G
Topic
First Poster
Last Poster
Integers on a cube
Rushil 6
N
by SomeonecoolLovesMaths
Source: Indian RMO 2004 Problem 2
Positive integers are written on all the faces of a cube, one on each. At each corner of the cube, the product of the numbers on the faces that meet at the vertex is written. The sum of the numbers written on the corners is 2004. If T denotes the sum of the numbers on all the faces, find the possible values of T.
6 replies
Tangents to circle concurrent on a line
Drytime 9
N
by Autistic_Turk
Source: Romania TST 3 2012, Problem 2
Let 
be a circle and 
a line in its plane. Let 
be a point on , located outside of . Let 
and 
be the tangents from to , where 
and 
are distinct points on . Let 
and 
be two points on . Lines 
and 
intersect line in two points 
and respectively 
. Lines 
and 
intersect the second time circle in points 
and 
. Prove that the tangents from and to are concurrent on line .
be a circle and 
a line in its plane. Let 
be a point on , located outside of . Let 
and 
be the tangents from to , where 
and 
are distinct points on . Let 
and 
be two points on . Lines 
and 
intersect line in two points 
and respectively 
. Lines 
and 
intersect the second time circle in points 
and 
. Prove that the tangents from and to are concurrent on line .
9 replies
Quadratic
Rushil 8
N
by SomeonecoolLovesMaths
Source: Indian RMO 2004 Problem 3
Let 
and 
be the roots of the equation 
where 
is an odd integer. Let 
Prove that
(A)
is an integer
(B) gcd (
) = 1 .
and 
be the roots of the equation 
where 
is an odd integer. Let 
Prove that
(A)
is an integer(B) gcd (
) = 1 .
8 replies
$n^{22}-1$ and $n^{40}-1$
v_Enhance 6
N
by BossLu99
Source: OTIS Mock AIME 2024 #13
Let denote the sum of all integers 
such that 
and exactly one of 
and 
is divisible by 
. Compute the remainder when is divided by 
.
Raymond Zhu
denote the sum of all integers 
such that 
and exactly one of 
and 
is divisible by 
. Compute the remainder when is divided by 
.Raymond Zhu
6 replies
Parallelogram in the Plane
Taco12 8
N
by lpieleanu
Source: 2023 Canada EGMO TST/2
Parallelogram 
is given in the plane. The incircle of triangle 
has center 
and is tangent to diagonal 
at 
. Let 
be the center of parallelogram . Show that 
and 
are parallel.
is given in the plane. The incircle of triangle 
has center 
and is tangent to diagonal 
at 
. Let 
be the center of parallelogram . Show that 
and 
are parallel.
8 replies
Combinatorial
|nSan|ty 7
N
by SomeonecoolLovesMaths
Source: RMO 2007 problem
How many 6-digit numbers are there such that-:
a)The digits of each number are all from the set
b)any digit that appears in the number appears at least twice ?
(Example:
is valid while 
is not)
[weightage 17/100]
a)The digits of each number are all from the set
b)any digit that appears in the number appears at least twice ?
(Example:
is valid while 
is not)[weightage 17/100]
7 replies
pairs (m, n) such that a fractional expression is an integer
cielblue 0
Find all pairs 
of positive integers such that 
is an integer.
of positive integers such that 
is an integer.
0 replies
the same prime factors
andria 6
N
by MathLuis
Source: Iranian third round number theory P4
are positive integers such that for every natural number the set of prime factors of 
are same. prove that 
.
6 replies
Prove that
Where 
Problem 4 of RMO 2006 (Regional Mathematical Olympiad-India)
makar 7
N
by SomeonecoolLovesMaths
Source: Combinatorics (Box Principle)
A 
square is dissected in to 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Prove that there are always two congruent rectangles.
square is dissected in to 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Prove that there are always two congruent rectangles.
7 replies
Simple FE
oVlad 52
N
by Sadigly
Source: BMO Shortlist 2022, A1
Find all functions 
such that
![\[f(x(x + f(y))) = (x + y)f(x),\]](//latex.artofproblemsolving.com/b/0/6/b0668edacd78de23f7611f47a7972a2dd148a4a0.png)
for all 
.
such that![\[f(x(x + f(y))) = (x + y)f(x),\]](http://latex.artofproblemsolving.com/b/0/6/b0668edacd78de23f7611f47a7972a2dd148a4a0.png)
for all 
.
52 replies
Cool Functional Equation
Warideeb 1
N
by maromex
Find all functions real to real such that
for all reals
.
for all reals
.
1 reply
