[/quote]
,
be an acute triangle, 
its orthocenter and 
the center of its nine point circle. Let 
be a point on the parallel through 
to 
such that 
and 
and 
a point on the parallel through 
such that 
and 
and 
are the reflections of 
are the intersections of 
and 
respectively with the circumcircle of 
and 
lies on 
.
,
is the angle bisector of 
.
intersects 
in 
respectively and the line through 
parallel to 
in 
respectively. Prove that point 
lie on a circle.
kids in a class. Is it true that for all positive integers 
it is possible to create several clubs each with 3 kids such that any pair of kids are both present in exactly one club?
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
lamps 
in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp 
and its neighbours (only one neighbour for 
or 
,
)
Prove that there are infinitely many integers 
for which all the lamps will eventually be off.
Prove that there are infinitely many integers 
integers 
,
,
be positive integers such that no one of them is divisible by any other member of the sequence. Then![\[
a_1 \geq 2^k,
\]](http://latex.artofproblemsolving.com/1/f/e/1fecb6d6811bfc4aff7ccefa20412f0a5b860066.png)
where 
is defined by the inequalities![\[
3^k < 2n < 3^{k+1}.
\]](http://latex.artofproblemsolving.com/2/8/f/28f12a76029727a828167f1cb8e118b84d710d62.png)
This estimate for 
is the best possible.
![\[
\max \left( (a_i, a_j) \right) > \frac{38n}{147} - c,
\]](http://latex.artofproblemsolving.com/6/5/d/65df2fa2c65d90d0548cb3c11dd36ea42c4060e6.png)
where 
is a constant independent of 
,
denotes the greatest common divisor of 
and 
.
satisfying 
and ![$ (x + 1)[f(x)]^2 - 1$](http://latex.artofproblemsolving.com/9/c/1/9c1e80d035c923dd13711bff624d8a1a3e6a3b15.png)
is an odd function.
be the inscribed circle of a triangle 
.
and 
be the tangency points of 
and 
and 
intersect 
and 
,
and 
intersects 
at 
,
intersects 
at 
, and the tangent line of 
at 
.
and 
Y by truffle, HamstPan38825, samrocksnature, megarnie
I was recently facing some problems regarding reading solutions. I have read some blogs of Evan Chen on that (1. For solution reading ; 2. General) and have got advice from some other expert people too. What I mainly understood is the following:
It is not necessary to understand every possible solution to a problem. One should look at a solution only if it is
But I have two doubts now:
1. Suppose I see a long solution. Then maybe I should first try to understand it properly, like understand all its arguments precisely. Then I should try to find the key idea behind the solution. But this process might take some time. Sometimes it might even take 20-30 minutes or so. So is this time worth spending? Since in general it is said that "It's going to be very counter-productive if seeing the sol takes too much time, most of the gain still comes from attempting the questions on your own."
2. Not all problems have official solution or some famous people posted solution to them. In that case mainly which solutions should I focus more on? Like the ones which seem shorter in length or ones which are on post #2? Of course the latter classification doesn't seem to be a nice idea. On the first classification, sometimes a long solution might have a short key idea and similarly, a short-looking solution might be harder to grasp.
Two recent examples of doubt 1. are as follows:
The following problem is Sharygin Finals 2018 Grade 10 P4
Following is its official solution.
The solution seems long and complicated. But after I read it, I understood that the overall idea behind it is quite clever. One doubt is the following:
Should I be trying to verify some of the facts written in the solution without proof, for example why the mentioned construction worked and why everything written in the following paragraph is true:
The second example is Sharygin Finals 2018 Grade 9 P8.
Below is the official solution:
It is not necessary to understand every possible solution to a problem. One should look at a solution only if it is
- fairly clean/short
- and idea behind it is very clever/unique
But I have two doubts now:
1. Suppose I see a long solution. Then maybe I should first try to understand it properly, like understand all its arguments precisely. Then I should try to find the key idea behind the solution. But this process might take some time. Sometimes it might even take 20-30 minutes or so. So is this time worth spending? Since in general it is said that "It's going to be very counter-productive if seeing the sol takes too much time, most of the gain still comes from attempting the questions on your own."
2. Not all problems have official solution or some famous people posted solution to them. In that case mainly which solutions should I focus more on? Like the ones which seem shorter in length or ones which are on post #2? Of course the latter classification doesn't seem to be a nice idea. On the first classification, sometimes a long solution might have a short key idea and similarly, a short-looking solution might be harder to grasp.
Two recent examples of doubt 1. are as follows:
The following problem is Sharygin Finals 2018 Grade 10 P4
Sharygin Finals 2018 Grade 10 P4 wrote:
We say that a finite set 
of red and green points in the plane is orderly if there exists a triangle 
such that all points of one colour lie strictly inside and all points of the other colour lie strictly outside of . Let 
be a finite set of red and green points in the plane, in general position. Is it always true that if every 
points in form a orderly set then is also orderly?
of red and green points in the plane is orderly if there exists a triangle 
such that all points of one colour lie strictly inside and all points of the other colour lie strictly outside of . Let 
be a finite set of red and green points in the plane, in general position. Is it always true that if every 
points in form a orderly set then is also orderly?Following is its official solution.
The solution seems long and complicated. But after I read it, I understood that the overall idea behind it is quite clever. One doubt is the following:
Should I be trying to verify some of the facts written in the solution without proof, for example why the mentioned construction worked and why everything written in the following paragraph is true:
The second example is Sharygin Finals 2018 Grade 9 P8.
Sharygin Finals 2018 Grade 9 P8 wrote:
Consider a fixed regular 
-gon of unit side. When a second regular -gon of unit size rolls around the first one, one of its vertices successively pinpoints the vertices of a closed broken line 
as in the figure.
![[asy]
int n=9;
draw(polygon(n));
for (int i = 0; i<n;++i) {
draw(reflect(dir(360*i/n + 90), dir(360*(i+1)/n + 90))*polygon(n), dashed+linewidth(0.4));
draw(reflect(dir(360*i/n + 90),dir(360*(i+1)/n + 90))*(0,1)--reflect(dir(360*(i-1)/n + 90),dir(360*i/n + 90))*(0,1), linewidth(1.2));
}
[/asy]](//latex.artofproblemsolving.com/d/a/4/da4418d91b2cde126bf559427bda331393273495.png)
Let be the area of a regular -gon of unit side, and let 
be the area of a regular -gon of unit circumradius. Prove that the area enclosed by equals 
.
-gon of unit side. When a second regular -gon of unit size rolls around the first one, one of its vertices successively pinpoints the vertices of a closed broken line 
as in the figure.![[asy]
int n=9;
draw(polygon(n));
for (int i = 0; i<n;++i) {
draw(reflect(dir(360*i/n + 90), dir(360*(i+1)/n + 90))*polygon(n), dashed+linewidth(0.4));
draw(reflect(dir(360*i/n + 90),dir(360*(i+1)/n + 90))*(0,1)--reflect(dir(360*(i-1)/n + 90),dir(360*i/n + 90))*(0,1), linewidth(1.2));
}
[/asy]](http://latex.artofproblemsolving.com/d/a/4/da4418d91b2cde126bf559427bda331393273495.png)
Let
be the area of a regular -gon of unit side, and let 
be the area of a regular -gon of unit circumradius. Prove that the area enclosed by equals 
.Below is the official solution:
without even coming close to finding 
in binomial coefficients?" rather than "does anyone have any good recommendations for IMO 3/6 level 
