/2,dir(215)); | ||
+ | label("b",(dir(0)+dir(40))/2,-dir(20)); | ||
+ | |||
+ | </asy> | ||
+ | |||
+ | ==Cyclic squares DOTS DTOS TDORS== | ||
+ | |||
+ | <asy> | ||
+ | draw(Circle((0,0),90)); | ||
+ | |||
+ | draw(Circle((30,40),10)); | ||
+ | |||
+ | dot((37,38)); | ||
+ | |||
+ | dot((25,39)); | ||
+ | |||
+ | dot((20,30),gray(0.5)); | ||
+ | |||
+ | dot((22,54),gray(0.6)); | ||
+ | dot((36,27),gray(0.5)); | ||
+ | dot((38,50),gray(0.4)); | ||
+ | |||
+ | dot((10,36),gray(0.8)); | ||
+ | |||
+ | dot((50,40),gray(0.75)); | ||
+ | |||
+ | dot((30,20),gray(0.7)); | ||
+ | |||
+ | dot((0,54),gray(0.85)); | ||
+ | dot((4,23),gray(0.85)); | ||
+ | dot((60,25),gray(0.9)); | ||
+ | dot((30,70),gray(0.9)); | ||
+ | </asy> |
Latest revision as of 21:25, 6 November 2014
Contents
- 1 Bobthesmartypants's Sandbox
- 2 Solution 1
- 3 Solution 2
- 4 Picture 1
- 5 Picture 2
- 6 physics problem
- 7 Solution
- 8 inscribed triangle
- 9 moar images
- 10 yay
- 11 solution reflection
- 12 origami
- 13 combos
- 14 circles
- 15 more circles
- 16 checkerboasrd
- 17 Fermat point
- 18 cenn driagrma
- 19 cyclic square
- 20 diagram
- 21 Cyclic squares DOTS DTOS TDORS
Bobthesmartypants's Sandbox
Solution 1
First, continue to hit
at
. Also continue
to hit
at
.
We have that . Because
, we have
.
Similarly, because , we have
.
Therefore, .
We also have that because
is a parallelogram, and
.
Therefore, . This means that
, so
.
Therefore, .
Solution 2
Note that is rational and
is not divisible by
nor
because
.
This means the decimal representation of is a repeating decimal.
Let us set as the block that repeats in the repeating decimal:
.
( written without the overline used to signify one number so won't confuse with notation for repeating decimal)
The fractional representation of this repeating decimal would be .
Taking the reciprocal of both sides you get .
Multiplying both sides by gives
.
Since we divide
on both sides of the equation to get
.
Because is not divisible by
(therefore
) since
and
is prime, it follows that
.
Picture 1
Two half-circles are drawn as shown above, with a line throught the two intersections points,
of the half-circles. Lines
for
parallel to the bases of the half-circles are drawn such that the distances between
and
and
and
are always the same for all
.
The intersection points of with one of the half-circles are labeled
, and with the other half-circle at
, as shown in the diagram.
Prove that
Picture 2
physics problem
Solution
inscribed triangle
moar images
yay
solution reflection
origami
combos
circles
more circles
checkerboasrd
Fermat point
cenn driagrma
cyclic square
diagram
Cyclic squares DOTS DTOS TDORS