Question on Balkan SL

by Fmimch, Apr 30, 2025, 12:13 AM

Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!

algebra

combinatorics

geometry

number theory

IMO Shortlist

hard inequalities

by pennypc123456789, Apr 30, 2025, 12:12 AM

Given $x,y,z$ be the positive real number. Prove that

$\frac{2xy}{\sqrt{2xy(x^2+y^2)}} + \frac{2yz}{\sqrt{2yz(y^2+z^2)}} + \frac{2xz}{\sqrt{2xz(x^2+z^2)}} \le \frac{2(x^2+y^2+z^2) + xy+yz+xz}{x^2+y^2+z^2}$

inequalities

Good divisors and special numbers.

by Nuran2010, Apr 29, 2025, 4:52 PM

$N$ is a positive integer. Call all positive divisors of $N$ which are different from $1$ and $N$ beautiful divisors.We call $N$ a special number when it has at least $2$ beautiful divisors and difference of any $2$ beautiful divisors divides $N$ as well. Find all special numbers.

number theory

Inspired by old results

by sqing, Apr 29, 2025, 12:29 PM

Let $a,b>0 , a^2+b^2+ab+a+b=5 .$ Prove that
$\frac{ 1 }{a+b+ab+1}+\frac{6}{a^2+b^2+ab+1}\geq \frac{7}{4}$ $\frac{ 1 }{a+b+ab+1}+\frac{1}{a^2+b^2+ab+1}\geq \frac{1}{2}$ $\frac{41}{a+b+2}+\frac{ab}{a^3+b^3+2} \geq \frac{21}{2}$

This post has been edited 1 time. Last edited by sqing, Yesterday at 12:49 PM

inequalities proposed

algebra

inequalities

Vasc = 1?

by Li4, Apr 26, 2025, 1:33 PM

Find all integer tuples $(a, b, c)$ such that
$(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1.$
Proposed by Li4, Untro368, usjl and YaWNeeT.

number theory

inequalities

hard problem

by Cobedangiu, Apr 21, 2025, 1:51 PM

Let $a,b,c>0$ and $a+b+c=3$ . Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$

inequalities

Counting graph theory

by MathSaiyan, Mar 17, 2025, 2:00 PM

Let $m$ and $n$ be positive integers. For a connected simple graph $G$ on $n$ vertices and $m$ edges, we consider the number $N(G)$ of orientations of (all of) its edges so that, in the resulting directed graph, every vertex has even outdegree.
Show that $N(G)$ only depends on $m$ and $n$ , and determine its value.

graph theory

combinatorics

H not needed

by dchenmathcounts, May 23, 2020, 11:00 PM

Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$ ). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar.

Robin Son

This post has been edited 2 times. Last edited by v_Enhance, Oct 25, 2020, 6:01 AM
Reason: backdate

\sqrt{(1^2+2^2+...+n^2)/n}$ is an integer.

by parmenides51, Mar 26, 2020, 5:08 PM

Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.

This post has been edited 1 time. Last edited by parmenides51, Mar 28, 2020, 3:42 AM

Intersection of circumcircles of MNP and BOC

by Djile, Apr 8, 2013, 3:13 PM

Let $M$ , $N$ and $P$ be midpoints of sides $BC, AC$ and $AB$ , respectively, and let $O$ be circumcenter of acute-angled triangle $ABC$ . Circumcircles of triangles $BOC$ and $MNP$ intersect at two different points $X$ and $Y$ inside of triangle $ABC$ . Prove that $\angle BAX=\angle CAY.$