The first problem I found via the Romantics of Geometry Facebook group: let be the point of tangency of the incircle of with and let be the foot of the perpendicular from the incentre of the to . Then show bisects .

The second problem is motivated by the above and problem 2 of the 2008 USAMO: this time let be a symmedian of and be the foot of the perpendicular from the circumcentre of to . Then show that bisects .

Here is a solution to the first problem inspired bythat of Vaggelis Stamatiadis. Let the line through the other two points of tangency of the incircle with intersect line at the point as shown below. Note that since and are tangents to the circle, line is the polar of with respect to the incircle.

Since is on the polar of , by La Hire’s theorem, is on the polar of . The polar of also passes through (as is a tangent to the circle at ). We conclude that the polar of is the line through and .

Next, let intersect at . By theorem 5(a) at this link, the points form a harmonic range. Since the cross ratio of collinear points does not change under central projection, considering the projection from , also form a harmonic range. (Alternatively, this follows from the theorems of Ceva and Menelaus using the Cevians intersecting at the Gergonne point and transveral ). Also, as both and are perpendicular to polar of .

Considering a central projection from of line to a line parallel to through , we see that form a harmonic range. Since is a point at infinity, this implies is the midpoint of and so triangles and are congruent (equality of two pairs of sides and included angle is ). Hence bisects as was to be shown.

For the second problem, we use the following characterisation of a symmedian: extended concurs with the lines of tangency of the circumcircle at and . (For three proofs of this see here.)

Define as the intersection of with and as the intersection of with the tangents at . Note that line is the polar of with respect to the circumcircle. By La Hire’s theorem, must be on the polar of . This polar is perpendicular to (the line joining to the centre of the circle) and as by construction of , it follows that line is the polar of . Again by theorem 5(a) in reference (2), form a harmonic range. Following the same argument as the previous proof, this together with imply bisects as required.

By similar arguments, one can prove the following, left to the interested reader. If is the -excentre of , the ex-circle’s point of tangency of , and the foot of the perpendicular from to line , then bisects .

(1) Alexander Bogomolny, __Poles and Polars__* from Interactive Mathematics Miscellany and Puzzles* http://www.cut-the-knot.org/Curriculum/Geometry/PolePolar.shtml, Accessed 19 March 2017

(2) Poles and Polars – Another Useful Tool! | The Problem Solver’s Paradise

(3) Yufei Zhao, Lemmas in Euclidean Geometry

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Test |
ODI |
T20I |
IPL |

200 (283) | 91 (97) | 90* (55) | 75 (51) |

44 (90) | 59 (67) | 59* (33) | 79 (48) |

3 (8) | 117 (117) | 50 (36) | 33 (30) |

4 (17) | 106 (92) | 7 (12) | 80 (63) |

9 (10) | 8 (11) | 49 (51) | 100* (63) |

18 (40) | 85* (81) | 56* (47) | 14 (17) |

9 (28) | 9 (13) | 41* (28) | 52 (44) |

45 (65) | 154* (134) | 23 (27) | 108* (58) |

211 (366) | 45 (51) | 55* (37) | 20 (21) |

17 (28) | 65 (76) | 24 (24) | 7 (7) |

40 (95) | 82* (51) | 109 (55) | |

49* (98) | 89* (47) | 75* (51) | |

167 (267) | 16 (9) | 113 (50) | |

81 (109) | 54* (45) | ||

62 (127) | 0 (2) | ||

6* (11) | 54 (35) | ||

235 (340) | |||

15 (29) | |||

1215 @ 75.9 |
739 @ 92.37, SR 100 |
641 @ 106.8, SR 140.3 |
973 @ 81.2, SR 152.0 |

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Bessel functions of the first () and second () kind of order satisfy:

.

Solutions for integer arise in solving Laplace’s equation in cylindrical coordinates while solutions for half-integer arise in solving the Helmholtz equation in spherical coordinates. Hence they come about in wave propagation, heat diffusion and electrostatic potential problems. The functions oscillate roughly periodically with amplitude decaying proportional to . Note that is the second linearly independent solution when is an integer (for integer , ). Also, for integer , has the generating function

the integral representations

and satisfies the orthogonality relation

where , Kronecker delta, and is the m-th zero of .

Modified Bessel functions of the first () and second () kind of order satisfy:

(replacing with in the previous equation).

The four functions may be expressed as follows.

(In the last formula we need to take a limit when is an integer.)

Note that and are singular at zero.

The Hankel functions and are also known as Bessel functions of the third kind.

