[/quote]
,
and let 
and 
be permutations of 
.
such that 
for all 
in 
is the number of possible pairs of permutations 
,
and 
Prove that 
be real numbers such that 
.
Prove that 
and 
Prove that
be pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
that satisfy the following property: if 
,
,
,
are all the positive divisors of 
with 
,
divides 
for every 
.
be an acute-angled triangle with 
.
be the circumcircle of 
of 
.
meets 
at 
and meets 
.
at 
.
by 
.
.
meets line 
.
circles arranged in an equilateral triangular shape such that for each 
,
,
,
row contains exactly 
circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of 
,
such that in each Japanese triangle there is a ninja path containing at least 
,
,
,![\[ P(x)=x^k+c_{k-1}x^{k-1}+\dots + c_1 x+c_0, \]](http://latex.artofproblemsolving.com/1/9/5/195d7c158bb2b4ccf4480d5d13c22583d3f4d711.png)
where 
,
,
are non-negative integers, such that ![\[ P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k} \]](http://latex.artofproblemsolving.com/e/9/7/e9725833bb88b0b27fdbfef3b9111b6fe6dad372.png)
for every integer 
.
be interior points of 
,
,
,
Let 
and 
meet at 
let 
and 
meet at 
and let 
and 
meet at 
is scalene, then the three circumcircles of triangles 
all pass through two common points.
medals cutoff?
specifies the greatest common divisor of 
and 
.
Y by
Welcome to the 1st idk12345678 Math Contest.
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You have 4 hours. You do not have to prove your answers.
Post \signup username to sign up. Post your answers in a hide tag and I will tell you your score.*
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*I mightve done them wrong feel free to ask about an answer
This post has been edited 1 time. Last edited by idk12345678, Apr 19, 2025, 12:47 AM
