for which, for all 
,![\[
f(x) f(yf(x)) + y f(xy) = \frac{f\left(\frac{x}{y}\right)}{y} + \frac{f\left(\frac{y}{x}\right)}{x}
\]](http://latex.artofproblemsolving.com/9/0/c/90c180110402e1a32b70edb2b0a03a28727457d1.png)
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
,![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
be a 
on it, line 
makes a fixed angle with the tangent to 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.
on the spiral, the polar of 
for 
.
.
,
,
divides 
.
,
.
,
.
,
,
.
,
,
.
and permutations.
works. Clearly it satisfies the conditions given. By the Euclidean Algorithm 
.
is not divisible by 5 or 7. However, note that 
,
so that 
.
,
so that 
.
and 
are coprime.
Adding all of the inequalities gives us the desired result. Equality holds only when 
.
Y by Adventure10, Mango247
The 30th Irish Mathematical Olympiad took place on Saturday the 6th of May 2017
This Thread contains all 10 problems asked over 2 papers. Solutions (particularly clear solutions) are missing for some of these problems so feel free to post your solutions!
Answer list :
By LateralUnits
By PepsiCola
by jasonhu4
By a1267ab
This Thread contains all 10 problems asked over 2 papers. Solutions (particularly clear solutions) are missing for some of these problems so feel free to post your solutions!
Answer list :
By LateralUnits
By PepsiCola
by jasonhu4
By a1267ab
This post has been edited 5 times. Last edited by LateralUnits, May 14, 2017, 3:56 PM
for 
and 
as diameters. The two circles passing through 
meet again at 
,
at 
, two circles at 
at 
and the two circles at 
at 
.
and 
is similar to 
,
.
and permutations.
.
denote the number of distinct colorings satisfying this condition.
.
is defined by 
and
Prove that 
is divisible by 
is divisible by 
and the two numbers 
and 
are co-prime?
draw both teams gain 
point, and for other draws 
e
is divided into 
and 
is a right angle. Prove that :
,
When is equality attained?
.
into 
,
.
are quadratic residues mod 
,
is a quadratic residue mod 
for some nonzero 
in the denominator of the expression. 
and 
are both integers mod 
t
mod 
,
is called 
for all positive integers 
-
is just casework and for 
,
,
be the smallest positive integer (if exists) that 
.
.
)
and 
must be zero. We can prove by induction that 
for all 
.
for all 
.
)
be real numbers and 
be positive real numbers. Then
Equality holds only when 
,
for some nonzero 
.
s.t. 
(by symmetry: 
)
with 
)
We know from Fermat's Little Theorem that 
and we know from the definition of the Quadratic Residue and Euler's Criterion that 
- but I don't see how to put it all together.
,
![\[a^3 -\frac{a}{2} - abc = 0 \]](http://latex.artofproblemsolving.com/c/c/8/cc8ab8489238acdbba833a57af560c258e230fd2.png)
![\[b^3 -\frac{b}{2} - abc = 0 \]](http://latex.artofproblemsolving.com/d/2/f/d2f90a02a2bc854feee71bc40dff7a8a754cb79d.png)
![\[c^3 -\frac{c}{2} - abc = 0 \]](http://latex.artofproblemsolving.com/0/6/e/06ee2af858df3e618cbe64314ed4d2a280ce63bc.png)
Adding these equations together (and taking into account that 
)![\[a^3+b^3+c^3-3abc=0\]](http://latex.artofproblemsolving.com/8/7/8/878081ca8ad1b20901fbf3c03153c03d5bfb0d9d.png)
Substituting 
above, 
,
