bartogian does the IMO (2014 Q4)

July 16, 2014

The fourth question on this year’s IMO is a Euclidean geometry question. And so I thought to myself, I wonder if I could get a computer algebra package (with Grobner bases, etc) to do the relevant algebraic manipulation to solve the problem for me, once I had converted the problem into coordinate geometry.

This is a desription of that journey. But first the problem:

Points P and Q lie on side BC of acute-angled triangle ABC such that $\angle PAB=\angle BCA$ and $\angle CAQ = \angle ABC$. Points M and N lie on lines AP and AQ, respectively, such that P is the midpoint of AM and Q is the midpoint of AN. Prove that the lines BM and CN intersect on the circumcircle of triangle ABC.

As every good student knows, it is a good idea to do a little bit of Euclidean geometry before starting on the trigonometry or coordinate geometry. So we will begin by noticing that APQ is isosceles.

We will set up our coordinates so that the line BC is the horizontal axis, with the origin at the midpoint of PQ. Then we introduce variables $h,x,y,z$ by $A=(0,h)$, $P=(x,0)$, $Q=(-x,0)$, $B=(y,0)$ and $C=(z,0)$. Hence $M=(2x,-h)$ and $N=(-2x,-h)$.

Let $X=(p,q)$ be the intersection of BM and CN and let $O=((y+z)/2,w)$ be the circumcentre of ABC.

So now for the equations that these seven variables satisfy.

Since X lies on BM, there is

$\displaystyle \frac{q}{p-y} = \frac{-h}{2x-y}$

and since X also lies on CN, there is

$\displaystyle \frac{q}{p-z} = \frac{-h}{-2x-z}$

From |AO|=|BO| there is

$\displaystyle \left(\frac{y+z}{2}\right)^2 + (w-h)^2=\left(\frac{y-z}{2}\right)^2+w^2$

and from the similarity of triangles ABC and QAC, in particular $AC^2=CQ\cdot BC$, we get

$\displaystyle h^2+z^2 = (z+x)(z-y).$

And what do we have to prove? Well nothing other than |AO|=|XO|, so it suffices to prove

$\displaystyle \left(p-\frac{y+z}{2} \right)^2+(q-w)^2 = \left(\frac{y+z}{2}\right)^2+(w-h)^2$

So, we go to sage (I skip the routine and boring simplification) and execute

sage: x,h,y,z,w,p,q=QQ['x,h,y,z,w,p,q'].gens() sage: I=ideal(q*(2*x-y)+h*(p-y),q*(2*x+z)-h*(p-z),y*z-2*w*h+h^2,h^2+z^2-(z+x)*(z-y)) sage: (p^2-p*y-p*z+(q-w)^2-(w-h)^2) in I False

Yikes!

sage: (p^2-p*y-p*z+(q-w)^2-(w-h)^2) in I.radical() False 

Yikes again!

While it is tempting to expect a typographical error, that is not in fact the case. The following computation reveals what is really going on.

sage: I.associated_primes() [Ideal (q, z, y, h) of Multivariate Polynomial Ring in x, h, y, z, w, p, q over Rational Field, Ideal (q, z, h, x) of Multivariate Polynomial Ring in x, h, y, z, w, p, q over Rational Field, Ideal (q, y, h, x) of Multivariate Polynomial Ring in x, h, y, z, w, p, q over Rational Field, Ideal (z, y, h, x) of Multivariate Polynomial Ring in x, h, y, z, w, p, q over Rational Field, Ideal (y*p + z*p - 2*p^2 - 2*h*q + 4*w*q - 2*q^2, y*z - p^2 - 2*h*q + 2*w*q - q^2, h*z - h*p + 2*x*q + z*q, 2*x*z - 2*h*w - 2*x*p - z*p + p^2 + h*q - 2*w*q + q^2, h*y - h*p - 2*x*q + y*q, 2*x*y + 2*h*w - 2*x*p - z*p + p^2 + h*q - 2*w*q + q^2, h^2 - 2*h*w + p^2 + 2*h*q - 2*w*q + q^2, z*p^2 - p^3 - 2*x*h*q + 4*x*w*q - h*p*q + 2*w*p*q - 2*x*q^2 - p*q^2, z^2*p - p^3 - 4*x*h*q + 8*x*w*q + 4*z*w*q - 2*h*p*q + 2*w*p*q - p*q^2, 2*h*w^2 - w*p^2 + 2*x^2*q - h*w*q + 2*w^2*q - w*q^2, h*w*p^2 - 2*x^2*h*q + 4*x^2*w*q - 2*x^2*q^2, w*p^4 - 4*x^2*h*w*q + 8*x^2*w^2*q - 2*x^2*p^2*q - 2*w^2*p^2*q + 2*x^2*h*q^2 - 8*x^2*w*q^2 + w*p^2*q^2 + 2*x^2*q^3) of Multivariate Polynomial Ring in x, h, y, z, w, p, q over Rational Field] 

And we see the issue. The ideal I has five associated primes, but only one of these primes corresponds to the geometric problem at hand, the other four correspond to highly degenerate triangles.

