Pythagoras...
by Hip1zzzil, May 17, 2025, 3:41 AM
Find the sum of all 
s such that:
There exists two odd positive integers
such that 
s such that:There exists two odd positive integers
such that 
This post has been edited 1 time. Last edited by Hip1zzzil, 6 hours ago
Reason: E
Reason: E
satisfying
Prove
Another Number Theory!
by matinyousefi, Apr 19, 2024, 5:23 PM
Find all natural numbers 
and primes 
that satisfy 
and primes 
that satisfy 
This post has been edited 2 times. Last edited by matinyousefi, Apr 19, 2024, 5:48 PM
IMO 2016 Problem 2
by shinichiman, Jul 11, 2016, 6:38 AM
Find all integers 
for which each cell of 
table can be filled with one of the letters 
and 
in such a way that:
table are each labelled 
to in a natural order. Thus each cell corresponds to a pair of positive integer 
with 
. For 
, the table has 
diagonals of two types. A diagonal of first type consists all cells for which 
is a constant, and the diagonal of this second type consists all cells for which 
is constant.
for which each cell of 
table can be filled with one of the letters 
and 
in such a way that:
- in each row and each column, one third of the entries are
, one third are 
and one third are ; and - in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are , one third are and one third are .
,
and one third are table are each labelled 
to in a natural order. Thus each cell corresponds to a pair of positive integer 
with 
. For 
, the table has 
diagonals of two types. A diagonal of first type consists all cells for which 
is a constant, and the diagonal of this second type consists all cells for which 
is constant.This post has been edited 2 times. Last edited by shinichiman, Jul 11, 2016, 6:40 AM
IMO Shortlist 2014 A2
by hajimbrak, Jul 11, 2015, 9:39 AM
Define the function 
by ![\[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\]](//latex.artofproblemsolving.com/c/b/d/cbd9b412893774283a438296cdeb05b80cc878bf.png)
Let 
and 
be two real numbers such that 
. We define the sequences 
and 
by 
, and 
, 
for 
. Show that there exists a positive integer such that ![\[(a_n - a_{n-1})(b_n-b_{n-1})<0.\]](//latex.artofproblemsolving.com/a/5/c/a5c8844d595bb1186d4804beadd69a3bc1d40ea3.png)
Proposed by Denmark
by ![\[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\]](http://latex.artofproblemsolving.com/c/b/d/cbd9b412893774283a438296cdeb05b80cc878bf.png)
Let 
and 
be two real numbers such that 
. We define the sequences 
and 
by 
, and 
, 
for 
. Show that there exists a positive integer such that ![\[(a_n - a_{n-1})(b_n-b_{n-1})<0.\]](http://latex.artofproblemsolving.com/a/5/c/a5c8844d595bb1186d4804beadd69a3bc1d40ea3.png)
Proposed by Denmark
This post has been edited 3 times. Last edited by djmathman, Jul 24, 2015, 8:08 PM
Reason: formatting
Reason: formatting
n = d2^2 + d3^3
by codyj, Jul 19, 2014, 6:01 PM
Let 
be the divisors of . Find all values of such that 
.
be the divisors of . Find all values of such that 
.3 numbers have their fractional parts lying in the interval
by orl, Aug 10, 2008, 1:01 AM
Let 
be positive integers satisfying the conditions 
and 
Show that there exists a real number 
with the property that all the three numbers 
have their fractional parts lying in the interval ![$ \left(\frac {1}{3}, \frac {2}{3} \right].$](//latex.artofproblemsolving.com/4/2/b/42ba58ddd6032a3176122f1c9b2015cb6f4ca925.png)
be positive integers satisfying the conditions 
and 
Show that there exists a real number 
with the property that all the three numbers 
have their fractional parts lying in the interval ![$ \left(\frac {1}{3}, \frac {2}{3} \right].$](http://latex.artofproblemsolving.com/4/2/b/42ba58ddd6032a3176122f1c9b2015cb6f4ca925.png)
This post has been edited 1 time. Last edited by Amir Hossein, May 11, 2011, 10:11 AM
Reason: Fixed, thanks math154!
Reason: Fixed, thanks math154!
sequence positive
by malinger, Apr 22, 2007, 2:36 AM
The sequence of real numbers 
is defined recursively by ![\[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]](//latex.artofproblemsolving.com/f/4/b/f4b8f95e49de7274cccd8ee498e0b496f31ddfea.png)
Show that 
for all 
.
Proposed by Mariusz Skalba, Poland
is defined recursively by ![\[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]](http://latex.artofproblemsolving.com/f/4/b/f4b8f95e49de7274cccd8ee498e0b496f31ddfea.png)
Show that 
for all 
.Proposed by Mariusz Skalba, Poland
This post has been edited 1 time. Last edited by djmathman, Jun 27, 2015, 12:05 AM
Reason: changed problem statement to match english version of ISL2006
Reason: changed problem statement to match english version of ISL2006
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