G
Topic
First Poster
Last Poster
Polynomial with real coefficients
electrovector 2
N
by Tamam
Source: Turkey National Mathematical Olympiad 2020 P5
Find all polynomials with real coefficients such that one can find an integer valued series 
satisfying 
for all 
real numbers.
satisfying 
for all 
real numbers.
2 replies
New factorial function - TT 2009 Senior-A4
Amir Hossein 6
N
by MathMaxGreat
Denote by ![$[n]!$](//latex.artofproblemsolving.com/6/7/f/67f271dbc18a6a34e5787ed1627e76fe8a007a22.png)
the product 
.(
factors in total). Prove that ![$[n + m]!$](//latex.artofproblemsolving.com/c/6/0/c607cb7d028197271d49bd6f21b3aa4bb42b959b.png)
is divisible by ![$ [n]! \times [m]!$](//latex.artofproblemsolving.com/b/a/5/ba5f1a557dfdd8ac96c19b8917e54b77efbe89bf.png)
(8 points)
![$[n]!$](http://latex.artofproblemsolving.com/6/7/f/67f271dbc18a6a34e5787ed1627e76fe8a007a22.png)
the product 
.(
factors in total). Prove that ![$[n + m]!$](http://latex.artofproblemsolving.com/c/6/0/c607cb7d028197271d49bd6f21b3aa4bb42b959b.png)
is divisible by ![$ [n]! \times [m]!$](http://latex.artofproblemsolving.com/b/a/5/ba5f1a557dfdd8ac96c19b8917e54b77efbe89bf.png)
(8 points)
6 replies
is it true c shortlist has 9 problems?
tastymath75025 19
N
by monval
Source: ISL 2019 N6
Let 
and let be a positive integer. Prove that there exists a constant 
such that, if 
satisfies 
, then there exist 
such that 
. (Here 
is the set of positive integers, and 
denotes the greatest integer less than or equal to 
.)
and let be a positive integer. Prove that there exists a constant 
such that, if 
satisfies 
, then there exist 
such that 
. (Here 
is the set of positive integers, and 
denotes the greatest integer less than or equal to 
.)
19 replies
Functional equation extension of angelstt problems
pco 16
N
by jasperE3
Source: f(xf(y))=f(x+y)
Find all functions 
such that ![\[ f(xf(y))=f(x+y)\]](//latex.artofproblemsolving.com/b/c/0/bc02441171db2c0bbbdf02f5739dc65e9f56c840.png)
for all positive reals and 
.
such that ![\[ f(xf(y))=f(x+y)\]](http://latex.artofproblemsolving.com/b/c/0/bc02441171db2c0bbbdf02f5739dc65e9f56c840.png)
for all positive reals and 
.
16 replies
Infinitely many n with a_n = n mod 2^2010 [USA TST 2010 5]
MellowMelon 15
N
by dno1467
Define the sequence 
by 
and, for 
,
![\[a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.\]](//latex.artofproblemsolving.com/a/5/8/a58bbcbb429bcada079bb810b286a7c243f25c41.png)
Prove that there are infinitely many such that 
.
by 
and, for 
,![\[a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.\]](http://latex.artofproblemsolving.com/a/5/8/a58bbcbb429bcada079bb810b286a7c243f25c41.png)
Prove that there are infinitely many
such that 
.
15 replies
Sums of powers with prime exponents, summing a square
Johann Peter Dirichlet 7
N
by Tera_Byte
Source: Problem 4, Brazilian MO, 2012
There exists some integers 
such that
![\[ n^2=\sum_{1 \leq i \leq 2012}{{a_i}^{p_i}} \]](//latex.artofproblemsolving.com/5/5/8/558c219f8ece333bc6abb4fac3f213834266cfe3.png)
where
is the i-th prime (
) and 
for all 
?
such that![\[ n^2=\sum_{1 \leq i \leq 2012}{{a_i}^{p_i}} \]](http://latex.artofproblemsolving.com/5/5/8/558c219f8ece333bc6abb4fac3f213834266cfe3.png)
where
is the i-th prime (
) and 
for all 
?
