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.![$f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$](http://latex.artofproblemsolving.com/c/a/1/ca172b912bf5bf7178cbbd5745fbc0a6a8b27503.png)
such that for all ![$x, y \in \mathbb{Q}[\sqrt{2}]$](http://latex.artofproblemsolving.com/9/f/e/9fede0b2fd22836372b8afc488ccd8a14db5d0c2.png)
,
where ![$\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$](http://latex.artofproblemsolving.com/f/a/c/fac8bf68761534a139d570bf66b1aa5dfb59e8db.png)
.
, a function 
is called interesting if the following two conditions hold:
such that 
., we have 
(where 
denotes the 
-th iteration of 
).
.
for which there is no interesting function
such that Bob can always find the two coins in at most guesses?
such that the following two conditions hold:
and 
satisfying 
, We have 
for all real numbers 
.
and 
, there exist real numbers and , such that and 
.
, 
, 
, and 
lie on a line in that order, and points 
and 
are located outside the line such that 
, 
and 
. Let the circumcircles of triangles 
and 
intersect at points 
and 
, and the circumcircles of triangles 
and 
intersect at points 
and 
. Prove that the lines 
and 
pass through the midpoint of segment 
.
that are relatively prime be called intertwined if among any two divisors of greater than 
, there exists a divisor of 
and among any two divisors of greater than , there exists a divisor of . For example, pair 
is intertwined. for which there exists an intertwined pair such that the product 
is equal to the product of the first prime numbers. such that the product is the product of 
distinct prime numbers. such that the number of divisors of is greater than .
and 
Find the value of 
, and 
be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
are collinear with betweeen and . 
is the circle with diameter 
, and 
is the circle with diameter 
. Another circle touches 
at and meets again at and again at . The line meets again at 
and again at 
. Show that the lines 
and 
meet on the perpendicular to at .
be a strictly increasing sequence of positive integers such that for all positive integers 
![\[d(2025^{a_n}-1)|a_{n+1}.\]](http://latex.artofproblemsolving.com/8/f/d/8fda531266d39c7f11c58d3c1077d2cab3357bea.png)
Show that for any positive real number 
there is a positive integers 
such that 
for all 
.
denotes the number of positive divisors of the positive integer .
be a quadrilateral with 
. Denote by the midpoint of the side , and assume that 
. Let 
be the foot of the perpendicular from the point to the line 
. The line 
meets 
at , and the line 
meets at . Show that 
and 
.
. By one question, we may take three different indices 
and find out the set of numbers 
(unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers 
.
. By one question, we may take three different indices and find out the set of numbers (unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers .
, actually, and, for general instead of 
, I think it's 
.
questions, where is the number of indices which have been included in our questions so far. After we have to stop, it means that we can start asking questions about the other 
indices, the ones we haven't touched so far, and the procedure is repeated. questions suffice, assume first that is even (but 
; for 
we need 
questions, 
don't suffice). We ask questions 
for odd 
between and 
, and after that we ask the question 
. For odd we do something very similar.
numbers after few questions concerning next group? For example: 123, then 234, then 567, then 514?
is the third one.
If there are more questions using numbers already used, just choose one of them. We do this until we no longer can, i.e. until there are no questions using the numbers previously used.
questions, where is the number of indices which have been included in our questions so far. After we have to stop, it means that we can start asking questions about the other indices, the ones we haven't touched so far, and the procedure is repeated.
. Ask 
. Let 
of be asked once. If 
are asked once, then they cannot be asked in the same set hence there are at least sets asked. Consider the bipartite graph whose vertices are 
and the asked sets. If one could find in 
questions, then by looking at degrees, we get 
or 
however, there must be at least sets which contradicts as desired.
Prove that
Prove that
?
questions 
where the 
'
'
'![$[1, 2005].$](http://latex.artofproblemsolving.com/b/a/0/ba07258b82d281889375fb4235b2b902ca265f90.png)
Call a positive integer 
sketchy if it only appears one time among the 
,
for 
,
is exactly equal to 
for all 
Indeed, if it was greater than 
are 
However, this means that the values of 
and 
are indistinguishable, impossible. Therefore, as 
contradiction. Hence, our assumption that we finished in 