[/quote]
,
be a positive integer, and let 
be nonnegative integers such that 
Prove that![\[
\sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}.
\]](http://latex.artofproblemsolving.com/c/4/9/c49d0ccfef69402d05dc44d721172498c605cf7e.png)
Note: 
for all nonnegative integers 
.
and a polynomial 
with real coefficients of degree 
)![\[
Q(x) \;=\; P(x) \;+\; a\,P'(x) \;+\; a^2\,P''(x) \;+\; \dots \;+\; a^n\,P^{(n)}(x)
\]](http://latex.artofproblemsolving.com/6/5/e/65e5e4eefcadd5d0019b4929b3472cdf203c13aa.png)
has no real roots.
,
the pair 
if for any elements 
we have 
.
be the greatest possible number of elements of a set that escapes the pair 
.
with incircle 
,
be the incenter and 
be the point where 
.
be the point on 
with 
and 
be the orthocenter of 
,
on 
also satisfies 
.
,
,
satisfying 
the following inequality holds:
of the triangle 
.
the orthogonal projection of 
on 
.
and 
are such that 
is a paralellogram, find the locus of point 
when 
.
be the circle tangent to 
passing through 
.
be the circle tangent to 
be the circle tangent to 
passing through 
.
its circumcenter and 
its circumcircle. Let 
and 
be points on 
,
.
,
.
.
and the tangent to 
are concurrent and that 
.
is tangent line to the circumcircle of triangle 
and line 
intersect 
and 
respectively. Line 
and line 
intersect the circumcircle of triangle 
respectively. Prove that 
Y by KevinYang2.71, aidan0626, MathRook7817, Sedro
Let 
be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective.
Note: A function is bijective if for every integer 
, there exists exactly one integer 
such that 
.
be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective.Note: A function
is bijective if for every integer 
, there exists exactly one integer 
such that 
.This post has been edited 1 time. Last edited by imagien_bad, Mar 20, 2025, 12:09 PM
be defined as![\[
d(x) = f(x+1) - f(x).
\]](http://latex.artofproblemsolving.com/9/6/1/9616c1dce04c5e6c8166a5c7183b29a1844b3245.png)
for some integer 
,![\[
\begin{aligned}
g(x+1) &= f(x+1) - (x+1)(f(x+1) - f(x)) \\
&= f(x+1) - (f(x+1) - f(x)) - x(f(x+1) - f(x)) \\
&= f(x) - x(f(x+1) - f(x)) \\
&= g(x),
\end{aligned}
\]](http://latex.artofproblemsolving.com/a/4/8/a48eab04cb0b085b0f1b20c4bfe336aeae36f5ef.png)
which implies that 
is not bijective.
be the minimum of 
.![\[
\begin{aligned}
g(x+1) &= g(x) + d(x) + c \\
&\geq g(x) + m + c \\
&\geq g(x) + 2,
\end{aligned}
\]](http://latex.artofproblemsolving.com/7/1/1/7110c01ce63fee4868242272d0b638edaa597ae2.png)
which implies that 
is bijective for all 
for some constant 
and 
,
for all nonnegative integers 
is strictly increasing.
for some integer 
.
gives![\[ g(n+1)=h(n+1)+m(n+1)=h(n)+mn=g(n), \]](http://latex.artofproblemsolving.com/7/6/7/7670d19785ce7d07f408dfb6412502173d45ab2c.png)
which contradicts our assumption that 
,
,![\[ g(x)\ge h(x)+1\ge h(0)+2, \]](http://latex.artofproblemsolving.com/f/9/5/f95080566f54d188f433e7a72b2fdd7b7af7500e.png)
while for all 
,![\[ g(x)\le h(x)\le h(0), \]](http://latex.artofproblemsolving.com/4/9/a/49a841cad1df45f76ae7ccdb892172de83c77bf7.png)
so 
,
how many mohs do yall think this is?
with 
)
.
,
to all of them, failing injectivity and thus failing bijectivity. Also just replace 
.
is much bigger than 
.
,
.
,
.
,
![\[
\mathcal D := \{f(n+1) - f(n): n\in\mathbb Z\}.
\]](http://latex.artofproblemsolving.com/a/6/b/a6b2ef3b6d0e60074aa239c7786700e0f83685b6.png)
If 
is infinite, then 
isn't injective for any 
.
.
implies 
is bijective.
.
gives 
which contradicts injectivity. 
; this clearly doesn't change the problem. By triangle inequality, the claim yields 
for every 
does not attain 
,
If 
where 
is not bijective.
for some constant 
.
and choose a 
-
,
so setting 
yields 
so 
then 
in 
violates this, contradiction. BOOM.
if and only if 
and thus from here we know that 
,
then 
for all integers 
and so on we get 
for all positive integers 
is not achieved on the positive side but also notice that 
and therefore 
for all negative 
,
.
and let 
.
.
.
.
When 
so 
.
we find 
.
.
which means 
.
.
which implies that when 
.
is linear. Then we can write 
so every integer attained by 
so the problem is clear.
It suffices to show that the RHS can attain infinitely many values. For the sake of a contradiction assume otherwise.
Then connecting this point to all other points 
and taking the union yields a finite set of lines (at least 2). But by the Pigeonhole principle, some line 
on 
can attain as 
Then 
so each increment is of the form 
and making 
so 
where 
as 
for some 
sum of elements of B 
,
(or is any element of 
and 
.
.
.
,
,
be the maximal element of 
produces 
for some 
.
,
.
for all y, and so 
so numbers are skipped.
.
for any pair 
.
gives 
for all 
has no maximum, then we cannot find a minimum value of 
for all x, so 
Then 
,
.
.
means 
