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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
1 viewing
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
An inequality
Rushil 11
N
by SomeonecoolLovesMaths
Source: Indian RMO 1994 Problem 8
If 
are positive real numbers such that 
, prove that ![\[ (1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) . \]](//latex.artofproblemsolving.com/0/b/e/0bedd9ed8362ac5202845f8a4d0d24519423e9b4.png)
are positive real numbers such that 
, prove that ![\[ (1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) . \]](http://latex.artofproblemsolving.com/0/b/e/0bedd9ed8362ac5202845f8a4d0d24519423e9b4.png)
11 replies
Minimum moves to reach 25
lkason 0
Source: Final of the XXI Polish Championship in Mathematical and Logical Games
Mateusz plays a game of erasing-writing on a large board. The board is initially empty.
In each move, he can either:
-- Write two numbers equal to
on the board.
-- Erase two numbers equal to and write instead the numbers 
and 
.
What is the minimal number of moves Mateusz needs to make for the number 25 to appear on the board?
Note: Numbers on the board retain their values; their digits cannot be combined or split.
Spoiler, answer:
In each move, he can either:
-- Write two numbers equal to
on the board.-- Erase two numbers equal to
and write instead the numbers 
and 
.What is the minimal number of moves Mateusz needs to make for the number 25 to appear on the board?
Note: Numbers on the board retain their values; their digits cannot be combined or split.
Spoiler, answer:
moves.0 replies
Reals to reals FE
a_507_bc 7
N
by jasperE3
Source: IMOC 2023 A2
Find all functions 
, such that 
, such that 
7 replies
,
for all 