The functions , , , and all satisfy the recurrence relations (using in place of any of these four functions)

Bessel functions of higher orders/derivatives can be calculated from lower ones via:

In particular, note that is the derivative of .

The Airy functions of the first () and second () kind satisfy

.

This arises as a solution to Schrödinger’s equation for a particle in a triangular potential well and also describes interference and refraction patterns.

Hermite polynomials (the probabilists’ defintion) can be defined by:

,

and are orthogonal with respect to weighting function on .

They satisfy the differential equation

(where is forced to be an integer if we insist be polynomially bounded at )

and the recurrence relation

.

The first few such polynomials are . The Physicists’ Hermite polynomials are related by and arise for example as the eigenstates of the quantum harmonic oscillator.

Laguerre polynomials are defined by

,

and are orthogonal with respect to on .

They satisfy the differential equation

,

recurrence relation

,

and have generating function

The first few values are . Note also that .

The functions come up as the radial part of solution to Schrödinger’s equation for a one-electron atom.

Legendre polynomials can be defined by

and are orthogonal with respect to the norm on .

They satisfy the differential equation

,

recurrence relation

and have generating function

.

The first few values are .

They arise in the expansion of the Newtonian potential (multipole expansions) and Laplace’s equation where there is axial symmetry (spherical harmonics are expressed in terms of these).

Chebyshev polynomials of the *1st kind* can be defined by

and are orthogonal with respect to weighting function in .

They satisfy the differential equation

,

the relations

and have generating function

The first few values are . These polynomials arise in approximation theory, namely their roots are used as nodes in piecewise polynomial interpolation. The function is the polynomial of leading coefficient 1 and degree n where the maximal absolute value on (-1,1) is minimal.

Chebyshev polynomials of the *2nd kind* are defined by

and are orthogonal with respect to weighting function in .

They satisfy the differential equation

,

the recurrence relation

and have generating function

The first few values are . (There are also less well known Chebyshev polynomials of the third and fourth kind.)

Bessel polynomials may be defined from Bessel functions via

.

They satisfies the differential equation

.

The first few values are .

**3. Integrals**

The error function has the form

.

This can be interpreted as the probability a normally distributed random variable with zero mean and variance 1/2 is in the interval .

The cdf of the normal distribution $\Phi(x)$ is related to this via . Hence the tail probability of the standard normal distribution is .

Fresnel integrals are defined by

They have applications in optics.

The exponential integral (used in heat transfer applications) is defined by

.

It is related to the logarithmic integral

by (for real ).

The incomplete elliptic integral of the first, second and third kinds are defined by

Setting gives the complete elliptic integrals.

Any integral of the form , where is a constant, is a rational function of its arguments and is a polynomial of 3rd or 4th degree with no repeated roots, may be expressed in terms of the elliptic integrals. The circumference of an ellipse of semi-major axis , semi-minor axis and eccentricity is given by , where is the complete integral of the second kind.

(Some elliptic functions are related to inverse elliptic integral, hence their name.)

The (upper) incomplete Gamma function is defined by

.

It satisfies the recurrence relation . Setting gives the Gamma function which interpolates the factorial function.

The digamma function is the logarithmic derivative of the gamma function:

.

Due the relation , this function appears in the regularisation of divergent integrals, e.g.

.

The incomplete Beta function is defined by

.

When setting this becomes the Beta function which is related to the gamma function via

.

This can be extended to the multivariate Beta function, used in defining the Dirichlet function.

.

The polylogarithm, appearing as integrals of the Fermi–Dirac and Bose–Einstein distributions, is defined by

Note the special case and the case is known as the dilogarithm. We also have the recursive formula

.

All the above functions can be written in terms of generalised hypergeometric functions.

where for or .

The special case is called a confluent hypergeometric function of the first kind, also written .

This satisfies the differential equation (Kummer’s equation)

.

The Bessel, Hankel, Airy, Laguerre, error, exponential and logarithmic integral functions can be expressed in terms of this.

The case is sometimes called Gauss’s hypergeometric functions, or simply hypergeometric functions. This satisfies the differential equation

.

The Legendre, Hermite and Chebyshev, Beta, Gamma functions can be expressed in terms of this.

