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Integer polynomial commutes with sum of digits
cjquines0 46
N
by cursed_tangent1434
Source: 2016 IMO Shortlist N1
For any positive integer 
, denote the sum of digits of in its decimal representation by 
. Find all polynomials 
with integer coefficients such that for any positive integer 
, the integer 
is positive and 
Proposed by Warut Suksompong, Thailand
, denote the sum of digits of in its decimal representation by 
. Find all polynomials 
with integer coefficients such that for any positive integer 
, the integer 
is positive and 
Proposed by Warut Suksompong, Thailand
46 replies
Weird expression being integer.
MarkBcc168 24
N
by AR17296174
Source: IMO Shortlist 2017 N5
Find all pairs 
of prime numbers which 
and
is an integer.
of prime numbers which 
and
is an integer.
24 replies
Find the value of n - ILL 1990 MEX1
Amir Hossein 3
N
by maromex
During the class interval, 
children sit in a circle and play the game described below. The teacher goes around the children clockwisely and hands out candies to them according to the following regulations: Select a child, give him a candy; and give the child next to the first child a candy too; then skip over one child and give next child a candy; then skip over two children; give the next child a candy; then skip over three children; give the next child a candy;...
Find the value of for which the teacher can ensure that every child get at least one candy eventually (maybe after many circles).
children sit in a circle and play the game described below. The teacher goes around the children clockwisely and hands out candies to them according to the following regulations: Select a child, give him a candy; and give the child next to the first child a candy too; then skip over one child and give next child a candy; then skip over two children; give the next child a candy; then skip over three children; give the next child a candy;...Find the value of
for which the teacher can ensure that every child get at least one candy eventually (maybe after many circles).
3 replies
Self-evident inequality trick
Lukaluce 14
N
by sqing
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let 
, and 
be positive real numbers, such that 
. Prove the inequality
![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](//latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
, and 
be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
14 replies
+2 w
maximum value
moldovan 4
N
by MathIQ.
Source: Austria 1989
Natural numbers 
satisfy 
. Find the maximum value of the product 
satisfy 
. Find the maximum value of the product 
4 replies
R to R, with x+f(xy)=f(1+f(y))x
NicoN9 5
N
by jasperE3
Source: Own.
Find all functions 
such that![\[
x+f(xy)=f(1+f(y))x
\]](//latex.artofproblemsolving.com/4/3/3/433f95836dad1224754d480aff8432ef033a37cb.png)
for all 
.
such that![\[
x+f(xy)=f(1+f(y))x
\]](http://latex.artofproblemsolving.com/4/3/3/433f95836dad1224754d480aff8432ef033a37cb.png)
for all 
.
5 replies
OTIS or MathWOOT 2
math_on_top 14
N
by Mathandski
Hey AoPS community I took MathWOOT 1 this year and scored an 8 on AIME (last year I got a 6). My goal is to make it to MOP next year through USAMO. It's gonna be a lot of work, but do you think that I should do MathWOOT 2 or OTIS? Personally, I felt that MathWOOT 1 taught me a lot but was more focused on computational - not sure how to split computation vs olympiad prep. So, for those who can address this question:
(1) How much compuational vs olympiad
(2) OTIS or MathWOOT 2 and why
(1) How much compuational vs olympiad
(2) OTIS or MathWOOT 2 and why
14 replies
1 viewing
Chain of floors
Assassino9931 1
N
by pi_quadrat_sechstel
Source: Vojtech Jarnik IMC 2025, Category I, P2
Determine all real numbers 
such that
![\[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \]](//latex.artofproblemsolving.com/3/1/9/319c9a2a80c4da5961cd5389af0606fa429ca8e2.png)
for any positive integer .
such that![\[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \]](http://latex.artofproblemsolving.com/3/1/9/319c9a2a80c4da5961cd5389af0606fa429ca8e2.png)
for any positive integer .
1 reply
2015 Taiwan TST Round 2 Quiz 1 Problem 2
wanwan4343 8
N
by Want-to-study-in-NTU-MATH
Source: 2015 Taiwan TST Round 2 Quiz 1 Problem 2
Let 
be the incircle of triangle 
and touches 
at 
. 
meets again at 
. Let 
be 
-excenter, and 
be the midpoint of 
, respectively. Prove that 
are concyclic.
be the incircle of triangle 
and touches 
at 
. 
meets again at 
. Let 
be 
-excenter, and 
be the midpoint of 
, respectively. Prove that 
are concyclic.
8 replies
Prove that
Triangle form by perpendicular bisector
psi241 52
N
by ihatemath123
Source: IMO Shortlist 2018 G5
Let be a triangle with circumcircle 
and incentre 
. A line 
intersects the lines 
, 
, and 
at points , 
, and 
, respectively, distinct from the points , 
, 
, and . The perpendicular bisectors 
, 
, and of the segments , 
, and 
, respectively determine a triangle 
. Show that the circumcircle of the triangle is tangent to .
be a triangle with circumcircle 
and incentre 
. A line 
intersects the lines 
, 
, and 
at points , 
, and 
, respectively, distinct from the points , 
, 
, and . The perpendicular bisectors 
, 
, and of the segments , 
, and 
, respectively determine a triangle 
. Show that the circumcircle of the triangle is tangent to .
52 replies
