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.
is written on the board. A move consists of adding either , 
or 
to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is 
, then in a move a player can't choose the number , but he can choose either or ). The player who first writes a perfect square on the board loses. Prove that one of the players has a winning strategy and determine who wins in the game.
and 
be the roots to Penchick's favorite polynomial:![\[x^3 - 2x^2 +3x - 4\]](http://latex.artofproblemsolving.com/4/e/6/4e60b82cfbb76fac9785cc7f4ae2b321e4d5f729.png)
Moments later, Penrick secretly tampers with the roots. Calculate:![\[\left( \frac{10}{a+1} \right)^4 + \left( \frac{10}{b+1} \right)^4 + \left( \frac{10}{c+1} \right)^4\]](http://latex.artofproblemsolving.com/1/d/7/1d79078122f848caea9eeb16ee90fe9456f10f5d.png)
identical cars on a circular track. Among all of them, they have just enough gas for one car to complete a lap. Show that there is a car that can complete a lap by collecting gas from the other cars on its way around
, which doesn't have right angle, let 
be its circumcenter. Let 
and 
be orthocenters of triangle 
and 
, respectively. Let 
be circumcenter of triangle 
. If circle 
and circle 
intersect at 
for the second time, prove that lies on 
.
-face dice is thrown three times. Calculate the probability that the number of dice points on all three throws is 
?
and 
with 
. Determine the minimum value of 
. such that the sum of the measures of the angles of the -gon, with 
is less than 
degrees.
and 
, then the number in question is 
(sum of corner number top left) cards and . If the numbers are in , , and 
, the number what is meant is 
or equal to 
(which is obtained by adding the numbers in the upper left corner of each card ,, and ). to with 
based on the rules above. Try making the cards.
is a polynomial with real coefficients such that![\[ P(x)^2 - 1 = 4 P(x^2 - 4x + 1). \]](http://latex.artofproblemsolving.com/b/4/2/b425a938a1dbbac25e0e0793b30c9e58845d45ab.png)
Find .
and 
be altitudes in an acute triangle 
which meet at 
. Suppose that 
meets the circumcircle of at 
and 
such that lies on the shorter arc of 
and lies on the shorter arc of 
. Let 
and 
meet at 
. Show that the circumcircles of 
and 
and the line 
concur.
be a diameter of a circle 
, and let 
be the tangent at 
. Furthermore, let 
be a fixed, positive real, and consider all pairs of points 
and 
lying on , on opposite sides of , such that 
. The lines 
and 
intersect at points 
and 
, respectively. Show that all the lines 
pass through a common point.
such that
for all 
satisfying 
, your right-neighbor’s is , and the next person over is 
, then for every trio in the circle they see
, the midpoint of the 
side is 
and 
is a point on the side such that 
is bisector of the angle 
. Show that is perpendicular to 
prove by Cauchy Inequality?
Prove that
by transforming 
:![\[(x-1)^3 - 2(x-1)^2 +3(x-1) - 4\]](http://latex.artofproblemsolving.com/5/b/8/5b8945a5a9c37d0e1b19e43b5544cb31f4d2fe27.png)
![\[= x^3 - 5x^2 + 10x - 10\]](http://latex.artofproblemsolving.com/5/6/1/5610dfc0a5413582501ec5d7117b5f317c27dc5d.png)
Then, we can construct another polynomial with roots 
by reversing the coefficients as follows:![\[-10x^3 + 10x^2 - 5x + 1\]](http://latex.artofproblemsolving.com/0/c/1/0c13c4050c413a83e875b31e892c53019a23cc4f.png)
We simply need to find the sum of each root raised to the 4th power multiplied by 
.![\[P_2 = S_1P_1 - 2S_2 = 1 - 2(\frac{1}{2}) = 0\]](http://latex.artofproblemsolving.com/d/1/8/d18ec5756c6fbd1ad0b91b987918c85f057d3aa5.png)
![\[P_3 = S_1P_2 - S_2P_1 + 3S_3 = 0 - (\frac{1}{2})(1) + 3(\frac{1}{10}) = -\frac{1}{5}\]](http://latex.artofproblemsolving.com/0/4/5/045d18536c017dbb806d8449b64ba9213584664c.png)
![\[P_4 = S_1P_3 - S_2P_2 + S_3P_1 = (1)(-\frac{1}{5}) - 0 + (\frac{1}{10})(1) = -\frac{1}{10}\]](http://latex.artofproblemsolving.com/0/8/d/08dfcaaa16090700f8e7ca46c75c5828a13bc4d8.png)
Hence, 
and 
.
.
is parallel to 
.
which implies that triangle 
is isosceles with 
.
,
with respect to 
be a point on 
such 
.
and 
.
.
and 
.
such that, 
.
intersect at 
is the angle bisector of 
of isosceles 
,
.
.
.
,
,
is cyclic. So, 
and 
.
.
.
is also cyclic. As, 
,
,
.
.
.
.
,
,
- cyclic with diameter 