So we try
 sage: J=I.associated_primes()[4] sage: (p^2-p*y-p*z+(q-w)^2-(w-h)^2) in J True 
and the proof is complete.

An alternative approach would be to artificially invert $h$ by creating a new variable $t$ satisfying the relation $ht=1$:
 sage: x,h,y,z,w,p,q,t=QQ['x,h,y,z,w,p,q,t'].gens() sage: I=ideal(q*(2*x-y)+h*(p-y),q*(2*x+z)-h*(p-z),y*z-2*w*h+h^2,h^2+z^2-(z+x)*(z-y),h*t-1) sage: (p^2-p*y-p*z+(q-w)^2-(w-h)^2) in I True 
and QED for the second time.

Bartogian’s guide to Seattle

May 3, 2014

Do: Underground Tour. I’ve only done the day tour and think night tour is essentially the same but with more emphasis on the history deemed inappropriate by the morality police (eg, prostitution). The ticket can give you some discounts on other touristy things.

Eat: Mod Pizza. Good pizza at a good price. Right across the road from WAC.

Sing: Hula Hula. Karaoke every evening. Was not crowded on a Thursday evening, can’t speak for other days of the week.

March 22, 2014

The following picture was taken at the intersection of Cardigan Street and Kilner Lane, Camperdown.

OK, so lets say you want to go to ‘main or ‘te Bay. You ignore the other two signs and continue as directed. Pretty soon you’re dumped at this intersection.

Yes, that is really telling you to turn right onto Parramatta road. But due to the timing of the traffic lights, you’ll likely have three lanes all to yourself, the leftmost of which is a bus lane.

At the first possible opportunity (Gordon St) to turn left, there is another cycling sign. Eager to get off the main road, you take it. Despite the fact that this sign is giving you two new destinations, Glebe and City.

Pretty soon, you’re dumped on Pyrmont Bridge Road, which could really do with both a high quality bike lane and a reduction in the speed limit from 60 to 50. (I don’t know myself how to fix these problems for you).

From here on, the trail goes cold as far as I can tell (as in there being no more signs to tell you where to go). From start to finish, we’ve managed to cover a grand total of just 280 metres.

\def, \newcommand, \span and \align

March 9, 2014

I learnt something about TeX/LaTeX today.

I got the following cryptic error message:
! Missing # inserted in alignment preamble.

 

 \crcr l.38 \end{align*}

Now when I went to google “Missing # inserted in alignment preamble”, I immediately got a lovely website which was able to answer my question. But hey, information survives by being copied, so if you’ll kindly allow me to continue, I shall.

Stripped down to a minimal example, here is the problematic code.

\documentclass[11pt, reqno]{amsart}

 \usepackage{amsmath} \def\span{\,\mbox{span}\,} \begin{document} \begin{align*} u &= \arctan x & dv &= 1 \, dx \\ du &= \frac{1}{1 + x^2} dx & v &= x. \end{align*} 

\end{document}

Actually stripping down to a minimal example kind of makes it obvious where the error is – it must be somewhere in the \def\span line.


The problem in the above code is that \span is already defined to be something important needed to internally execute the \align command. By using \def, we’re redefining what \span means. If we used \newcommand instead, we wouldn’t be able to redefine \span and we’d get an instructive error message.

So the moral of the story is: \newcommand is safer.

Walking between Sydney Airport Domestic Terminal and Mascot Station

February 28, 2014

OK, this one is really easy.

Just follow Google’s directions. You can’t get lost.

If you happen to be travelling at the right time of day between the airport and the CBD, there is also the option of the 305 bus.

Walking between Sydney International and Wolli Creek station

February 9, 2014

First the why:

One way Central-International train ticket: 16.40AUD.

One way Central-Wolli Creek train ticket: 3.80AUD.

[Warning: Never buy a ticket to the City. It prices out to the most expensive option of all city train stations. Always buy a ticket to the specific train station you want to get off on]

The reason for the difference in the train fares is the existence of a station access fee at Sydney airport locations. Or more bluntly, they’re price gouging becasue they think they can get away with it. There is currently an inquiry into this practise by the NSW parliament.

It is very possible and easy to walk from Wolli Creek train station to the international terminal at Sydney airport in less than half an hour.

First the map:

Fortunately there are no fences at Sydney airport waiting to trap you along this route. This makes getting in and out of Sydney airport easy.

The main thing about this route to remember is the loop. Walking to the airport, after travelling along the north side of the bridge, one loops down below at the first opportunity to continue to the airport.

The only possibly tricky part is to orient yourself when leaving Wolli Creek station. Exit the station from the exit closer to the lower platform and you will arrive at the starting point on the map above. Turn right out of the station to begin your walk.

