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:![\[
a^2 + b^2 \mid a^3 + b^3
\]](http://latex.artofproblemsolving.com/a/1/d/a1ddbd20c854e291da5670de449229d1664fa56c.png)
be an isosceles triangle with 
. The circle 
, passing through 
and 
, intersects segment 
at 
. The circle 
is tangent to 
at and passes through 
. Let 
and 
be the midpoints of segments and 
, respectively. The line 
intersects and at points 
and 
, respectively, where and are the intersections closer to . Prove that 
.
such that for any 
:
, then either travels [list] units vertically (up or down) and 
units horizontally (left or right); or units horizontally (left or right) and units vertically (up or down)., the ant can choose to go to one of eight possible points.
and 
, the ant can travel from 
to 
using at most 
moves.
, the sides and 
are parallel and the angles 
and 
are acute. Show that it is possible to divide the triangle into 4 disjoint triangle 
and the triangle 
into 4 disjoint triangles 
such that the triangles 
and 
are congruent for all 
.
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
, let 
be the reflection of in sides 
respectively. Define a function 
such that 
is the circumcircle of 
. Is 
injective? is called injective if for any 
, 
board, some squares are painted red. Let be the number of rows that have an odd number of red squares and let be the number of columns that have an odd number of red squares. Find all possible values of 
. For each value found, give a example of how the board can be painted.
sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following:
join two vertices with a segment, without cutting another already marked segment; or delete a vertex that does not belong to any marked segment.
be the set of positive real numbers. Determine all functions 
such that ![\[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]](http://latex.artofproblemsolving.com/e/7/1/e71b6e0d0b858eb46b81149f1e6be8c41e13d301.png)
for every 
.
friends, 
people are seated around a circular table in a fixed order. Ali places apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction). such that after some number of steps, the situation reaches a point where each person has exactly one apple.
be an isosceles triangle, and point 
in its interior such that 
.
.
measure of the 
. By the Law of Sines in 
and 
we have that
works
. Which is clear, since 
. 
is define in unique way by the condition of the problem, so we conclude that answer is 
be the point on the extension of 
such that 
. Thus, 
.
, and 
.
is an equilateral triangle and so 
.
is twice of 
, so is the circumcenter of 
.
, so 
. Finally, 
.
be an isosceles triangle, and point in its interior such that ..
. Point is on the bicector of angle 
.
are agnle bisectors in triangle 
, so point 
in the incenter of triangle .
.
![\[\angle ABC=\angle DBA+\angle DBC=80\]](http://latex.artofproblemsolving.com/3/d/7/3d75a25394e642ffb5968ae0c9021bdc6358e9bd.png)
and since it's isoceles we're done.
and 
(sides where B lies are equal) (sides where A lies are equal) (sides where C lies are equal)
and since it's isoceles we're done. and , will be outside triangle, so there is no question!
such that: 
.
