Hungary Math Olympiad Problem | Best Math Olympiad Problems | Geometry Problem 

The result does not depend on the value of the circle radius. For r = 0, we have: AO²+OC² = BO²+DO² 33²+r²+x²+r²=85²+r²+35²+r² 33²+x²=85²+35² José Osorio from Brazil.

I really enjoyed this - and so many of your other puzzle problems. You have a special knack for choosing interesting challenges for your audience (like this one!) and your solutions are ingenious.: we are all lucky to have found you here. Thank you for brightening my day!

R=0 British Flag theorem: x²+33²=35²+85² x= 85,796 cm ( Solved √ )

this is the 4th time i have written a code for this but yet i don't know why some numbers for r=... won't lead to a solution: 10 dim x(3),y(3):r=20:l1=35:l3=85:l4=33:sw=.1:xm=r*1.3+sw:goto 40 20 ym=sqr(l1^2+r^2-xm^2):lh=sqr(l4^2+r^2-xm^2)+ym 30 lb=sqr(l3^2+r^2-(ym-lh)^2)+xm:return 40 gosub 20 50 if lh0 then 130 150 xs1=(xs11+xs12)/2:gosub 100:if dg1*dg>0 then xs11=xs1 else xs12=xs1 160 if abs(dg)>1E-10 then 150:rem den thalessatz anwenden 170 xmt=(xm+lb)/2:ymt=ym/2:rt=sqr((xm-xmt)^2+(ym-ymt)^2) 180 xs2=xm-r:goto 220 190 ys2=ym-sqr(r^2-(xs2-xm)^2):l2=sqr((xs2-lb)^2+ys2^2) 200 dgu1=(xs2-xmt)^2/l1^2:dgu2=(ys2-ymt)^2/l1^2:dgu3=rt^2/l1^2: 210 dg=dgu1+dgu2-dgu3: return 220 gosub 190 230 xs21=xs2:dg1=dg:xs2=xs2+sw:if xs2>10*lb then stop 240 xs22=xs2:gosub 190:if dg1*dg>0 then 230 250 xs2=(xs21+xs22)/2:gosub 190:if dg1*dg>0 then xs21=xs2 else xs22=xs2 260 if abs(dg)>1E-10 then 250 else print "der gesuchte abstand=";l2 270 xs3=xm:goto 310 280 disy=l3^2-(xs3-lb)^2:if disy0 then 330 350 xs3=(xs31+xs32)/2:gosub 280:if dg1*dg>0 then xs31=xs3 else xs32=xs3 360 if abs(dg)>1E-10 then 350 370 xmt=xm/2:ymt=(lh+ym)/2: xs4=xm-r:rt=sqr(xmt^2+(ymt-lh)^2):goto 410 380 disy=rt^2-(xs4-xmt)^2 :if disy0 then 430 450 xs4=(xs41+xs42)/2:gosub 380:if dg1*dg>0 then xs41=xs4 else xs42=xs4 460 if abs(dg)>1E-10 then 450 470 x(0)=0:y(0)=0:x(1)=lb:y(1)=0:x(2)=x(1):y(2)=lh:x(3)=0:y(3)=y(2):mass=1E3/(l1+l2+l3+l4)*4 480 goto 500 490 xbu=x*mass:ybu=y*mass:return 500 xba=0:yba=0:for a=1 to 4:ia=a:if ia=4 then ia=0 510 x=x(ia):y=y(ia):gosub 490:xbn=xbu:ybn=ybu:goto 530 520 line xba,yba,xbn,ybn:xba=xbn:yba=ybn:return 530 gosub 520:next a:gcol 4:x=xm:y=ym:gosub 490:circle xbu,ybu,r*mass :gcol 8 540 xba=0:yba=0:x=xs1:y=ys1:gosub 490:xbn=xbu:ybn=ybu:gosub 520 550 gcol7:x=lb:y=0:gosub 490:xba=xbu:yba=ybu:x=xs2:y=ys2:gosub 490:xbn=xbu:ybn=ybu:gosub 520 560 gcol8: x=lb:y=lh:gosub 490:xba=xbu:yba=ybu:x=xs3:y=ys3:gosub 490:xbn=xbu:ybn=ybu:gosub 520 570 x=0:y=lh:gosub 490:xba=xbu:yba=ybu:x=xs4:y=ys4:gosub 490:xbn=xbu:ybn=ybu:gosub 520 108.666398% 59.142812 xm=26.1 ym=30.7211653 der gesuchte abstand=85.7962703 > run in bbc basic sdl and hit ctrl tab to copy from the results window

DA =a + b, AB =c + d (separation at the center of the circle), r - radius of the circle #1 a^2 + c^2 - r^2 = 35^2 #2 b^2 + c^2 - r^2 = 33^2 #3 b^2 + d^2 - r^2 = 85^2 a^2 + d^2 - r^2 = #1 - #2 + #3 = x^2 => x =sqrt( 35^2-33^2+85^2 )

Find X? --- X is in the lower right in the diagram. Easy!

