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.
st sees the other three, the 
nd sees the 
rd and 
th, and the rd sees only the th. The devil has 
hats, numbered from to , he puts one hat on each wise man, and hides the extra 
hats. After that, each wise man (in turn: first the first, then the second, etc.) loudly calls a number, trying to guess the number of his hat. The numbers mentioned should not be repeated. When all the wise men have spoken, they take off their hats and check which one of them has guessed. Can the sages to act in such a way that at least three of them knowingly guessed?
+1 w
be an acute triangle with circumcircle 
. Let 
be the midpoint of 
and let 
be the midpoint of 
. Let 
be the foot of the altitude from 
and let 
be the centroid of the triangle . Let 
be a circle through and that is tangent to the circle at a point 
. Prove that the points 
and 
are collinear.
be the incentre of acute-angled triangle . Let the incircle meet 
, and at 
, and 
respectively. Let line 
intersect the circumcircle of the triangle at 
and 
, such that 
lies between 
and . Prove that 
.
be an acute-angled triangle with circumcircle . A circle 
is internally tangent to at and also tangent to 
at . Let and intersect at and respectively. Let 
and 
be points on line such that 
is the midpoint of 
and 
is the midpoint of 
. Lines 
and 
meet at 
and intersect again at and 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.
.
the circumcircle of the acute triangle . Point is the midpoint of the arc of not containing . Circle centered at is tangent to the segment at point . Tangents to the circle passing through point intersect line at points and 
such that points 
lie on the line in that order. Circle 
is tangent to the segments 
and 
and to the circle at point . Circle 
is tangent to the segments 
and 
and to the circle at point . Lines 
and 
intersect at point . Prove that 
.
be the set of all positive integers. Find all functions 
such that 
divides 
for all positive integers 
and 
.
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each , determine the largest number of dominoes that can be placed in this way.
or 
. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)
and 
are given. Losyash likes some of the cells of the 
checkerboard. In addition, he is interested in any checkered rectangle with a perimeter of 
, the upper-left corner of which coincides with the upper-left corner of the board (there are such rectangles in total). Given and 
, determine whether Losyash can color each cell he likes in one of colors so that in any rectangle of interest to him the number of cells of any two colors differ by no more than .
is the midpoint of side of an acute—angled triangle . The point 
is symmetric to the orthocenter relative to its circumcenter. The point 
inside triangle is such that 
. Prove that 
.
is given. Suppose that 
, 
and 
are different positive integer numbers such that 
. Prove that 
.
to 
are arranged in some order in the cells of the 
strip. Let's call a flip an operation that takes two arbitrary cells of a strip and swaps the numbers written in them, but only if the larger of these numbers is located to the left of the smaller one. A flop is a set of several flips that do not contain common cells that are executed simultaneously. (For example, a simultaneous flip between the 2nd and 8th cells and a flip between the 5th and 101st cells.) Prove that there exists a sequence of 
flops such that for any initial arrangement, applying this sequence of flops to it will result in the numbers being ordered from left to right in ascending order.
, a line 
to tangent , and another circle disjoint from such that and lie on opposite sides of . The tangents to from a variable point on meet at 
and 
. Prove that, as varies over , the circumcircle of 
is tangent to two fixed circles.
![\[ p = \frac{2ab}{a + b} \]](http://latex.artofproblemsolving.com/0/f/4/0f4297ec59fe97c92fda75af4cc835be6b9a2507.png)
, ![\[ p, a, b \in \mathbb{C} \]](http://latex.artofproblemsolving.com/b/4/4/b4499970ac49d770a810d7345fedef7773cd03f5.png)
be a parallelogram such that 
Let and be the midpoints of and 
respectively. Assuming that 
is a cyclic quadrilateral, find 
intersect at , 
and 
. I draw 
perpendicular to .
, thus 
and finally 
.
,
.
.
,