The Proof is Trivial!

Problem 71**

Find all $x,y \in \mathbb{C}$ such that $x^y = y^x$.

I quite like it The only real disadvantage is trying to get a compass that doesn't screw up your circle.

Seems like you are alone in this

Initial thoughts:

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I'm fairly sure no such functions exist.

Not ones that aren't multivalued anyway.

Solution 71
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Make sure you have a look at STEP and MAT, assuming you apply to Oxford / Cambridge / Imperial / Warwick.

Also, you'll be pleased to know that there is no Euclidean plane geometry of the kind that's useful for IMO at university!

I quite like it

My profound compliments!




Ceva, Menelaus, Lehmus, Brianchon, Pappus, Feuerbach - I am scared!
On one hand, projective geometry is somewhat interesting; on the other hand, barycentric coordinates and complex numbers, for example, are such a computational disaster.
Not to mention the problems where we have to deal with hundreds of circles..




I assume you relish doing pure geometry. Here you are:

Let $k$ be the circumscribed circle of the triangle $ABC$. $D$ is an arbitrary point on the segment $AB$. Let $I$ and $J$ be the centers of the circles which are tangent to the side $AB$, the segment $CD$ and the circle $k$. Assume that $A$, $B$, $I$, and $J$ are concyclic. The excircle of the triangle $ABC$ is tangent to the side $AB$ in the point $M$. Then $M \equiv D$.

Note that Thebault's theorem was not allowed without proof.

I have looked at some STEP papers. The problems seem quite accessible;

Btw when I said I liked geometry; that doesn't necessarily mean I'm a whizz at it!

For the love of god you have to apply. That's just :O

Solution 71

Not sure if this is 100% correct but it seems ok

Let $x=ay$

we then get $(ay)^y=y^{ay}$

rearanging gives $a^y=(y^y)^{a-1}$

so

$yln(a)=(a-1)ln(y^y)=(a-1)yln(y)$

so

$y=a^{\frac{1}{a-1}}$
and
$x=a^{\frac{a}{a-1}}$

This should only fail if x or y are 0, but that cannot give a solution unless we accept x=y=0 (which would be given by a=0 anyway).

I have looked at some STEP papers. The problems seem quite accessible; one surprising thing is that help is given for a considerable number of questions.
I would like to know what is your opinion - are there any substantial differences between the colleges within, for example, the University of Cambridge (I mean academic differences)?

Problem 72: *

Bit of an effort: - But hey ho.
Prove by considering: $\ I =\displaystyle\int^1_0 \dfrac{x^4(1 - x^4)}{1+x^2}\ dx$ that $\pi < \frac{22}{7}$

Find $\ I = \displaystyle\int \sqrt{1+e^x}\ dx$