Wikipedia: List of mathematical functions

Wikipedia: List of special functions and eponyms

Wikipedia Category: Orthogonal polynomials

Weisstein, Eric W. “Laplace’s Equation.” From *MathWorld*–A Wolfram Web Resource. http://mathworld.wolfram.com/LaplacesEquation.html

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Duration of daylength (hrs) | Dates | Frequency |

< 10 | 19 May-24 July | 67 |

10-10.5 | 25 July-10 August, 2-18 May | 34 |

10.5-11 | 11-24 August, 18 April-1 May | 28 |

11-11.5 | 25 August-6 September, 5-17 April | 26 |

11.5-12 | 6-19 September, 24 March-4 April | 25 |

12-12.5 | 20 September-1 October, 12-23 March | 24 |

12.5-13 | 1-13 October, 28 February-11 March | 25 |

13-13.5 | 14-26 October, 16-27 February | 25 |

13.5-14 | 9 Nov, 2-15 Feb | 28 |

14-14.5 | 10 Nov-27 Nov, 16 Jan-1Feb | 35 |

>14.5 | 28 Nov-15 Jan | 49 |

What surprised me the most about this was that only 100 days of the year have daylength between 11 and 13 hours and we have a good 84 days with light longer than 14 hours.

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Here are the numbers below 2160 also with 36 factors:

The first integer with more than 40 factors is $2520 = 2^3 \times 3^2 \times 5 \times 7$ (48 factors).

[1] N. J. A. Sloane and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane), A000005 – OEIS.

[2] Highly composite number – Wikipedia, the free encyclopedia

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February set the record of greatest anomaly from mean monthly temperatures, beating the previous record (set only the previous month) by more than 0.2°C. The map here shows that the vast majority of the planet had above-average temperatures, with the greatest deviation in the arctic region. As an example, check out the temperatures of Salekhard, Russia on the arctic circle during this time (this is a place that registers temperatures below -40 during winters). Over the month its average was 12.5°C above the mean! Data is from [6].

[1] Record Warmth in February : Image of the Day

[2] NOAA National Centers for Environmental Information, State of the Climate: Global Analysis for February 2016, published online March 2016, retrieved on March 27, 2016 from http://www.ncdc.noaa.gov/sotc/global/201602.

[4] Ogimet: Synop report summary for Svalbard airport

[5] Longyearbyen February Weather 2016 – AccuWeather

[6] Погода и Климат – Климатический монитор: погода в Салехарде (pogodaiklimat.ru)

[8] Record-Shattering February Warmth Bakes Alaska, Arctic 18°F Above Normal | ThinkProgress

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For example (from [1], p18), if we wish to show that for real numbers with sum that

we may write (equivalent to ), but this implies and so the sign goes the wrong way.

A way around this is to write

Summing this over then gives as desired.

Here are a few more examples demonstrating this technique.

**2.** (p9 of [2]) If are positive real numbers with , then

To prove this we write

Next we have as this is equivalent to . This means . Putting everything together,

as required.

**3.** (based on p8 of [2]) If for and then

By the AM-GM inequality, , so

Summing this over gives

**4**. (from [3]) If are positive, then

Once again, focusing on the denominator,

Hence,

as desired.

**5**. (from the 1991 Asian Pacific Maths Olympiad, see [4] for other solutions) Let be positive numbers with . Then

Here we write

as required.

[1] Zdravko Cvetkovski, *Inequalities: Theorems, Techniques and Selected Problems*, Springer, 2012.

[2] Wang and Kadaveru, *Advanced Topics in Inequalities*, available from http://www.artofproblemsolving.com/community/q1h1060665p4590952

[3] Cauchy Reverse Technique: https://translate.google.com.au/translate?hl=en&sl=ja&u=http://mathtrain.jp/crt&prev=search

[4] algebra precalculus – Prove that – Mathematics StackExchange

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Day | Jurassic World | Star Wars: The Force Awakens |

1 | 786 | 464 |

2 | 275 | 183 |

3 | 143 | 96 |

4 | 108 | 68 |

5 | 79 | 40 |

6 | 69 | 29 |

7 | 54 | 22 |

8 | 37 | 11 |

9 | 28 | 6 |

10 | 18 | 5 |

11 | 15 | 5 |

12 | 11 | 5 |

13 | 10 | 4 |

14 | 9 | 3 |

15 | 7 | 2 |

16-19 | 5 | 2 |

20-21 | 5 | 1 |

22-44 | 4 | 1 |

45+ | 3 | 1 |

I was amazed by Jurassic World’s summer run and then that of The Force Awakens simply blew my mind.

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Clearly if that point is either a midpoint or one of the vertices, the answer is a median of the triangle. A median cuts a triangle in half since the two pieces have the same length side and equal height.

So what if the point is not a midpoint or a vertex? Referring to the diagram below, if is our desired point closer to than , the end point of the area-bisecting segment would need to be on side so that area(BPQ) = area(ABC)/2.

