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find all function F: IR->IR such that 
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IMO 2008, Question 2
delegat 63
N
by ezpotd
Source: IMO Shortlist 2008, A2
(a) Prove that
![\[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\]](//latex.artofproblemsolving.com/f/a/6/fa6c46c12e7e6bf0b04f6f9a910df6c2b452fe6a.png)
for all real numbers 
, 
, 
, each different from 
, and satisfying 
.
(b) Prove that equality holds above for infinitely many triples of rational numbers, , , each different from , and satisfying .
Author: Walther Janous, Austria
![\[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\]](http://latex.artofproblemsolving.com/f/a/6/fa6c46c12e7e6bf0b04f6f9a910df6c2b452fe6a.png)
for all real numbers 
, 
, 
, each different from 
, and satisfying 
.(b) Prove that equality holds above for infinitely many triples of rational numbers
, , , each different from , and satisfying .Author: Walther Janous, Austria
63 replies
d(2025^{a_i}-1) divides a_{n+1}
navi_09220114 2
N
by mickeymouse7133
Source: TASIMO 2025 Day 2 Problem 5
Let 
be a strictly increasing sequence of positive integers such that for all positive integers 
![\[d(2025^{a_n}-1)|a_{n+1}.\]](//latex.artofproblemsolving.com/8/f/d/8fda531266d39c7f11c58d3c1077d2cab3357bea.png)
Show that for any positive real number 
there is a positive integers 
such that 
for all 
.
Note. Here
denotes the number of positive divisors of the positive integer 
.
be a strictly increasing sequence of positive integers such that for all positive integers 
![\[d(2025^{a_n}-1)|a_{n+1}.\]](http://latex.artofproblemsolving.com/8/f/d/8fda531266d39c7f11c58d3c1077d2cab3357bea.png)
Show that for any positive real number 
there is a positive integers 
such that 
for all 
.Note. Here
denotes the number of positive divisors of the positive integer 
.
2 replies
Funky function
TheUltimate123 22
N
by jasperE3
Source: CJMO 2022/5 (https://aops.com/community/c594864h2791269p24548889)
Find all functions 
such that for all real numbers 
and 
, ![\[f(f(xy)+y)=(x+1)f(y).\]](//latex.artofproblemsolving.com/d/1/f/d1f369109456464bc8b0102763211802770d983d.png)
Proposed by novus677
such that for all real numbers 
and 
, ![\[f(f(xy)+y)=(x+1)f(y).\]](http://latex.artofproblemsolving.com/d/1/f/d1f369109456464bc8b0102763211802770d983d.png)
Proposed by novus677
22 replies
R+ FE f(f(xy)+y)=(x+1)f(y)
jasperE3 0
Source: p24734470
Find all functions 
such that for all positive real numbers and :
edit oops sorry I misinterpreted that the original poster had solved it, solvable tho
such that for all positive real numbers and :
edit oops sorry I misinterpreted that the original poster had solved it, solvable tho
0 replies
Inequality with x+y+z=1.
FrancoGiosefAG 1
N
by Blackbeam999
Let 
be positive real numbers such that 
. Show that
![\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]](//latex.artofproblemsolving.com/e/a/1/ea1f56cc594862673040411c3ad322cb00cfe904.png)
be positive real numbers such that 
. Show that![\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]](http://latex.artofproblemsolving.com/e/a/1/ea1f56cc594862673040411c3ad322cb00cfe904.png)
1 reply
Find all numbers
Rushil 10
N
by SomeonecoolLovesMaths
Source: Indian RMO 1994 Problem 3
Find all 6-digit numbers 
formed by using the digits 
once each such that the number 
is divisible by 
for 
.
formed by using the digits 
once each such that the number 
is divisible by 
for 
.
10 replies
Some number theory
EeEeRUT 3
N
by MathLuis
Source: Thailand MO 2025 P9
Let 
be an odd prime and 
Assume that
is a bijection and 
is an integer such that 
Show that 
is a multiple of .
be an odd prime and 
Assume that
is a bijection and 
is an integer such that 
Show that 
is a multiple of .
3 replies
Gcd
Rushil 5
N
by SomeonecoolLovesMaths
Source: Indian RMO 1994 problem 5
Let 
be a set of 
positive integers with the property that the product of any two distinct members of will not exceed 1994. Show that there are numbers 
and 
in the set such that the gcd of and is greater than 1.
be a set of 
positive integers with the property that the product of any two distinct members of will not exceed 1994. Show that there are numbers 
and 
in the set such that the gcd of and is greater than 1.
5 replies
Solve the system
Rushil 20
N
by SomeonecoolLovesMaths
Source: 0
Solve the system of equations for real and : 
and : 
20 replies
Angles made with the median
BBNoDollar 1
N
by Ianis
Determine the measures of the angles of triangle 
, knowing that the median 
makes an angle of 
with side 
and an angle of 
with side 
.
, knowing that the median 
makes an angle of 
with side 
and an angle of 
with side 
.1 reply
Find all rationals s.t..
Rushil 12
N
by SomeonecoolLovesMaths
Source: Indian RMO 1994 Problem 7
Find the number of rationals 
such that
(i)
;
(ii) and 
are relatively prime;
(iii)
.
such that(i)
;(ii)
and 
are relatively prime;(iii)
.
12 replies
interesting function equation (fe) in IR
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Source: mine
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or 
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Y by
skellyrah wrote:
find all function F: IR->IR such that
If IR means irrational numbers, and the problem is to find all
such that 
for all 
then setting 
gives a contradictionThis post has been edited 1 time. Last edited by jasperE3, Apr 23, 2025, 9:43 PM
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