Question on Balkan SL

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0 replies

jlacosta

May 1, 2025

0 replies

2-var inequality

sqing 4

N 2 minutes ago by sqing

Source: Own

Let $a,b>0$ and $\frac{a}{b+1}+ \frac{b}{a+1}\geq 1$ . Prove that
$\frac{a}{a^2+b+1}+ \frac{b}{b^2+a+1} \leq \frac{2}{3}$ Let $a,b>0$ and $\frac{1}{a+1}+ \frac{1}{b+1}\geq 1$ . Prove that
$\frac{a}{a^2+b+1}+ \frac{b}{b^2+a+1} \leq \frac{2}{3}$

4 replies

1 viewing

sqing

43 minutes ago

sqing

2 minutes ago

3-var inequality

sqing 1

N 10 minutes ago by lbh_qys

Source: Own

Let $a,b,c>0$ and $\frac{1}{a+1}+ \frac{1}{b+1}+\frac{1}{c+1} \geq \frac{a+b +c}{2}$ . Prove that
$\frac{1}{a+2}+ \frac{1}{b+2} + \frac{1}{c+2}\geq1$

1 reply

+1 w

sqing

34 minutes ago

lbh_qys

10 minutes ago

Combinatorics from EGMO 2018

BarishNamazov 27

N 15 minutes ago by HamstPan38825

Source: EGMO 2018 P3

The $n$ contestant of EGMO are named $C_1, C_2, \cdots C_n$ . After the competition, they queue in front of the restaurant according to the following rules.
[list]
[*]The Jury chooses the initial order of the contestants in the queue.
[*]Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$ .
[list]
[*]If contestant $C_i$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions.
[*]If contestant $C_i$ has fewer than $i$ other contestants in front of her, the restaurant opens and process ends.
[/list]
[/list]
[list=a]
[*]Prove that the process cannot continue indefinitely, regardless of the Jury’s choices.
[*]Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.
[/list]

27 replies

BarishNamazov

Apr 11, 2018

HamstPan38825

15 minutes ago

Do you have any idea why they all call their problems' characters "Mykhailo"???

mshtand1 1

N 18 minutes ago by sarjinius

Source: Ukrainian Mathematical Olympiad 2025. Day 2, Problem 10.7

In a row, $1000$ numbers $2$ and $2000$ numbers $-1$ are written in some order.
Mykhailo counted the number of groups of adjacent numbers, consisting of at least two numbers, whose sum equals $0$ .
(a) Find the smallest possible value of this number.
(b) Find the largest possible value of this number.

Proposed by Anton Trygub

1 reply