In other words, we require area(BPQ) = area(BDQ), or, subtracting the areas of triangle BDQ from both sides,

Since these two triangles share the common base , this tells us that we require them to have the same height. In other words, we require to be parallel to . This tells us how to construct the point given on :

- Construct the midpoint D of .
- Draw parallel to .

See [1] for an animation of this construction.

In turns out that the set of all area-bisecting lines are tangent to three hyperbolas and enclose a deltoid of area times the original triangle. [2,3,4]

[1] Jaime Rangel-Mondragon, “Bisecting a Triangle” http://demonstrations.wolfram.com/BisectingATriangle/ from the Wolfram Demonstrations Project Published: July 10, 2013

[2] Ismail Hammoudeh, “Triangle Area Bisectors” http://demonstrations.wolfram.com/TriangleAreaBisectors/ from the Wolfram Demonstrations Project

[3] Ed Pegg Jr,“Halving a Triangle” http://demonstrations.wolfram.com/HalvingATriangle/ from the Wolfram Demonstrations Project Published: September 28, 2007

[4] Henry Bottomley, Area bisectors of a triangle, January 2002

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1 | Hirwani, Massie (16) | F Martin (Eng), Krejza (12) | several tied on 11 |

2 | Hirwani (24) | Bedser (22) | Massie (21) |

3 | Hirwani (31) | Turner (29) | Hogg (27) |

4 | Turner (39) | Hirwani (36) | Hogg, Valentine (33) |

5 | Turner (45) | Hogg (40) | Valentine (39) |

6 | Turner (50) | Philander (45) | Valentine (43) |

7 | Richardson (53) | Philander, Turner (51) | Valentine (49) |

8 | Richardson (66) | Turner (56) | Valentine (55) |

9 | Richardson (66) | Turner (63) | Ferris (61) |

10 | Richardson (71) | Turner (69) | Tate (65) |

11 | Richardson, Turner (72) | Yasir Shah (69) | Grimmett, Peel, Roberts, Tate, Valentine (65) |

