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.
from the interval ![$[-2,10]$](http://latex.artofproblemsolving.com/d/6/8/d68464eed4c2fe38f134793ffca4a671614f8634.png)
such that their sum is 
. Prove that the sum of their squares is smaller than 
.
. Show that there exists 
such that![\[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\geqslant (a_1x_1+a_2x_2+\ldots +a_nx_n)^2.\]](http://latex.artofproblemsolving.com/4/d/2/4d269e5b682be6778e89237aa91da5e0399042a7.png)
, inscribed in the circle 
, satisfies 
and 
, and 
is the intersection point of the diagonals 
and 
. The circle with center 
and radius 
intersects in two points 
and 
. Prove that the line 
is tangent to the circles with diameters 
and 
.
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 
.
be non-negative real numbers such that 
Prove that
such that 
and 
for any real numbers 
and 
.
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.
and 
as ![\[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\]](http://latex.artofproblemsolving.com/4/d/5/4d5049d5c3b26757a25c5dd34462c5462228b83d.png)
where each 
or 
is a digit between 0 and 9. Prove that there exist at least 1000 values of 
between 
and 
such that 
.
of positive integers with 
is called good if we can colour each integer red, blue, green or purple, in such a way that of each 
consecutive integers at least one is coloured red;
of each 
consecutive integers at least one is coloured blue;
of each 
consecutive integers at least one is coloured green;
of each 
consecutive integers at least one is coloured purple.
. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was 
, what is the fifth term of the sequence?
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
such that :
for all prime numbers 
.
for all positive integers 
such that 
is a set of 
points in the plane that are not all collinear. Prove that contains three points that form a triangle whose circumcentre is not a point in .
- such points, that all another points lies on line 
but not on segment or in the one side from line . There are some points that are not on line . Let point is not on and has minimal circumradius among triangles with side , and 
is circumcenter of . But then circumradius of 
is less then circumradius of 
- contradiction.
is . As 
lie on the perpendicular bisector of and is on same side as , then it means that 
less 
where is altitude of .
and 
so 
so the circumradius of is not necessary the great but the idea remain good : Fortunately ,except in the case of equilateral triangle, one of the angles has less than 
we consider its opposite side instead of . and the circumcenter and the circumcenter of which ends the proof
satisfying the equation![\[
(f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/2/5/e/25eed911f303ed7cc4e03765a2f943ad4641c451.png)
