G
Topic
First Poster
Last Poster
powers sums and triangular numbers
gaussious 4
N
by kiyoras_2001
prove 1^k+2^k+3^k + \cdots + n^k \text{is divisible by } \frac{n(n+1)}{2} \text{when} k \text{is odd}
4 replies
complex bashing in angles??
megahertz13 2
N
by ali123456
Source: 2013 PUMAC FA2
Let 
and 
be the incircle and incenter of triangle 
. Let 
, 
, 
be the tangency points of to 
, 
, 
and let 
be the reflection of about . Assume 
intersects the tangents to at and at points 
and 
. Show that 
.
and 
be the incircle and incenter of triangle 
. Let 
, 
, 
be the tangency points of to 
, 
, 
and let 
be the reflection of about . Assume 
intersects the tangents to at and at points 
and 
. Show that 
.2 replies
f(x+y+f(y)) = f(x) + f(ay)
the_universe6626 5
N
by deduck
Source: Janson MO 4 P5
For a given integer 
, find all functions 
such that
![\[f(x+y+f(y))=f(x)+f(ay)\]](//latex.artofproblemsolving.com/7/5/d/75d1ac71fa0c2e53fdfc8e198b109d18e935fe74.png)
holds for all 
.
(Proposed by navi_09220114)
, find all functions 
such that![\[f(x+y+f(y))=f(x)+f(ay)\]](http://latex.artofproblemsolving.com/7/5/d/75d1ac71fa0c2e53fdfc8e198b109d18e935fe74.png)
holds for all 
.(Proposed by navi_09220114)
5 replies
a, b subset
MithsApprentice 19
N
by Maximilian113
Source: USAMO 1996
Determine (with proof) whether there is a subset 
of the integers with the following property: for any integer 
there is exactly one solution of 
with 
.
of the integers with the following property: for any integer 
there is exactly one solution of 
with 
.
19 replies
Hard Polynomial
ZeltaQN2008 1
N
by kiyoras_2001
Source: IDK
Let ?(?) be a polynomial with integer coefficients. Suppose there exist infinitely many integer pairs (?,?) such that
?(?) + ?(?) = 0. Prove that the graph of ?(?) is symmetric about a point (i.e., it has a center of symmetry).
?(?) + ?(?) = 0. Prove that the graph of ?(?) is symmetric about a point (i.e., it has a center of symmetry).
1 reply
Arrangement of integers in a row with gcd
egxa 1
N
by Rohit-2006
Source: All Russian 2025 10.5 and 11.5
Let 
be a natural number. The numbers 
are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the 
GCDs obtained?
be a natural number. The numbers 
are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the 
GCDs obtained?1 reply
Grasshoppers facing in four directions
Stuttgarden 2
N
by biomathematics
Source: Spain MO 2025 P5
Let 
be a finite set of cells in a square grid. On each cell of we place a grasshopper. Each grasshopper can face up, down, left or right. A grasshopper arrangement is Asturian if, when each grasshopper moves one cell forward in the direction in which it faces, each cell of still contains one grasshopper.
[list]
[*] Prove that, for every set, the number of Asturian arrangements is a perfect square.
[*] Compute the number of Asturian arrangements if is the following set:
be a finite set of cells in a square grid. On each cell of we place a grasshopper. Each grasshopper can face up, down, left or right. A grasshopper arrangement is Asturian if, when each grasshopper moves one cell forward in the direction in which it faces, each cell of still contains one grasshopper.[list]
[*] Prove that, for every set
, the number of Asturian arrangements is a perfect square.[*] Compute the number of Asturian arrangements if
is the following set:2 replies
Number Theory
Fasih 0
Find all integer solutions of the equation 
for all x, y 
0, where 
is the prime number.
author @Fasih
for all x, y 
0, where 
is the prime number.author @Fasih
0 replies
Polynomial functional equation
Fishheadtailbody 1
N
by Sadigly
Source: MACMO
P(x) is a polynomial with real coefficients such that
P(x)^2 - 1 = 4 P(x^2 - 4x + 1).
Find P(x).
P(x)^2 - 1 = 4 P(x^2 - 4x + 1).
Find P(x).
1 reply
Bijection on the set of integers
talkon 19
N
by AN1729
Source: InfinityDots MO 2 Problem 2
Determine all bijections 
satisfying
for all integers 
.
Note:
, and for any positive integer 
, 
means 
applied times to , and 
means 
applied times to .
Proposed by talkon
satisfying
for all integers 
.Note:
, and for any positive integer 
, 
means 
applied times to , and 
means 
applied times to .Proposed by talkon
19 replies
