AoPS Academy
rows of coins. There is 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on, up to coins in the th row. Initially, all of the coins show heads (H). Carley plays a game in which, on each turn, she chooses three mutually adjacent coins and flips these three coins over. To win the game, all of the coins must be showing tails (T) after a sequence of turns. An example game with 4 rows of coins after a sequence of two turns is shown. for which it is possible to win the game starting with rows of coins.
satisfying 
, prove that![\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]](http://latex.artofproblemsolving.com/f/e/8/fe85cf52b883c214806ab837a2eb25cebb511d7b.png)
such that there exists a unique integer 
such that 
with the following property:![\[ n!\mid a^n + 1
\]](http://latex.artofproblemsolving.com/e/2/3/e23d39aab685888993d355d8c93d4e6475f53487.png)
(use in a problem for junior) to practice. Can anyone share to me?
be a convex quadrilateral with a point 
in the interior and points 
on the sides 
respectively such that 
are parallelograms. Prove that ![\[\sqrt{S_{BQOP}}+\sqrt{S_{DSOR}}\le \sqrt{S_{ABCD}}.\]](http://latex.artofproblemsolving.com/9/8/d/98da4ba2d1beb64298f1e3940ca829ef34f32725.png)
? This problem is elementary.
? This problem is elementary.
![\[ S_{BPOQ}+S_{DSOR}\le S_{APOS}+S_{CROQ} \]](http://latex.artofproblemsolving.com/b/a/4/ba4a2ea3c8637736d0d661b5ab910f251e9f50a6.png)
be a convex quadrilateral with a point in the interior and points on the sides respectively such that are parallelograms. Prove that ![\[ \sqrt{a^2+x^2}+\sqrt{b^2+y^2}\le\sqrt{(a+b)^2+(x+y)^2}\]](http://latex.artofproblemsolving.com/b/2/a/b2a2c42606c97a8430bd1ba44bbb0782cb3674f8.png)
Can we use that?
