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.
cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from 
to . In a move, Madina is allowed to choose any 
cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?
be a triangle with incenter 
such that 
. The second intersections of 
, 
, and 
with the circumcircle of triangle are 
, 
, and 
, respectively. Lines and 
intersect at 
and lines 
and 
intersect at 
. Suppose the circumcircle of triangles 
and 
intersect again at 
. Lines 
and 
intersect the circumcircle of triangle 
again at 
and 
, respectively.
lies on 
.
grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.
+2 w
such that for every pair of distinct positive integers 
, 
is a divisor of 
.
is divisible by 
for every positive integer . that is not a polynomial in (i.e., there does not exist a polynomial ![\(P(X) \in \mathbb{R}[X]\)](http://latex.artofproblemsolving.com/9/8/4/984bd826ad67d4445e27fd211182f8e3d2a13d5f.png)
such that 
for all 
) and satisfies the given condition?
have the Dirichlet propriety, if 
, 
.
with 
subset of 
and have the Dirichlet propriety.
?
?
with the following property: for all positive divisors 
of , we have 
or 
is prime.
and 
we define 
as the set of all products of an element of and an element of . For example, if 
and 
then 
We call a set 
of positive integers good if there do not exist sets 
of positive integers, each with at least two elements and such that the sets 
and are the same. Prove that the set of perfect powers greater than or equal to 
is good.
, 
, no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form 
, where 
and 
are integers.)
,then prove the following![\[
a\mid c, b\mid c \implies lcm(a,b)\mid c.
\]](http://latex.artofproblemsolving.com/2/7/a/27a4f2847b3aa9791d71736c0ddd61cfb67e579d.png)
And also prove that![\[
c\mid a,c\mid b \implies c\mid gcd(a,b)
\]](http://latex.artofproblemsolving.com/2/6/4/2644ac4b07062dfca3a456011ecacaf96fbd2059.png)
And by the help of God I finish this question.
less than 2024 for which 
and 
are relatively prime for all integers .
so that the sum of its edges' lengths is minumum.
as the set of all numbers of the form 
for some nonnegative 
and 
. Find (with proof) all pairs 
such that 
and 
.
over 
.
as the set of all numbers of the form for some nonnegative and . Find (with proof) all pairs such that and . over .
Prove that![$$a+\frac{1}{2}\sqrt{b}+\frac{1}{3}\sqrt[3]{c}\leq \frac{491}{432}$$](http://latex.artofproblemsolving.com/9/0/9/9094d04344e062c69e3b5557de871a5fa266c828.png)
![$$a+\frac{1}{3}\sqrt{b}+\frac{1}{4}\sqrt[4]{c} \leq \frac{37}{36}+\frac{3}{32\sqrt[3]{2}}$$](http://latex.artofproblemsolving.com/9/e/b/9eb55c582906974d2415bd559005a8d2b57dc260.png)
![$$a+\frac{1}{2}\sqrt{b}+\frac{1}{3}\sqrt[4]{c} \leq \frac{51+2\sqrt[3]{18}}{48}$$](http://latex.artofproblemsolving.com/8/7/9/87938dd3993a37fb696cafd62a6d5aa443314692.png)