I think I need subtitles. By the way, what kind of genius must you be to solve this?! Maximum respect from me.

It has a shortcut on that given any length Suppose you has a rectangle ABCD or square with circle inside givem that the origin is at any point inside the triangle with unknown radius. Then each verteces of the rectangle was connected with a of tangency thru the circle. Given that there was a missing value of line C (from the vertex of the triangle to the tabgent point of the circle. The solution is always: A²+C²=B²+D² Example A=9 B=x C=4 D=1 So using the formula B= 4 times sqrt of 6 Is this right? Kindly validate please

Bravo! Well done.

Thanks for your job

Practically Marlen Theorem

Greetings from Poland

This problem is very very beautiful.

How do you think, what country has best geometry scholary. Maybe Hungary , Austria

Ура. Спасибо за интересные задачи!

Leaving your fraction with the square root of 2 is not solving this problem.

Nice application and derivation of the British Flag Theorem.

оце я розумію "завдання". 👍

Bhi 16minat mea q hal hoga short turm mea bato

Great

so 85....basically....got it in like 2 seconds.

So similar to Turkey NMO 1st Round

That is almost 85.8!

Gran problema de olimpiadas excelente like

Darimana kamu bisa mendapatkan semua soal ini 🤔

شما استاد بزرگی در ریاضی هستی❤

13:10 23^2 (?)

Awesome

It's the best program.

asnwer=95 isit

I think we can also use trigonometric ratio Ex: x/ sin × : y /sin y 😊

87

i like

哈哈 听印度朋友讲数学

who's hungry

ahm....from the drawn you are assuming that u can make a 90 degree angle line to the center of the circle...but that isn't "evident" on the draw, but i guess it's not possible to solve if that is not the case.

I took one look at this and realized immediately the answer must be 87. No matter the radius of the circle or where it is placed within the rectangle, the lengths of the corners A and C tangents will always equal the lengths of the corners B and D tangents. A + C = B + D 85 + 35 = 120 thus: 120 - 33 = 87 being the length of X (the tangent of corner C) The overly complicated methodology of Math Booster gave the incorrect answer of √7361 which is 85.79627032 Squaring the tangent lengths will not give the correct answer, that is of use only if you are looking for the length of each corner ABCD to the centre of the circle, e.g the hypotenuse of the right angle triangles in the video, but you would also need to know the radius of the circle.

CLAP CLAP CLAP !

Seramalatuhgn enkuwan yanten x lifelglh qerto righthnm alfelg deffar azza eski leqeqegnna braseh zabbiya bicha nurrr ynenen x ante aydelehm mitnegregn kante shi etf ymibeltu ngrewgnalna begeza ejh atqlel bayhon yanten x let me tell you yante x yaahya sega alga silut amed ytebalew is best fit for you ok hule khusr ywurre tilq yelewm ytebalew haq new tilq millionair bileh litakebrew sitl bymnderu ltelkasha were endahyaw sega silkesekes setagegnew yane new ekkhkhkjjhkhkhkhjgeeeey miyasegnhh ene badaf endashagn tefeqdolnal ekhkjkjkhkhk allah zerzuryahn ezaaae allah yargewna ameen😮😮😮

眼看了，但？？？？？？？？？？？？／～～！！＠＃＃＄＄％％

Why? I don't care!

Just a little too tedious, along with the not-so-easy-on-the-ear English pronunciation - the video could be almost half its length without losing everything, and not every note written must also be narrated. The solution is nice and beautiful - but I think you could also give 10-20 seconds before solving, on the way you looked at the problem, and how you attacked it, and show the viewers your "war plan", not just the technical details narrated so indifferently - that one KNOWS you had the solution complete to start with. Give it some life! let children not only see how you solve, but develop an appetite for STRUGGLING with such questions. Thank you anyway.

Bro, it's right there.

87

87