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Advanced Floor Function in Number Theory
kisah_sangjuara 10
N
by littleduckysteve
Problem. Prove or disprove that for any 
, the value of 
is always even!
Notes. Here
denotes the largest integer that is less than or equal to 
, the value of 
is always even!Notes. Here
denotes the largest integer that is less than or equal to 
10 replies
floor function
hoangvu1009 2
N
by Pal702004
\item [(a)] For every positive integer 
, define:
![\[ S_n = (5 + 2\sqrt{6})^n + (5 - 2\sqrt{6})^n. \]](//latex.artofproblemsolving.com/1/6/1/161190c006da14c4fa7b912792e3953f3ca3e033.png)
Prove that 
and 
are integers with the same last digit.
\item [(b)] Find the last three digits of:
![\[ \lfloor (\sqrt{3} + \sqrt{2})^{48} \rfloor. \]](//latex.artofproblemsolving.com/6/b/b/6bb617a8b8982cf2b4c8a935a308e053cd5328a3.png)
\item [(c)] Find the last digit of:
![\[ \lfloor (\sqrt{3} + \sqrt{2})^{250} \rfloor. \]](//latex.artofproblemsolving.com/3/d/3/3d39bd55298cdb07c8804a155767be4f17d6c7fe.png)
\end{enumerate}
, define:![\[ S_n = (5 + 2\sqrt{6})^n + (5 - 2\sqrt{6})^n. \]](http://latex.artofproblemsolving.com/1/6/1/161190c006da14c4fa7b912792e3953f3ca3e033.png)
Prove that 
and 
are integers with the same last digit.\item [(b)] Find the last three digits of:
![\[ \lfloor (\sqrt{3} + \sqrt{2})^{48} \rfloor. \]](http://latex.artofproblemsolving.com/6/b/b/6bb617a8b8982cf2b4c8a935a308e053cd5328a3.png)
\item [(c)] Find the last digit of:
![\[ \lfloor (\sqrt{3} + \sqrt{2})^{250} \rfloor. \]](http://latex.artofproblemsolving.com/3/d/3/3d39bd55298cdb07c8804a155767be4f17d6c7fe.png)
\end{enumerate}
2 replies
[PMO25 Qualifying II.8] A Square Can't Be A Floor
kae_3 1
N
by Andyluo
Determine the largest perfect square less than 
that cannot be expressed as 
for some positive real number .
that cannot be expressed as 
for some positive real number .
1 reply
[PMO27 Qualis] III.6 "All floors are not made equal"
BinariouslyRandom 1
N
by Demonstration2121
Consider the equation
![\[ 3 \lfloor x \rfloor^2 = x \lfloor x \rfloor + 21 \{x\}^2, \]](//latex.artofproblemsolving.com/9/c/1/9c1ad1b1c41e8ee0a53a3b125f010b7b2690b6ef.png)
where and 
are the greatest integer part of and the fractional part of , respectively. The sum of all solutions to the equation can be written in the form 
, where 
and 
are relatively prime positive integers. What is the value of 
?
![\[ 3 \lfloor x \rfloor^2 = x \lfloor x \rfloor + 21 \{x\}^2, \]](http://latex.artofproblemsolving.com/9/c/1/9c1ad1b1c41e8ee0a53a3b125f010b7b2690b6ef.png)
where and 
are the greatest integer part of and the fractional part of , respectively. The sum of all solutions to the equation can be written in the form 
, where 
and 
are relatively prime positive integers. What is the value of 
?
1 reply
floor function
hoangvu1009 1
N
by aidan0626
Find all positive integers such that is not a perfect square and 
is divisible by 
.
such that is not a perfect square and 
is divisible by 
.
1 reply
floor function
hoangvu1009 1
N
by mondayleftmebroken
Given a positive integer such that is not a perfect square and is divisible by 
.
Prove that at least one of the numbers
and 
is a perfect square.
such that is not a perfect square and is divisible by 
.Prove that at least one of the numbers
and 
is a perfect square.
1 reply
floor function
hoangvu1009 1
N
by dan09
Let 
.
*) Compute
.
*) Prove that
is an odd natural number for all positive integers .
.*) Compute
.*) Prove that
is an odd natural number for all positive integers .
1 reply
floor function
hoangvu1009 1
N
by dan09
Find all positive integers that satisfy
![\[
\left\lfloor \sqrt{n^4 + 2n^3 + 3n^2 + 2n + 2} \right\rfloor - \left\lfloor \sqrt{16n^2 - 8n + 3} \right\rfloor
\]](//latex.artofproblemsolving.com/3/a/a/3aaea04188789a39cfca52261a41a8d0fe809423.png)
is a prime number.
that satisfy![\[
\left\lfloor \sqrt{n^4 + 2n^3 + 3n^2 + 2n + 2} \right\rfloor - \left\lfloor \sqrt{16n^2 - 8n + 3} \right\rfloor
\]](http://latex.artofproblemsolving.com/3/a/a/3aaea04188789a39cfca52261a41a8d0fe809423.png)
is a prime number.
1 reply
floor function
hoangvu1009 1
N
by dan09
Given two positive integers 
satisfying
![\[
\lfloor (14 + 8\sqrt{3})m \rfloor = \lfloor (14 - 8\sqrt{3})n \rfloor.
\]](//latex.artofproblemsolving.com/9/8/7/987040417aa725788a2d81dd5b0621ec48e50d21.png)
Prove that 
is not divisible by 
.
satisfying![\[
\lfloor (14 + 8\sqrt{3})m \rfloor = \lfloor (14 - 8\sqrt{3})n \rfloor.
\]](http://latex.artofproblemsolving.com/9/8/7/987040417aa725788a2d81dd5b0621ec48e50d21.png)
Prove that 
is not divisible by 
.1 reply
inequalities math
hoangvu1009 0
Given positive numbers 
satisfying 
. Find the minimum value of the expression
![\[
A = \left\lfloor \frac{1}{x} + \frac{1}{y} \right\rfloor + \left\lfloor \frac{1}{y} + \frac{1}{z} \right\rfloor + \left\lfloor \frac{1}{z} + \frac{1}{x} \right\rfloor.
\]](//latex.artofproblemsolving.com/5/a/d/5ad2d607e2b5f794f0618a9766df31bdc6e23f43.png)
satisfying 
. Find the minimum value of the expression![\[
A = \left\lfloor \frac{1}{x} + \frac{1}{y} \right\rfloor + \left\lfloor \frac{1}{y} + \frac{1}{z} \right\rfloor + \left\lfloor \frac{1}{z} + \frac{1}{x} \right\rfloor.
\]](http://latex.artofproblemsolving.com/5/a/d/5ad2d607e2b5f794f0618a9766df31bdc6e23f43.png)
0 replies
