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.
such that![\[
f(f(x)+y) = x+f(f(y))
\]](http://latex.artofproblemsolving.com/9/d/7/9d780faa5901646f7035e88400a9e3d8023f04ff.png)
for all 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
, centroid 
and orthocenter 
. Prove that 
is constant.
-
plane, given a parabola 
passing through 3 points 
and 
.
vary in the range of 
, draw the domain swept out by .
is defined by 
and 
Show that there exist infinitely many prime number that divide at least one number in this sequences
be an acute scalene triangle , 
be the orthogonal projection of 
on 
, 
and 
are the midpoints of 
and 
respectively. Let 
and 
are points on the minor arcs 
and 
of the circumcircle of triangle respectively such that 
. Show that the circumcircles of triangles 
and 
are tangent if and only if lies on 
.
be 
given points in the plane, and let 
be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points , i.e., she connects some of the points with edges so that from any point you can reach any other point by moving along the edges.Then, Alice assigns to each vertex 
a non-negative real number 
, for 
, such that 
. Bob then selects a sequence of distinct vertices 
such that 
and 
are connected by an edge for every 
. (Note that the length 
is not fixed and the first selected vertex always has to be 
.) Bob wins if![\[
\frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r;
\]](http://latex.artofproblemsolving.com/1/7/9/1795db8e4a509dd465f6ff462093ae75b04de2b4.png)
otherwise, Alice wins. Depending on 
, determine the largest possible value of 
for which Bob has a winning strategy.
such that![\[
f(x^2 \cdot f(x) + f(y)) = f(f(x^3)) + y
\]](http://latex.artofproblemsolving.com/8/6/5/865b10bbc5ba7721078b1c60eaeede2abfeeb1a2.png)
.
be an acute triangle and let be an interior point for it such that 
. Prove that
When does equality hold?
, with 
, 
is the incenter, 
is the intersection of -excircle and . Point 
lies on the external angle bisector of 
such that and lieas on the same side of the line 
and 
. Point lies on such that 
. Circle 
is tangent to 
and at 
, circle 
is tangent to and at 
and both circles pass through the inside of triangle . if is the Midpoint od the arc , which does not contain , prove that lies on the radical axis of and . be a triangle with incircle 
, which touches sides 
and at points and , respectively. Choose points and on the line 
such that 
and 
. Let be the intersection of lines 
and 
. Define and 
as the intersections of 
and 
with lines 
and 
, respectively. Assume that 
is the intersection of 
and . Prove that 
.
converge?
is a polynomial with integer coefficients, with 
odd. Suppose that 
for all 
Prove that 
is nonzero for all 
and 
define
if 
& 
if 
for a positive integer r, such that
where 
and 
are positive integers.
and respectively can be
is continuous)![\[
F(c) = \frac{1}{c} \int_0^c F(x) \,dx = F(d)
\]](http://latex.artofproblemsolving.com/3/b/f/3bfc29ca2f8e48b786c29f4b32827a7b872f9129.png)
such that 
for ![$x \in [0, 0.5]$](http://latex.artofproblemsolving.com/6/3/4/634744b90be18039a0025700eaaee1d6d21dd32e.png)
,
, so there exists a 
such that 
, ie 
, 
and we are done.
![$F:[0,1] \rightarrow \mathbb{R}$](http://latex.artofproblemsolving.com/1/8/7/187134a27a96f71ad2928bcdb6b458994f3e6073.png)
is a continuous function such that 
, then by Rolle's Theorem 
such that 
. Suppose that 
for some 
. If , then:
.
holds, then we obtain:![$F(1/2) = 2\int_{0}^{1/2} F(x) dx \Rightarrow B = 2\left[\frac{A}{6}(x-1/2)^3+Bx\right]_{0}^{1/2} \Rightarrow B = B +\frac{A}{24} \Rightarrow A=0$](http://latex.artofproblemsolving.com/a/2/b/a2b08a87d49b064cc807fa4675349503c9fc6c84.png)
;
such that 
for constant function .
is a continuous function such that , then by Rolle's Theorem such that . Suppose that for some . If , then:. holds, then we obtain:; such that for constant function .
is continuous) such that for ,, so there exists a such that , ie , and we are done.
is differentiable, then you assume that 
and worst of all, you assume exactly what you are supposed to prove. Please reconsider the way you are contributing, because this type of posts doesn't help anyone, but actually creates confusions.
that you get becomes fixed, so you cannot choose it to be 
has minimum and maximum, so let ![$a= \sup \{ s : F(s)= \min_{x \in [0,1]} F(x) \}$](http://latex.artofproblemsolving.com/2/a/5/2a52abe4328b1bb4c7d2efffca7db0477595862f.png)
and ![$b= \sup \{ s : F(s)= \max_{x \in [0,1]} F(x) \}.$](http://latex.artofproblemsolving.com/a/4/8/a4854f7fd4a4c7c30aaeb0f5f31b78934b3c2ae1.png)
By continuity, we get ![$F(a)=\min_{x \in [0,1]} F(x)$](http://latex.artofproblemsolving.com/f/2/d/f2d96cbb7dd166cca597801367b97fc79c3183a7.png)
and ![$F(b)=\max_{x \in [0,1]} F(x).$](http://latex.artofproblemsolving.com/9/2/7/927b844225e072a4204706d90de456ac1dfd1124.png)
then ![$F(0)= \min_{x \in [0,1]} F(x),$](http://latex.artofproblemsolving.com/5/e/8/5e85dbeb11ad94ba4118e7ee0ebba6bb60cf84da.png)
so ![$F(1)= \min_{x \in [0,1]} F(x),$](http://latex.artofproblemsolving.com/2/0/c/20cbead3d7063cc0c90d46a09916b90427193101.png)
contradicting the definition of 
Therefore 
and similarly we get 
![$G:(0,1] \to \mathbb{R},$](http://latex.artofproblemsolving.com/7/9/c/79c58b06d34f657174c1fbc92ee3757381521af6.png)
We have 
If 
then 
which combined with 
and the continuity of 
implies ![$F(t)=F(a), \forall t \in [0,a].$](http://latex.artofproblemsolving.com/0/9/7/0977356964063a315214dfcf875002a083d0e0d2.png)
In this case we can choose any 
for example 
and if 
then ![$F(t)=F(b), \forall t \in [0,b],$](http://latex.artofproblemsolving.com/0/9/9/09962ff194f42ad46348f43c8baf610597455b9b.png)
and we can choose 
and 
then by IVT there is 
such that 
![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
such that 
,
such that:![\[
F(c) = \frac{1}{c} \int_0^c F(x) \,dx
\]](http://latex.artofproblemsolving.com/8/f/e/8fea4d84bc98b6f4953e19976cfa554945a37ea6.png)