12 | Turner (80) | Richardson, Yasir Shah (76) | Grimmett (71) |

13 | Yasir Shah (86) | Turner (81) | Richardson (78) |

14 | Richardson (88) | Yasir Shah (87) | Turner (83) |

15 | Yasir Shah (90) | Lohmann (89) | Barnes, Grimmett, Philander (87) |

16 | Lohmann (101) | Yasir Shah (95) | Barnes, Turner (94) |

17 | Lohmann (109) | Yasir Shah (102) | Barnes, Grimmett, Turner (101) |

18 | Barnes, Lohmann, Yasir Shah (112) | Grimmett (105) | Ashwin (104) |

19 | Yasir Shah (116) | Barnes (112) | Grimmett (109) |

20 | Barnes (122) | Yasir Shah (116) | Grimmett (112) |

21 | Barnes (122) | Grimmett (120) | Yasir Shah (119) |

22 | Barnes (135) | Grimmett (128) | Botham (118) |

23 | Grimmett (142) | Barnes (140) | Botham (122) |

24 | Barnes (150) | Grimmett (142) | Waqar Younis (134) |

25 | Barnes (167) | Grimmett, Waqar Younis (143) | Botham (139) |

26 | Barnes (175) | Waqar Younis (148) | Grimmett (144) |

27 | Barnes (189) | Waqar Younis (154) | Grimmett (147) |

28 | Waqar Younis (159) | Grimmett (155) | Botham (149) |

29 | Waqar Younis (166) | Grimmett (156) | Ashwin, Saeed Ajmal (153) |

30 | Waqar Younis (169) | Saeed Ajmal (159) | Ashwin, Grimmett (157) |

31 | Waqar Younis (180) | Ashwin (169) | Grimmett (164) |

32 | Waqar Younis (187) | Ashwin (176) | Grimmett (172) |

33 | Waqar Younis (190) | Ashwin (183) | Grimmett (177) |

34 | Waqar Younis (191) | Ashwin (189) | Grimmett (183) |

35 | Waqar Younis (194) | Grimmett (193) | Ashwin (192) |

36 | Grimmett (203) | Waqar Younis (196) | Ashwin (193) |

37 | Grimmett (216) | Ashwin (203) | Waqar Younis (199) |

38 | Ashwin (207) | Lillee (206) | Waqar Younis (200) |

39 | Ashwin (220) | Waqar Younis (208) | Lillee (206) |

40 | Ashwin (223) | Lillee (206) | Steyn (205) |

41 | Ashwin (231) | Waqar Younis (216) | Steyn (211) |

42 | Ashwin (235) | Waqar Younis (217) | Lillee (214) |

43 | Ashwin (247) | Lillee, Waqar Younis (222) | Steyn (217) |

44 | Ashwin (248) | Lillee (229) | Waqar Younis (227) |

45 | Ashwin (254) | Steyn (232) | Lillee (230) |

46 | Ashwin (261) | Steyn (238) | Lillee (237) |

47 | Ashwin (269) | Steyn (244) | Lillee (243) |

48 | Ashwin (271) | Lillee (251) | Steyn (249) |

49 | Lillee (259) | Steyn (255) | Donald (248) |

50 | Lillee (262) | Steyn (260) | Donald (251) |

51 | Lillee (269) | Steyn (263) | Donald, Waqar Younis (254) |

52 | Lillee (273) | Steyn (265) | Muralitharan (261) |

53 | Lillee (279) | Steyn (270) | Muralitharan (265) |

54 | Lillee (290) | Steyn (272) | Donald, Muralitharan (265) |

55 | Lillee (296) | Steyn (279) | Muralitharan (278) |

56 | Lillee (305) | Muralitharan (283) | Steyn (282) |

57 | Lillee (305) | Muralitharan (291) | Steyn (287) |

58 | Lillee (315) | Muralitharan (302) | Marshall (290) |

59 | Lillee (321) | Muralitharan (303) | Marshall (296) |

60 | Lillee (321) | Muralitharan (310) | Hadlee, Marshall, Steyn (299) |

61 | Lillee (321) | Muralitharan (315) | Steyn (304) |

62 | Lillee (325) | Muralitharan (317) | Steyn (312) |

63 | Lillee (328) | Muralitharan (325) | Steyn (323) |

64 | Lillee (332) | Muralitharan (329) | Steyn (327) |

65 | Muralitharan (340) | Lillee (335) | Hadlee, Steyn (332) |

66 | Muralitharan (350) | Lillee, Steyn (336) | Hadlee (334) |

67 | Muralitharan (361) | Steyn (340) | Hadlee, Lillee (336) |

68 | Muralitharan (371) | Hadlee, Lillee (342) | Steyn (341) |

69 | Muralitharan (374) | Hadlee (351) | Steyn (350) |

70 | Muralitharan (382) | Steyn (356) | Hadlee, Lillee (355) |

71 | Muralitharan (395) | Steyn (361) | Hadlee (358) |

72 | Muralitharan (404) | Hadlee (363) | Steyn (362) |

73 | Muralitharan (412) | Hadlee (373) | Steyn (371) |

74 | Muralitharan (417) | Steyn (375) | Hadlee (373) |

75 | Muralitharan (420) | Steyn (383) | Hadlee (378) |

76 | Muralitharan (430) | Steyn (389) | Hadlee (388) |

77 | Muralitharan (433) | Hadlee (391) | Steyn (389) |

78 | Muralitharan (437) | Steyn (396) | Hadlee (395) |

79 | Muralitharan (442) | Steyn (399) | Hadlee (396) |

80 | Muralitharan (450) | Hadlee (403) | Steyn (402) |

81 | Muralitharan (455) | Hadlee (406) | Steyn (402) |

82 | Muralitharan (459) | Hadlee (408) | Steyn (406) |

83 | Muralitharan (470) | Hadlee (415) | Steyn (408) |

84 | Muralitharan (478) | Hadlee (419) | Steyn (416) |

85 | Muralitharan (485) | Hadlee (423) | Steyn (417) |

86 | Muralitharan (496) | Hadlee (431) | Kumble (415) |

87-101 | Muralitharan | Kumble | McGrath |

102-126 | Muralitharan | Kumble | Warne |

127 | Muralitharan (770) | Warne (611) | Kumble (608) |

128 | Muralitharan (777) | Warne (623) | Kumble (608) |

129 | Muralitharan (783) | Warne (629) | Kumble (613) |

130 | Muralitharan (786) | Warne (634) | Kumble (616) |

131 | Muralitharan (788) | Warne (638) | Kumble (616) |

132 | Muralitharan (792) | Warne (645) | Kumble (619) |

133 | Muralitharan (800) | Warne (651) | Kallis (258) |

134-145 | Warne | Kallis | Waugh |

146-166 | Kallis | Waugh | Tendulkar |

167-168 | Waugh | Tendulkar | Ponting |

169-200 | Tendulkar |

[1] Most wkts in consec Tests from debut – Google Groups

[2] Top 10 bowlers with most wickets in 10 Tests or less (crictracker.com)

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