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Greatest algebra ever
EpicBird08   13
N 18 minutes ago by Assassino9931
Source: ISL 2024/A2
Let $n$ be a positive integer. Find the minimum possible value of
\[
S = 2^0 x_0^2 + 2^1 x_1^2 + \dots + 2^n x_n^2,
\]where $x_0, x_1, \dots, x_n$ are nonnegative integers such that $x_0 + x_1 + \dots + x_n = n$.
13 replies
EpicBird08
Today at 3:00 AM
Assassino9931
18 minutes ago
Hard sequence
straight   0
19 minutes ago
Source: Own
Consider a sequence $(a_n)_n, n \rightarrow \infty$ of real numbers.

Consider an infinite $\mathbb{N} \times \mathbb{N}$ grid $a_{i,j}$. In the first row of this grid, we place $a_0$ in every square ($a_{0,n} = a_0)$. In the first column of this grid, we place $a_n$ in the $n$-th square ($a_{n,0} = a_n)$.
Next, fill up the grid according to the following rule: $a_{i,j} = a_{i-1,j} + a_{i,j-1}$.

If $\lim_{i \rightarrow \infty} a_{i,j} = \infty$ for all $j = 0,1,...$, does this mean that $a_n = 0$ for all $n$?

Hint?
0 replies
straight
19 minutes ago
0 replies
The inekoalaty game
sarjinius   26
N 40 minutes ago by straight
Source: 2025 IMO P5
Alice and Bazza are playing the inekoalaty game, a two‑player game whose rules depend on a positive real number $\lambda$ which is known to both players. On the $n$th turn of the game (starting with $n=1$) the following happens:
[list]
[*] If $n$ is odd, Alice chooses a nonnegative real number $x_n$ such that
\[
    x_1 + x_2 + \cdots + x_n \le \lambda n.
  \][*]If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that
\[
    x_1^2 + x_2^2 + \cdots + x_n^2 \le n.
  \][/list]
If a player cannot choose a suitable $x_n$, the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players.

Determine all values of $\lambda$ for which Alice has a winning strategy and all those for which Bazza has a winning strategy.

Proposed by Massimiliano Foschi and Leonardo Franchi, Italy
26 replies
+1 w
sarjinius
Today at 3:18 AM
straight
40 minutes ago
BMO Shortlist 2021 G6
Lukaluce   23
N an hour ago by Rayvhs
Source: BMO Shortlist 2021
Let $ABC$ be an acute triangle such that $AB < AC$. Let $\omega$ be the circumcircle of $ABC$
and assume that the tangent to $\omega$ at $A$ intersects the line $BC$ at $D$. Let $\Omega$ be the circle with
center $D$ and radius $AD$. Denote by $E$ the second intersection point of $\omega$ and $\Omega$. Let $M$ be the
midpoint of $BC$. If the line $BE$ meets $\Omega$ again at $X$, and the line $CX$ meets $\Omega$ for the second
time at $Y$, show that $A, Y$, and $M$ are collinear.

Proposed by Nikola Velov, North Macedonia
23 replies
Lukaluce
May 8, 2022
Rayvhs
an hour ago
quadrilateral geo with length conditions
OronSH   9
N an hour ago by player-019
Source: IMO Shortlist 2024 G1
Let $ABCD$ be a cyclic quadrilateral such that $AC<BD<AD$ and $\angle DBA<90^\circ$. Point $E$ lies on the line through $D$ parallel to $AB$ such that $E$ and $C$ lie on opposite sides of line $AD$, and $AC=DE$. Point $F$ lies on the line through $A$ parallel to $CD$ such that $F$ and $C$ lie on opposite sides of line $AD$, and $BD=AF$.

Prove that the perpendicular bisectors of segments $BC$ and $EF$ intersect on the circumcircle of $ABCD$.

Proposed by Mykhailo Shtandenko, Ukraine
9 replies
OronSH
Today at 3:13 AM
player-019
an hour ago
k What happened to 2025 IMO P4 post?
sarjinius   3
N an hour ago by AltruisticApe
Contest is over, why deleted?
3 replies
1 viewing
sarjinius
Today at 4:30 PM
AltruisticApe
an hour ago
Periodic sequence
EeEeRUT   5
N an hour ago by dangerousliri
Source: Isl 2024 A5
Find all periodic sequence $a_1,a_2,\dots$ of real numbers such that the following conditions hold for all $n\geqslant 1$:$$a_{n+2}+a_{n}^2=a_n+a_{n+1}^2\quad\text{and}\quad |a_{n+1}-a_n|\leqslant 1.$$
Proposed by Dorlir Ahmeti, Kosovo
5 replies
EeEeRUT
Today at 3:01 AM
dangerousliri
an hour ago
Bonza functions
KevinYang2.71   51
N an hour ago by Oznerol1
Source: 2025 IMO P3
Let $\mathbb{N}$ denote the set of positive integers. A function $f\colon\mathbb{N}\to\mathbb{N}$ is said to be bonza if
\[
f(a)~~\text{divides}~~b^a-f(b)^{f(a)}
\]for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\leqslant cn$ for all bonza functions $f$ and all positive integers $n$.

Proposed by Lorenzo Sarria, Colombia
51 replies
+1 w
KevinYang2.71
Yesterday at 3:38 AM
Oznerol1
an hour ago
Next term is sum of three largest proper divisors
vsamc   3
N an hour ago by KevinYang2.71
Source: 2025 IMO P4
A proper divisor of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.

The infinite sequence $a_1, a_2, \cdots$ consists of positive integers, each of which has at least three proper divisors. For each $n\geq 1$, the integer $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$.

Determine all possible values of $a_1$.
3 replies
vsamc
an hour ago
KevinYang2.71
an hour ago
2024 International Math Olympiad Number Theory Shortlist, Problem 3
brainfertilzer   10
N 2 hours ago by vsamc
Source: 2024 ISL N3
Determine all sequences $a_1, a_2, \dots$ of positive integers such that for any pair of positive integers $m\leqslant n$, the arithmetic and geometric means
\[ \frac{a_m + a_{m+1} + \cdots + a_n}{n-m+1}\quad\text{and}\quad (a_ma_{m+1}\cdots a_n)^{\frac{1}{n-m+1}}\]are both integers.
10 replies
brainfertilzer
Today at 3:00 AM
vsamc
2 hours ago
a