Oh we meet again sweet Aime

by shiningsunnyday, Jun 21, 2016, 4:19 PM

1987 AIME P15 wrote:
Squares $S_1$ and $S_2$ are inscribed in right triangle $ABC$, as shown in the figures below. Find $AC + CB$ if area$(S_1) = 441$ and area$(S_2) = 440$.

[asy]
size(250);
real a=15, b=5;
real x=a*b/(a+b), y=a/((a^2+b^2)/(a*b)+1);
pair A=(0,b), B=(a,0), C=origin, X=(y,0), Y=(0, y*b/a), Z=foot(Y, A, B), W=foot(X, A, B);
draw(A--B--C--cycle);
draw(W--X--Y--Z);
draw(shift(-(a+b), 0)*(A--B--C--cycle^^(x,0)--(x,x)--(0,x)));
pair point=incenter(A,B,C);
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$A$", (A.x-a-b,A.y), dir(point--A));
label("$B$", (B.x-a-b,B.y), dir(point--B));
label("$C$", (C.x-a-b,C.y), dir(point--C));
label("$S_1$", (x/2-a-b, x/2));
label("$S_2$", intersectionpoint(W--Y, X--Z));
dot(A^^B^^C^^(-a-b,0)^^(-b,0)^^(-a-b,b));[/asy]
Solution
Tidbit
1998 IMO P2 wrote:
In a competition, there are $\color[rgb]{0.35,0.35,0.35}a$ contestants and $\color[rgb]{0.35,0.35,0.35}b$ judges, where $\color[rgb]{0.35,0.35,0.35}b \ge 3$ is an odd integer. Each judge rates each contestant as either “pass” or “fail.” Suppose that $\color[rgb]{0.35,0.35,0.35}k$ is a number such that, for any two judges, their ratings coincide for at most $\color[rgb]{0.35,0.35,0.35}k$ contestants. Prove that $\color[rgb]{0.35,0.35,0.35}\dfrac{k}{a} \ge \dfrac{b-1}{2b}$.
Solution
Tidbit
This post has been edited 3 times. Last edited by shiningsunnyday, Jun 22, 2016, 3:30 PM

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4 Comments

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1988 #14 is super nice

Yes and so are you.
#Burned
This post has been edited 1 time. Last edited by shiningsunnyday, Jun 22, 2016, 1:52 AM

by DeathLlama9, Jun 21, 2016, 5:39 PM

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I don't understand the 1998 IMO P2 question, can you explain it?

Errr my sol is the best I can explain it. You can look up the thread to it.
This post has been edited 1 time. Last edited by shiningsunnyday, Jun 22, 2016, 1:52 AM

by Sun13, Jun 21, 2016, 7:59 PM

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Unfortunately I have done all the AIMEs in my all-nighters during school so I mostly remember the solution when I see one. Especially with the lower numbered ones.

Same lol. Do them again though!
This post has been edited 1 time. Last edited by shiningsunnyday, Jun 22, 2016, 3:09 AM

by MathAwesome123, Jun 22, 2016, 2:38 AM

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wait this is 1987

Darn lol
This post has been edited 1 time. Last edited by shiningsunnyday, Jun 22, 2016, 3:30 PM

by DeathLlama9, Jun 22, 2016, 1:46 PM

To share with readers my favorite problem I came across today :) (Shout for contrib.)

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