be the orthocenter of acute triangle 
,
be the foot of the altitude from 
to 
,
be the reflection of 
.
intersects line 
and 
.
.
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 bijective.
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 
.
.
with the property that 
implies 
is bijective.
.
gives 
which contradicts injectivity. 
so that 
; 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 
where for all 
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 
be the set of all non-zero 
where 
as 
for some 
sum of elements of B 
,
is some subset of 
(or is any element of 
and 
.
.
given by all numbers of the form 
.
,
,
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 
