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.
of positive integers, with 
prime, such that 
is a perfect square.
be an integer. Two players, Alice and Bob, play the following game on the complete graph 
: They take turns to perform operations, where each operation consists of coloring one or two edges that have not been colored yet. The game terminates if at any point there exists a triangle whose three edges are all colored.
, Alice has a strategy such that, no matter how Bob colors the edges, the game terminates with the number of colored edges not exceeding![\[
\left( \frac{1}{4} - \varepsilon \right) n^2 + n.
\]](http://latex.artofproblemsolving.com/a/9/c/a9cca00e87c643af32eb75a2501b59060f3504fb.png)
+1 w
, 
(where 
). For any given integer 
, consider a graph 
with 
vertices that contains a Hamiltonian cycle 
and has more than 
edges. For which lengths 
must the graph necessarily contain a cycle of length 
?
be a given integer, 
. Consider a graph 
with 
vertices satisfying the condition: for any two non-adjacent vertices 
and 
in graph , the sum of their degrees must satisfy 
. Please answer the following questions and prove your conclusions. is 
satisfying the inequality 
, does graph necessarily contain a cycle of length 
? (The length of a path or cycle refers to the number of edges that make up the path or cycle.)
and graph is connected, can we determine that the length of the longest path in graph , 
?
, is it necessary for graph to have a path of length 
(i.e., a Hamiltonian path)?
be a set of 2025 positive real numbers. For a subset 
, define 
as the median of 
when all elements of are arranged in increasing order, with the convention that 
. Define![\[
P(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ odd}}} M_T, \quad Q(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ even}}} M_T.
\]](http://latex.artofproblemsolving.com/1/d/f/1df2ea8487289111939a53dcc49576fbe6fdef37.png)
Find the smallest real number 
such that for any set of 2025 positive real numbers, the following inequality holds:![\[
P(A) - Q(A) \leq C \cdot \max(A),
\]](http://latex.artofproblemsolving.com/c/d/e/cdea343d11c722a423e857aeb01202863405773e.png)
where 
denotes the largest element in .
, 
, if a positive integer leaves a remainder (the smallest non-negative remainder) that belongs to 
when divided by 19, then that positive integer is called an 
number. If a positive integer leaves a remainder that belongs to 
when divided by 19, then that positive integer is called a 
number., among all its positive divisors, are the numbers of divisors and divisors equal?, are the numbers of divisors less than the numbers of divisors? For which positive integers , are the numbers of divisors greater than the numbers of divisors?
be the Euler's totient function.
, does there exist 
such that for any 
, 
and 
, 
is a non-negative power of 
?
, are there integers 
and 
satisfying:![\[
\varphi(k_i) \in \left ( \frac{x}{a} ,x \right ], i = 1,2; \quad \varphi(k_1) \neq \varphi(k_2).
\]](http://latex.artofproblemsolving.com/6/c/b/6cbe59f15b51439d1202267f521b573a52f1fe67.png)
And these two different 
correspond to the same 
and 
as described in (1), yet 
.
as the number of elements in set 
. For integer , let 
and 
. Compare 
with 
.
such that for all 
, we have 
.
such that, for all integers 
that satisfy 
, the following equality holds:![\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]](http://latex.artofproblemsolving.com/a/9/e/a9edf1a3207d29facc970b6ee288d4fabdd43a4b.png)
denotes the set of integers.)
be a 
board. Some of its cells are colored black in such a way that every 
board of has at most 
black cells. Find the maximum amount of black cells that the board may have.
such that 
is unbounded and![\[2f(m)f(n)-f(n-m)-1\]](http://latex.artofproblemsolving.com/0/f/7/0f77b5addbf8f5e603eb1b6923cdac795cd132e9.png)
is a perfect square for all integer 
, 
and 
with real coefficients and leading coefficient 
, such that each of the polynomials![\[
P(Q(x)), \quad Q(R(x)), \quad R(P(x))
\]](http://latex.artofproblemsolving.com/5/5/4/55478b1313c73a45ed64be4e9f4075ac61475545.png)
has at least one real root, while each of the polynomials![\[
Q(P(x)), \quad R(Q(x)), \quad P(R(x))
\]](http://latex.artofproblemsolving.com/f/f/b/ffb016543e621fec0a5e8a668a9702318ec1f1d7.png)
has no real roots., , has at least one zero.
, WLOG has an odd degree, then it is surjective and so 
must have a zero.
, 
and 
, 
and 
and imply can not reach 
and so 
and since 
, we get 
and imply can not reach 
and so 
and since 
, we get 
and imply can not reach 
and so 
and since 
, we get 
, , can not be true together.
