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.
+3 w
be positive integers and 
be prime. We know that 
and 
. Prove that 
is a perfect square.
be a set of 
residues 
. Prove that there exists a set 
of of residues such that 
contains at least half of all the residues .
+1 w
such that for all 
the following equality holds ![\[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \]](http://latex.artofproblemsolving.com/f/3/a/f3ac860e90d671f9fb592a6bd55c102a3000daec.png)
where 
is greatest integer not greater than 
of people, we define 
to be the least number of tables needed such that each person from the group sits in one of them with no two friends sitting on the same table, and 
to be the least number of tables needed such that each person from the group sits in one of them with no two enemies sitting on the same table.
such that for each group of 
people we have![\[ f(n) \leqslant f(a(G)) + f(b(G)) \quad \text{and} \quad f(2) = 1.\]](http://latex.artofproblemsolving.com/0/3/d/03dfc700f9284152b39e32031c92c33283d65da6.png)
Find all possible values of 
.
is the set of all positive integers.
such that,
for all non-negative integers 
and 
.
is the set of all non-negative integers and 
is the set of all real numbers. cells is a connected figure consisting of equal-sized cells[1].
cells. It is said to be primitive it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur. cells, one with 
cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
such that 
and 
are coprime if and only if 
?
are positive reals. Consider the following polynomial for each 
in 
:
where indices are taken modulo 
, i.e., 
for any 
in 
. Show that it is impossible that each of these polynomials has all its roots real.
. By Vieta we have
Let WLOG 
. Using 
we have 
. Using 
, 
and equation 
in equation 
we have ![\[ 62(r_1+r_2)-32(r_3+r_4)=-200\tag{$5$} \]](http://latex.artofproblemsolving.com/c/1/c/c1c24e8ffcdadc268e0b9cb490b09d39846f2725.png)
Rewriting the equation we have ![\[ (r_1+r_2)(r_3+r_4)+30=k\tag{$6$} \]](http://latex.artofproblemsolving.com/7/5/1/7519a0798962f665f8ed987da0174beb1b77a95a.png)
Solving the equation and 
in terms of variable 
and 
we get 
. Putting these value in the equation 
we get out answer 
.
for some 
. Equating the 
and 
coefficients yields
so 
, and 
.
by vieta's we have 
, and 
are 
Given product of any two roots of this equation is -32. By Vietas Relation we get 
Lets assume 
from product of roots we get 
From these we also get 
Therefore, 
Is the correct answer...
. By Vieta, if , then 
. Additionally, Vieta gives 
But notice that 
again by Vieta. Then, solving we get that 
and 
. Finally, by Vieta, we see have 
and we're done.
, 
, 
and 
. WLOG, let 
, and by Vieta's, we have that 
, meaning that 
. Let 
and 
. By Vieta's, we have that![\[18=a+b+c+d=(a+b)+(c+d)=x+y,\]](http://latex.artofproblemsolving.com/c/9/6/c96f4716894eeeb77ba664f249fdd9fa36438fb1.png)
and![\[-200=abc+abd+acd+bcd=cd(a+b)+ab(c+d)=62x-32y.\]](http://latex.artofproblemsolving.com/1/c/1/1c19e783b755b05bfa2548d7013a00ab624f0b04.png)
Solving the system of equations gives us that 
and 
. Again using Vieta's, we have that![\[k=ac+ad+bc+bd+ab+cd=(a+b)(c+d)+ab+cd=xy-32+62,\]](http://latex.artofproblemsolving.com/e/a/d/eada5dc32b39f0ec39f12eba269545985560097b.png)
or 
, giving us that is , finishing the problem.
, 2) 
, 3) 
, 4) 
. Now we can assume 
= -32 
= 62. We now plug this into 3) and we get 
and after using 1) we get 
. Let 
, we get 
a = 4 
and 
. Now we plug in everything we know in 2) so 
k = 86 the answer is 86.
. Since we know the product of one pair of roots, we can find the product of the other pair of roots as 
. We can split the original polynomial into two quadratic polynomials:![\[
x^4 - 18x^3 + kx^2 + 200x - 1984 = (x^2 + ax - 32)(x^2 + bx + 62)
\]](http://latex.artofproblemsolving.com/9/8/7/9876a06d8427ffc083c9b6b2d3ed976efdde5798.png)
and :![\[
(a, b) = (-4, -14).
\]](http://latex.artofproblemsolving.com/8/7/b/87b47dbc46bf7e1bcb055349f144b96baf31fd8b.png)
, we find:![\[
k = \boxed{86}.
\]](http://latex.artofproblemsolving.com/6/1/a/61a14b71f7ec8ba8dd9aa51fdf1f082c8b028f9b.png)
with 
By Vieta's, we get the following:
The last equation implies that 
and 
Thus,
is 
.
.
*
.
.
and as a result, 
.
.
and 
.
as 
s
.
,
,
,
.
.
,
.![\[a(c+d)+b(c+d)=(a+b)(c+d)\]](http://latex.artofproblemsolving.com/a/8/3/a83ac093e8d99e425c2585b08196d0ea5b8608a4.png)
![\[-32c-32d+62a+62b=-32(c+d)+62(a+b)\]](http://latex.artofproblemsolving.com/e/d/0/ed00fbabdf6f2e16f848b7478c37862dc0ae34e2.png)
and 
.
.
and 
,
Without loss of generality, assume that 
Letting 
and 
,
,
,
and 
.
.
Note that:
We can factor 
as 
.
Note that:
We know that 
so 
.
and 
.
(or vice versa).
.
.
,
.
,
,
.
,
,
![\[abc+abd+acd+bcd=62(a+b)-32(c+d)=-200,\]](http://latex.artofproblemsolving.com/9/f/9/9f99c9235d8e31ea14afe8608bda592f11d42599.png)
and that 
,
and 
.![\[k=ab+ac+ad+bc+bd+cd=-32+62+(a+b)(c+d)=30+4\cdot 14=\boxed{86}.\]](http://latex.artofproblemsolving.com/0/3/4/0344796137ab70bc67e65774bf966ca6338fd3a3.png)
which is equivalent to finding 
.![\[ab(c+d)+(a+b)cd=62(a+b)-32(c+d)=-200\]](http://latex.artofproblemsolving.com/a/0/0/a00a431a73c63a4fa64bbd13cf027911fc6e9731.png)
![\[\implies 31(a+b)-16(c+d)=-100\]](http://latex.artofproblemsolving.com/a/8/0/a8036f2f37e8ac127422da80876dec5ff915ac46.png)
and 
,
,
,
.
![\begin{align*}
\sigma_4=&-1984 \underbrace{\implies}_{ab=-32}cd=62\\
\sigma_3=&-200=ab(c+d)+cd(a+b)=-32(c+d)+62(a+b)\\
\sigma_2=k=&(a+b)(c+d)+\underbrace{ab+cd}_{-32+62=30}\\
\sigma_1=&18 \implies a+b=18-(c+d). \\
-200=& -32\left [18-(a+b)\right ]+62(a+b) \implies a+b=4; \; c+d=14\\
\implies & k=4 \cdot 14+30=\boxed{86}.
\end{align*}](http://latex.artofproblemsolving.com/2/c/a/2cae37a4ecc53401a680e2f76c16f5154de012e5.png)
