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Combo resources
Fly_into_the_sky 1
N
by Fly_into_the_sky
Ok so i never did combinatorics in my life :oops: and i am willing to be able to do P1/P4 combos (or even more)
So yeah how can i start from scratch?
Remark:i don't want compuational combo resources :noo:
So yeah how can i start from scratch?
Remark:i don't want compuational combo resources :noo:
1 reply
Polynomial Application Sequences and GCDs
pieater314159 46
N
by cursed_tangent1434
Source: ELMO 2019 Problem 1, 2019 ELMO Shortlist N1
Let 
be a polynomial with integer coefficients such that 
, and let 
be an integer. Define 
and 
for all integers 
. Show that there are infinitely many positive integers 
such that 
.
Proposed by Milan Haiman and Carl Schildkraut
be a polynomial with integer coefficients such that 
, and let 
be an integer. Define 
and 
for all integers 
. Show that there are infinitely many positive integers 
such that 
.Proposed by Milan Haiman and Carl Schildkraut
46 replies
such that 
coprime, 
and 
.
![\[
\text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\
f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right)
\]](http://latex.artofproblemsolving.com/2/a/7/2a73a5e58eab4a994fbd83160fb79a0b00152951.png)
Serbian selection contest for the IMO 2025 - P6
OgnjenTesic 16
N
by JARP091
Source: Serbian selection contest for the IMO 2025
For an 
table filled with natural numbers, we say it is a divisor table if:
- the numbers in the
-th row are exactly all the divisors of some natural number 
,
- the numbers in the
-th column are exactly all the divisors of some natural number 
,
-
for every 
.
A prime number
is given. Determine the smallest natural number , divisible by , such that there exists an divisor table, or prove that such does not exist.
Proposed by Pavle Martinović
table filled with natural numbers, we say it is a divisor table if:- the numbers in the
-th row are exactly all the divisors of some natural number 
,- the numbers in the
-th column are exactly all the divisors of some natural number 
,-
for every 
.A prime number
is given. Determine the smallest natural number , divisible by , such that there exists an divisor table, or prove that such does not exist.Proposed by Pavle Martinović
16 replies
equal segments on radiuses
danepale 8
N
by zuat.e
Source: Croatia TST 2016
Let 
be an acute triangle with circumcenter 
. Points 
and 
are chosen on segments 
and 
such that 
. If 
is the midpoint of the arc 
and 
is the midpoint of the arc 
, prove that 
.
be an acute triangle with circumcenter 
. Points 
and 
are chosen on segments 
and 
such that 
. If 
is the midpoint of the arc 
and 
is the midpoint of the arc 
, prove that 
.
8 replies
Inequality
SunnyEvan 8
N
by arqady
Let 
, 
, 
be non-negative real numbers, no two of which are zero. Prove that :
, 
, 
be non-negative real numbers, no two of which are zero. Prove that :
8 replies
Inequality conjecture
RainbowNeos 2
N
by RainbowNeos
Show (or deny) that there exists an absolute constant 
that, for all and positive real numbers 
, there is
![\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]](//latex.artofproblemsolving.com/4/8/b/48b9fc4af5a20de7fd9dfc45a74d5aa27e615e46.png)
that, for all and positive real numbers 
, there is![\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]](http://latex.artofproblemsolving.com/4/8/b/48b9fc4af5a20de7fd9dfc45a74d5aa27e615e46.png)
2 replies
2- player game on a strip of n squares with two game pieces
parmenides51 2
N
by Gggvds1
Source: 2023 Austrian Mathematical Olympiad, Junior Regional Competition , Problem 3
Alice and Bob play a game on a strip of 
squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of 
squares.
IMAGE
The players alternate. In each move, they advance their own game piece by one or two squares in the direction of the opponent’s piece. The piece has to land on an empty square without jumping over the opponent’s piece. Alice makes the first move with her own piece. If a player cannot move, they lose.
For which can Bob ensure a win no matter how Alice plays?
For which can Alice ensure a win no matter how Bob plays?
(Karl Czakler)
squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of 
squares.IMAGE
The players alternate. In each move, they advance their own game piece by one or two squares in the direction of the opponent’s piece. The piece has to land on an empty square without jumping over the opponent’s piece. Alice makes the first move with her own piece. If a player cannot move, they lose.
For which
can Bob ensure a win no matter how Alice plays?For which
can Alice ensure a win no matter how Bob plays?(Karl Czakler)
2 replies
