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.
denote the set of positive integers. For any positive integer 
, a function 
is called -good if 
for all 
. Find all such that there exists a -good function.
be real numbers such that 
Prove that
Let be real numbers such that 
Prove thatLet be real numbers such that 
Prove that
is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
, if and only if 
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
. Let 
be a point on the arc 
(not containing ) such that 
and 
. Let the lines 
and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point 
, and intersects the circumcircle of again at point 
. lies on the circumcircle of 
.
such that for all positive integer 
, we have 
Where 
denotes the number of divisors of . Define 
.
, there exists a positive constant 
such that for all positive integer , the th smallest prime is at most 
.
then![\[\sum |\pm z_1\pm z_2\pm z_3|^2=2^3\sum_{i=1}^3|z_k|^2.\]](http://latex.artofproblemsolving.com/6/d/1/6d1cd28a47704aa0cac2a6c6b572816fc1db790a.png)
Generalize the problem. and be isogonal conjugates with respect to . Let 
and 
be their respective pedal triangles. Let![\[ X_1=P_2Q_3\cap P_3Q_2,\quad X_2=P_1Q_3\cap P_3Q_1,\quad X_3=P_1Q_2\cap P_2Q_1 \]](http://latex.artofproblemsolving.com/7/f/e/7fe3ec4f69ec074635b2202daad9371e3bb8c72d.png)
Prove that the points 
, 
and 
lie on the line 
.
be an equilateral triangle, be a point inside it, and 
be the intersections of 
with the sides of . If 
are the midpoints of , 
, 
, show that there is a triangle with sides 
, 
and 
.
such that 
and 
for any real numbers and 
.
lies inside the rhombus 
. It is known that 
, 
, 
. What can the angle 
be equal?
lie onto diagonals of the rhombus, hence the 2 possible values are 
.
and 
Find minimum 
![$ a,b\in[\frac{1}{2},1] $](http://latex.artofproblemsolving.com/0/d/2/0d24f742e1f7db553b3a8795c061c796f07a897f.png)
.
Let ![$ a,b\in[\frac{1}{2},2] $](http://latex.artofproblemsolving.com/a/8/d/a8d4c8205bac0e34129f582a8f255355eda12e0b.png)
.
be real numbers satisfying![\[x^2 +y^2+z^2 = 1 \]](http://latex.artofproblemsolving.com/d/1/8/d18e31dce3f312d786cca935d497743ba76091ef.png)
Prove that![\[ Q = xy + yz +2xz \le \dfrac{1+\sqrt{3}}{2}\]](http://latex.artofproblemsolving.com/e/5/6/e563c1223d410c64506a55e1ced878df96b7971c.png)
be non-negative real numbers.Prove that :![\[ \sqrt{x^4+y^4 } +(2+\sqrt{2})xy \geq x^2+y^2 \]](http://latex.artofproblemsolving.com/6/2/3/623d50afffd81b2783ff98568791164b53747567.png)