AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be the set of integers. Determine all functions 
such that, for all integers 
and 
, 
Proposed by Liam Baker, South Africa
, there exists some positive integer 
that makes the following equation have no integer root 
.
and the segment is 3. Find the possible minimum value of 
such that for any polynomial 
of degree 
with real coefficients, there exists a polynomial 
of degree at most with real coefficients such that the graphs of 
and 
intersect at exactly points?
be a convex quadrilateral. If 
are the midpoints of the segments 
, and 
, respectively, and there is also a point inside the quadrilateral such that, 
and 
. Show that 
.
that satisfy the following condition: for all 
and 
relatively prime to 
we have ![\[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]](http://latex.artofproblemsolving.com/e/8/5/e85c5b249ac8e75b45a525b8786414798a4921fe.png)
by the expression
and 1.
Find the range of such that 
such that 
is an integer.
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
sports, each awarding one gold medal to a winner. athletes participate, each in all sports. There are also experts, each of whom must predict the number of gold medals each athlete will win. In each prediction, the medal counts must be non-negative integers summing to . An expert is called competent if they correctly guess the number of gold medals for at least one athlete. What is the maximum number such that the experts can make their predictions so that at least of them are guaranteed to be competent regardless of the outcome?
has exactly two real roots, one of which is 
where 
and 
are integers, 
and are relatively prime, and 
Find 
to the system of equations![\[\begin{aligned}
\begin{cases}
x^2+y^2-xy-3x+3=0,\\
x^2+y^2+z^2-xy-yz-2zx-3x+3=0.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/4/a/b/4ab8e94e2e39be219c80264aa5ea23456b74b3e8.png)
intersect at point 
. Prove that the tangent at point to the circle circumscribed around the triangle 
is parallel to .
intersect at point . Prove that the tangent at point to the circle circumscribed around the triangle is parallel to .
intersect at point . Prove that the tangent at point to the circle circumscribed around the triangle is parallel to .
Prove that
and 
Prove that