The entire route is on a paved surface (partly on a joint pedestrian/bike path). I’ve only ever done it with a backpack, but it should be easy enough with luggage on wheels as well.

2013 Statistics

February 1, 2014

Countries visited: 10.
New: Denmark, Vatican City, Argentina, Brazil.

Flights: 40
Total distance (great circles): 195,133km
Average flight length: 4878km
Longest flight: SFO-SYD 11937km
Shortest flight: MSY-BHM 517km
Airlines flown: 12
New: WN, SQ, HA, CZ, AR
Airports visited: 31
New: NRT, SAN, MSY, BHM, LAS, EWR, PEK, CPH, HNL, CAN, BKK, EZE, USH, AEP, IGR

(a) I don’t have data for flight length in terms of time.
(b) Hopefully the headline figure of 195,133km will decrease in 2014.
(c) I don’t really feel that the Vatican City is a real country.
(d) I visited the old BKK in 1999. The airport code has shifted to the new airport in the interim.

2013 Federal Election Senate Results

October 2, 2013

Only six years ago, the three most popular states presented a snooze-fest when it came to the senate – 3 each to the ALP and the Coalition, thank you all for voting and we’ll continue with our nice little duopoly.

What a different world we have this time around. While barring a recount the Sports party ultimately failed to win a seat off their 0.23% primary vote, we’ve still got ten senators from outside the majors. With all the talk of preference deals, hopefully we’ll get to see some reform that will get rid of ticket voting before the next time around.

But hey, let’s get to the results:

1st place: South Australia (2 Liberal, 1 Xenophon, 1 ALP, 1 Greens, 1 Family First)

Oh dear, what is going on? How can we possibly give the South Australians first place when they’ve gone ahead and elected a member of a right wing Christian party? This should really be considered an indictment on how well the election actually went. The Croweaters get first place on the back of the elections of Sarah Hanson-Young and Nick Xenophon.

2nd place: Tasmania (2 Liberal, 2 ALP, 1 Greens, 1 Palmer)

Those Taswegians usually put in a good performance at elections and their high occurrence of below the line voting sets a good example to the rest of the country. Disappointing to see Peter Whish-Wilson have to go to preferences to win his seat. But more disappoing is the election of Jacqui Lambie. Clive Palmer is a one-percenter. ’nuff said.

3rd place: Victoria (2 Liberal, 2 ALP, 1 Greens, 1 Motoring)

Looking really good until you see Ricky Muir of the Australian Motoring Enthusiasts Party winning a seat. Sadly, this is just the latest installment from the state that has previously elected members from Family First and the DLP. With Tony Abbott already preferring a 1950s era roads only transport policy, this can’t end well for the nation.

4th place: Queensland (3 LNP, 2 ALP, 1 Palmer)

Poor. Actually downgrade that to terrible. Won’t see much support for a mining tax here.

5th place: New South Wales (2 Liberal, 2 ALP, 1 National, 1 Liberal Democrat)

When you first listen to the libertarian position, it can appear alluring. The government shouldn’t interfere in personal life, so therefore censorship shouldn’t happen and marijuana should be legalised. OK but those positions are the easy ones to get to. It’s when you delve a bit further that problems start to appear – there’s that classic line about ending up advocating for the abolition of fire departments. (Actually they’d probably try to privatise them). Fundamentally, you probably shouldn’t get someone to run a government when their core belief is that the government shouldn’t exist.

6th place* Western Australia (3 Liberal, 2 ALP, 1 Palmer)

The failure to reelect Scott Ludlam puts WA into last place here. I’m ashamed of my home state.

*subject to change if the recount changes the outcome

World Cup 2015

September 21, 2013

Hidden amongst the media coverage of the Ashes and the Essendon saga was an announcement about the format of the 2015 World Cup.

Fourteen teams, two groups of seven, top four in each group go on to play a knockout tournament.

Yes. A knockout tournament. Geez. What a crap shoot.

Of course, just like four years ago, we know exactly which eight teams will get to go on to the crap shoot stage. And unlike eight years go, with six group games each, a single upset is unlikely to result in any change to which eight teams progress.

So why even bother with the group stage?

They could at least have a final eight like in the AFL, which would give an advantage to the top two teams in each pool, making the group stages have at least some meaning.

Oh, and a bonus: Then we’d have nine high-revenue matches in the finals stage, rather than seven, which for a money obsessed governing body is surely the preferred position.

As it stands, I cannot respect the 2015 World Cup, or its winner as a worthy champion due to the format, and sincerely hope that the schedule for the 2014/15 season is fixed to correct this atrocity (and the other atrocity of no WACA test – why can’t we play five against India?)

How to ascertain your home phone number

August 7, 2013

(In Australia): Dial 127 22 123.

Yes this sound silly. But it also works to determine the last connected number at the premises, which is information a telecommunications company can find useful in connecting a phone or internet service.