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or 
tile.
be an integer. Dominoes are placed on an 
board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some 
such that each row and each column has a value of 
. Prove that a balanced configuration exists for every , and find the minimum number of dominoes needed in such a configuration.
be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer 
such that 
. such that 
be the circumcircle of triangle 
with center 
, and the 
inmixtilinear circle is tangent to 
at 
respectively. 
is the intersection of 
and 
and 
is the intersection of 
and . Prove that the isogonal conjugate of lies on the line passing through the midpoint of 
and .
![$f \colon [0,1] \to \mathbb{R} $](http://latex.artofproblemsolving.com/2/1/8/21859d37ed035abd07d0be112f49d52598fa477e.png)
be a differentiable function such that its derivative is an integrable function on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
, and 
. Prove that ![\[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]](http://latex.artofproblemsolving.com/2/3/e/23e33eed39fc1e20486bbb86261d3566e8fd2017.png)
be an odd prime number, and be an odd number not divisible by . Consider a field 
be a field with 
elements, and 
be the set of elements of 
, whose order is not in the multiplicative group 
. Prove that the polynomial 
has at least coefficients equal to 
.
and 
be a continuous function for which there exists an antiderivative 
, such that 
, for any 
, and
holds for any 
. Prove that 
for all .
be a positive integer, ![$g \in \mathbb{R}[X]$](http://latex.artofproblemsolving.com/0/1/e/01e7ebcebcb9f16935716ee859f7acfaa66c29d4.png)
, 
be a polynomial with all of its roots being real, and a polynomial function such that 
for any . Prove that 
for all .
be two matrices such that 
. Prove that:
is odd, then 
;
, then .
, the following 
statements are equivalent:
is differentiable, with continuous first derivative.
and for any two sequences 
, convergent to 
, such that 
for any positive integer , the sequence 
is convergent.
matrices which has only 
and for their entries?
, with their second derivative being continuous, such that the following holds for all 
: ![\[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]](http://latex.artofproblemsolving.com/0/2/9/029a935e28ca0a9b36e65702689cca91eeb8f614.png)
![$f\colon[0,1]\rightarrow \mathbb{R}$](http://latex.artofproblemsolving.com/f/d/b/fdb1fb0216ab9d17007b1b7b03d20d1060b88531.png)
be a continuous function. Suppose that for each 
, the function ![\[f_t\colon[0,1-t]\rightarrow\mathbb{R}, f_t(x)=f(x+t)-f(x)\]](http://latex.artofproblemsolving.com/4/f/c/4fc6a66d0600a999c000136a26ea23dfcc269cc3.png)
has an unique global minimum point, which we will denote by 
. Prove that if 
, then 
is constant zero.
and be positive integers. The square in the 
th row and 
th column of an -by- grid contains the number 
. For which and is it possible to select squares from the grid, no two in the same row or column, such that the numbers contained in the selected squares are exactly 
?
on the board gets replaced by 
, where is the largest prime divisor of .
on the board gets replaced by , where is the largest prime divisor of .
, then the prime factor won't decrease.
, then the prime factor won't decrease., then still in the case it doesn't decrease, and actually might increase.
seconds we get number 
then 
and so power of two will be at 
or 
that power of two can not be reached in seconds
is number after second, then has a prime 
for every 
where is largest prime than 
,so 
and 
is increasing sequence and so 
or 
which is contradiction
seconds is enough because 
seconds to get power of two for any 
digits number
,
.
is greater than 
is a multiple of 
.
,
seconds, a power of two will appear on the board for the first time.
after one second and 
after two seconds.
seconds or