7 replies
Nonnegative integer sequence containing floor(k/2^m)?
polishedhardwoodtable 9
N
by cursed_tangent1434
Source: ELMO 2024/4
Let be a positive integer. Find the number of sequences 
of integers in the range ![$[0,n]$](//latex.artofproblemsolving.com/1/1/e/11ef480c3077b93f3ac006b998c520ece6100116.png)
such that for all integers 
and all nonnegative integers 
, there exists an integer 
such that 
Andrew Carratu
be a positive integer. Find the number of sequences 
of integers in the range ![$[0,n]$](http://latex.artofproblemsolving.com/1/1/e/11ef480c3077b93f3ac006b998c520ece6100116.png)
such that for all integers 
and all nonnegative integers 
, there exists an integer 
such that 
Andrew Carratu
9 replies
AIME Mock HAM-004 Number Theory: Floor function fun
haihaibaba 3
N
by haihaibaba
Source: own creation
Let 
denote the positive part of 
. Let 
denote the largest integer 
such that 
. Define a function on the integers by
![\[ f(i) = \left \lfloor \left( \frac{2024.2025 - i^2}{ 1 + i^2} \right)^+ \right \rfloor .\]](//latex.artofproblemsolving.com/7/c/b/7cbb6a9519316e0209929c3ecab12f804d61cdcc.png)
Evaluate
![\[
\sum_{i = -\infty}^\infty \left \lfloor \sqrt{f(i)} \right \rfloor.
\]](//latex.artofproblemsolving.com/a/1/6/a16a7c3df5cb00fabe0aac20f2c5293df5e24b31.png)
Note: Give a thumb-up if you like the problem.
denote the positive part of 
. Let 
denote the largest integer 
such that 
. Define a function on the integers by![\[ f(i) = \left \lfloor \left( \frac{2024.2025 - i^2}{ 1 + i^2} \right)^+ \right \rfloor .\]](http://latex.artofproblemsolving.com/7/c/b/7cbb6a9519316e0209929c3ecab12f804d61cdcc.png)
Evaluate![\[
\sum_{i = -\infty}^\infty \left \lfloor \sqrt{f(i)} \right \rfloor.
\]](http://latex.artofproblemsolving.com/a/1/6/a16a7c3df5cb00fabe0aac20f2c5293df5e24b31.png)
Note: Give a thumb-up if you like the problem.
3 replies
Complicated floor definition
pi271828 10
N
by ihatemath123
Source: USA Team Selection Test for IMO 2023, Problem 4
Let 
denote the floor function. For nonnegative integers 
and 
, their bitwise xor, denoted 
, is the unique nonnegative integer such that 
is even for every 
. Find all positive integers such that for any integers 
, we have ![\[ x\oplus ax \neq y \oplus ay. \]](//latex.artofproblemsolving.com/3/b/1/3b16f78e94778d386091ca82a62d5a99f6391f87.png)
Carl Schildkraut
denote the floor function. For nonnegative integers 
and 
, their bitwise xor, denoted 
, is the unique nonnegative integer such that 
is even for every 
. Find all positive integers such that for any integers 
, we have ![\[ x\oplus ax \neq y \oplus ay. \]](http://latex.artofproblemsolving.com/3/b/1/3b16f78e94778d386091ca82a62d5a99f6391f87.png)
Carl Schildkraut
10 replies
USAMO Inequality chain for getting started on the contest
orl 36
N
by dolphinday
Source: USAMO 2006, Problem 1, proposed by Kiran Kedlaya
Let 
be a prime number and let 
be an integer with 
Prove that there exist integers and with 
and
![\[ \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p} \]](//latex.artofproblemsolving.com/9/a/b/9ab00a059f9f1119a6b8a698e5ed713f19163f89.png)
if and only if is not a divisor of 
.
Note: For a real number, let 
denote the greatest integer less than or equal to , and let 
denote the fractional part of x.
be a prime number and let 
be an integer with 
Prove that there exist integers and with 
and![\[ \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p} \]](http://latex.artofproblemsolving.com/9/a/b/9ab00a059f9f1119a6b8a698e5ed713f19163f89.png)
if and only if
is not a divisor of 
.Note: For
a real number, let 
denote the greatest integer less than or equal to , and let 
denote the fractional part of x.
36 replies
