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.
players. A round involves splitting the players into two different teams and having every member of one team play with every member of the other team. A round is called balanced if both teams have an equal number of players. A tournament consists of several rounds at the end of which any two players have played each other. The committee organised a tournament last year which consisted of 
rounds. Prove that the committee can organise a tournament this year with balanced rounds.
and 
to the circumcircle of an acute angled triangle 
meet the tangent at 
at 
and 
respectively. 
meets 
at 
, and 
is the midpoint of 
; 
meets 
at 
, and 
is the midpoint of 
. Prove that 
. Determine, in terms of ratios of side lengths, the triangles for which this angle is a maximum.
satisfying
Prove
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
satisfying![\[f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),\]](http://latex.artofproblemsolving.com/c/6/2/c6237d13efcb3f2ef74cfdbede1a6583ce970e79.png)
For all 
.
+1 w
to 
face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.
questions.
be a point in the interior of 
such that 
, 
, 
, 
, 
, 
. Show that 
.
. I couldn't solve this and was wondering if any of you can find the pattern
. I couldn't solve this and was wondering if any of you can find the pattern
. I couldn't solve this and was wondering if any of you can find the pattern
= (next Catalan number)
= (next Catalan number)
(next Catalan number), 26 + 14 = 40
(next Catalan number), 1238 + 42 = 1280
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
,![\[\frac{2}{3}\left(\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\right)\leq\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+d}+\sqrt{d+a}\leq 2(\sqrt{2}-1)\left(\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\right).\]](http://latex.artofproblemsolving.com/2/d/4/2d48c32aff8f3b3992202f149d835e4ee0982c1b.png)
.
,
,
![\[
1, 2, 5, 40, 1280, ?
\]](http://latex.artofproblemsolving.com/a/b/d/abdadcc9d1511e0bc50630d0764932c6723f6123.png)
![\[
a_n = \frac{381}{8}n^{4} - \frac{1885}{4}n^{3} + \frac{13103}{8}n^{2} - \frac{9313}{4}n + 1115
\]](http://latex.artofproblemsolving.com/3/a/5/3a5bf7857ee63dfefc85efdd2f73062889f38dd4.png)
:![\[
a_1 = \frac{381}{8}(1)^4 - \frac{1885}{4}(1)^3 + \frac{13103}{8}(1)^2 - \frac{9313}{4}(1) + 1115 = 1
\]](http://latex.artofproblemsolving.com/a/4/b/a4b8690544780a4953ff30b7e9bfd650d5c05c64.png)
:![\[
a_2 = \frac{381}{8}(2)^4 - \frac{1885}{4}(2)^3 + \frac{13103}{8}(2)^2 - \frac{9313}{4}(2) + 1115 = 2
\]](http://latex.artofproblemsolving.com/4/0/1/401ac0e6b9e04a3bbb18b93adf3afd46aa401c2a.png)
:![\[
a_3 = \frac{381}{8}(3)^4 - \frac{1885}{4}(3)^3 + \frac{13103}{8}(3)^2 - \frac{9313}{4}(3) + 1115 = 5
\]](http://latex.artofproblemsolving.com/7/f/c/7fc785dbf6d9a0992d3bb5e2011bc4db67ef1c02.png)
:![\[
a_4 = \frac{381}{8}(4)^4 - \frac{1885}{4}(4)^3 + \frac{13103}{8}(4)^2 - \frac{9313}{4}(4) + 1115 = 40
\]](http://latex.artofproblemsolving.com/2/1/b/21b4e2e4d19b6d5c983417f70abf485ad3eb155a.png)
:![\[
a_5 = \frac{381}{8}(5)^4 - \frac{1885}{4}(5)^3 + \frac{13103}{8}(5)^2 - \frac{9313}{4}(5) + 1115 = 1280
\]](http://latex.artofproblemsolving.com/b/d/9/bd9f2528ec68ad7182630c86181261f2eeabad17.png)
:![\[
a_6 = \frac{381}{8}(6)^4 - \frac{1885}{4}(6)^3 + \frac{13103}{8}(6)^2 - \frac{9313}{4}(6) + 1115
\]](http://latex.artofproblemsolving.com/4/a/8/4a8c57a4df9d6246743b76544bd06d424c99c27f.png)
![\[
a_6 = \frac{381}{8} \times 1296 - \frac{1885}{4} \times 216 + \frac{13103}{8} \times 36 - \frac{9313}{4} \times 6 + 1115
\]](http://latex.artofproblemsolving.com/b/a/d/bade9652d9835650aa5c7ae840e9564696946715.png)
![\[
a_6 = 69420
\]](http://latex.artofproblemsolving.com/a/9/c/a9cc1cb8407e92daee7dff378e21ddbde47fd9fd.png)
![\[
\boxed{69420}
\]](http://latex.artofproblemsolving.com/2/f/1/2f16609b47f00089d42077ac2f80a849752c8766.png)
.
