[/quote]
,
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)
from the interval ![$[-2,10]$](http://latex.artofproblemsolving.com/d/6/8/d68464eed4c2fe38f134793ffca4a671614f8634.png)
such that their sum is 
.
.
.
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)
,
,
and 
,
is the intersection point of the diagonals 
and 
.
and radius 
intersects 
and 
.
is tangent to the circles with diameters 
and 
.
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)
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
.
be a point on the arc 
and 
.
and 
intersect at point 
.
be the reflection of 
and 
intersect at point 
,
.
.
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 
,
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
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.
.
,
Y by lian_the_noob12, Adventure10, Mango247
let 
an equilateral triangle with circum circle 
let
a point on arc 
( point 
is on the other side )
pass a tangent line
through point such that 
and 
let
the center of the circle
prove that
an equilateral triangle with circum circle 
let
a point on arc 
( point 
is on the other side )pass a tangent line
through point such that 
and 
let
the center of the circle prove that
This post has been edited 2 times. Last edited by acupofmath, Sep 3, 2015, 5:18 PM
be the midpoints of 
respectively. then by angle casing we get 
so it's enough to prove that 
.
arc 
equals 
and it's clear that arc 
lies inside 
hence 
.
and 
,
.
and 
,
is the polar of point 
is the polar of point 
wrt 
since 
because the point 
(this circle passes through points 
