[/quote]
,
be an acute triangle with circumcircle 
.
be the midpoint of 
and let 
be the midpoint of 
.
be the foot of the altitude from 
and let 
be the centroid of the triangle 
be a circle through 
.
and 
are collinear.
be the incentre of acute-angled triangle 
,
,
respectively. Let line 
intersect the circumcircle of the triangle at 
and 
,
lies between 
and 
.
is internally tangent to 
at 
and 
be points on line 
is the midpoint of 
and 
is the midpoint of 
.
and 
meet at 
and intersect 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.
.
such that points 
lie on the line 
is tangent to the segments 
and 
and to the circle 
is tangent to the segments 
and 
and to the circle 
and 
intersect at point 
.
be the set of all positive integers. Find all functions 
such that 
divides 
for all positive integers 
and 
.
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
are given. Losyash likes some of the cells of the 
checkerboard. In addition, he is interested in any checkered rectangle with a perimeter of 
,
,
.
is symmetric to the orthocenter 
inside triangle 
.
.
is given. Suppose that 
,
and 
are different positive integer numbers such that 
.
.
n
r
t
hats, numbered from 
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?
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.
to tangent 
and 
.
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)
point 
and 
.
passes from the intersection of diagonals of 
.
Y by GeoKing, itslumi
On 
squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.
squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.This post has been edited 1 time. Last edited by Kei0923, Feb 11, 2023, 11:22 AM
tetronominos such that each but the central square are covered.
is not possible, consider the following coloring:
is the lower left corner, colour all 
fields with both coordinates even red and all 
tetrominos but then it is impossible to cover all 
![[asy]
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0));
draw((1,0)--(1,5));
draw((2,0)--(2,5));
draw((3,0)--(3,5));
draw((4,0)--(4,5));
draw((0,1)--(5,1));
draw((0,2)--(5,2));
draw((0,3)--(5,3));
draw((0,4)--(5,4));
fill((1,1)--(2,1)--(2,2)--(1,2)--cycle, black);
fill((1,3)--(2,3)--(2,4)--(1,4)--cycle, black);
fill((3,3)--(4,3)--(4,4)--(3,4)--cycle, black);
fill((3,1)--(4,1)--(4,2)--(3,2)--cycle, black);
[/asy]](http://latex.artofproblemsolving.com/e/d/5/ed523383a151dc7858c31be9aba9dc63ec0c6c22.png)
![[asy]
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0));
draw((1,0)--(1,5));
draw((2,0)--(2,5));
draw((3,0)--(3,5));
draw((4,0)--(4,5));
draw((0,1)--(5,1));
draw((0,2)--(5,2));
draw((0,3)--(5,3));
draw((0,4)--(5,4));
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle, blue);
fill((0,2)--(0,3)--(1,3)--(1,2)--cycle, blue);
fill((0,4)--(0,5)--(1,5)--(1,4)--cycle, blue);
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle, blue);
fill((2,2)--(3,2)--(3,3)--(2,3)--cycle, blue);
fill((2,4)--(3,4)--(3,5)--(2,5)--cycle, blue);
fill((4,0)--(5,0)--(5,1)--(4,1)--cycle, blue);
fill((4,2)--(5,2)--(5,3)--(4,3)--cycle, blue);
fill((4,4)--(4,5)--(5,5)--(5,4)--cycle, blue);
[/asy]](http://latex.artofproblemsolving.com/0/2/2/022e835a6bf0ebc5991a2b7de3fc605524d7020d.png)
![[asy]
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0));
draw((1,0)--(1,1)--(0,1));
draw((0,3)--(1,3)--(1,2)--(2,2)--(2,0));
draw((2,2)--(2,4)--(1,4)--(1,5));
draw((5,0)--(4,0)--(4,1)--(3,1)--(3,5));
draw((4,5)--(4,4)--(5,4));
draw((3,3)--(4,3)--(4,2)--(5,2));
[/asy]](http://latex.artofproblemsolving.com/d/5/d/d5dd0831f1b90f124251bbc4963d66c765187315.png)
![[asy]
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0));
draw((0,1)--(1,1)--(1,2)--(3,2)--(3,1)--(2,1)--(2,0));
draw((3,2)--(5,2));
draw((5,1)--(4,1)--(4,0));
draw((1,5)--(1,4)--(0,4));
draw((0,3)--(2,3)--(2,4)--(3,4)--(3,5));
draw((2,3)--(4,3)--(4,4)--(5,4));
[/asy]](http://latex.artofproblemsolving.com/9/8/6/9861d75ee61886019065bbbb921043ecfc4f257c.png)
rectangle that has two corners missing
