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inequality
danilorj   1
N an hour ago by arqady
Let $a, b, c$ be nonnegative real numbers such that $a + b + c = 3$. Prove that
\[ \frac{a}{4 - b} + \frac{b}{4 - c} + \frac{c}{4 - a} + \frac{1}{16}(1 - a)^2(1 - b)^2(1 - c)^2 \leq 1, \]and determine all such triples $(a, b, c)$ where the equality holds.
1 reply
danilorj
Yesterday at 9:08 PM
arqady
an hour ago
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A4 BMO SHL 2024
mihaig   0
an hour ago
Source: Someone known
Let $a\ge b\ge c\ge0$ be real numbers such that $ab+bc+ca=3.$
Prove
$$3+\left(2-\sqrt 3\right)\cdot\frac{\left(b-c\right)^2}{b+\left(\sqrt 3-1\right)c}\leq a+b+c.$$Prove that if $k<\sqrt 3-1$ is a positive constant, then
$$3+\left(2-\sqrt 3\right)\cdot\frac{\left(b-c\right)^2}{b+kc}\leq a+b+c$$is not always true
0 replies
mihaig
an hour ago
0 replies
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Nice one
imnotgoodatmathsorry   5
N 2 hours ago by arqady
Source: Own
With $x,y,z >0$.Prove that: $\frac{xy}{4y+4z+x} + \frac{yz}{4z+4x+y} +\frac{zx}{4x+4y+z} \le \frac{x+y+z}{9}$
5 replies
imnotgoodatmathsorry
May 2, 2025
arqady
2 hours ago
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Equal segments in a cyclic quadrilateral
a_507_bc   4
N 2 hours ago by AylyGayypow009
Source: Greece JBMO TST 2023 P2
Consider a cyclic quadrilateral $ABCD$ in which $BC = CD$ and $AB < AD$. Let $E$ be a point on the side $AD$ and $F$ a point on the line $BC$ such that $AE = AB = AF$. Prove that $EF \parallel BD$.
4 replies
a_507_bc
Jul 29, 2023
AylyGayypow009
2 hours ago
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functional equation
hanzo.ei   3
N 2 hours ago by jasperE3

Find all functions \( f : \mathbb{R} \to \mathbb{R} \) satisfying the equation
\[ (f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}. \]
3 replies
hanzo.ei
Apr 6, 2025
jasperE3
2 hours ago
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Geometry
AlexCenteno2007   0
2 hours ago
Source: NCA
Let ABC be an acute triangle. The altitudes from B and C intersect the sides AC and AB at E and F, respectively. The internal bisector of ∠A intersects BE and CF at T and S, respectively. The circles with diameters AT and AS intersect the circumcircle of ABC at X and Y, respectively. Prove that XY, EF, and BC meet at the exsimilicenter of BTX and CSY
0 replies
AlexCenteno2007
2 hours ago
0 replies
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Inspired by xytunghoanh
sqing   2
N 3 hours ago by sqing
Source: Own
Let $ a,b,c\ge 0, a^2 +b^2 +c^2 =3. $ Prove that
$$ \sqrt 3 \leq a+b+c+ ab^2 + bc^2+ ca^2\leq 6$$Let $ a,b,c\ge 0, a+b+c+a^2 +b^2 +c^2 =6. $ Prove that
$$ ab+bc+ca+ ab^2 + bc^2+ ca^2 \leq 6$$
2 replies
sqing
4 hours ago
sqing
3 hours ago
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Based on IMO 2024 P2
Miquel-point   1
N 3 hours ago by MathLuis
Source: KoMaL B. 5461
Prove that for any positive integers $a$, $b$, $c$ and $d$ there exists infinitely many positive integers $n$ for which $a^n+bc$ and $b^{n+d}-1$ are not relatively primes.

Proposed by Géza Kós
1 reply
Miquel-point
Yesterday at 6:15 PM
MathLuis
3 hours ago
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egmo 2018 p4
microsoft_office_word   29
N 3 hours ago by math-olympiad-clown
Source: EGMO 2018 P4
A domino is a $ 1 \times 2 $ or $ 2 \times 1 $ tile.
Let $n \ge 3 $ be an integer. Dominoes are placed on an $n \times n$ board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some $k \ge 1 $ such that each row and each column has a value of $k$. Prove that a balanced configuration exists for every $n \ge 3 $, and find the minimum number of dominoes needed in such a configuration.
29 replies
microsoft_office_word
Apr 12, 2018
math-olympiad-clown
3 hours ago
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Tangents to a cyclic quadrilateral
v_Enhance   24
N 3 hours ago by hectorleo123
Source: ELMO Shortlist 2013: Problem G9, by Allen Liu
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$.

Proposed by Allen Liu
24 replies
v_Enhance
Jul 23, 2013
hectorleo123
3 hours ago
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integer functional equation
ABCDE   152
N 3 hours ago by pco
Source: 2015 IMO Shortlist A2
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\]holds for all $x,y\in\mathbb{Z}$.
152 replies
ABCDE
Jul 7, 2016
pco
3 hours ago
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confusing inequality
giangtruong13   5
N Apr 20, 2025 by arqady
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{b+c}{a} \geq 2\sqrt{3(ab+bc+ca)}$$
5 replies
giangtruong13
Apr 18, 2025
arqady
Apr 20, 2025
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confusing inequality
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giangtruong13
146 posts
giangtruong13
#1 h  Apr 18, 2025, 2:07 PM
Y by
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{b+c}{a} \geq 2\sqrt{3(ab+bc+ca)}$$
This post has been edited 3 times. Last edited by giangtruong13, Apr 20, 2025, 3:02 PM
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arqady
30251 posts
arqady
#2 h  Apr 18, 2025, 4:14 PM
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giangtruong13 wrote:
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{a}{b+c} \geq 2\sqrt{3(ab+bc+ca)}$$
It's $$\sum_{cyc}\frac{a}{b+c}\geq2\sqrt{\frac{(ab+ac+bc)(a^2b^2+a^2c^2+b^2c^2)}{a^2b^2c^2}}.$$Are you sure that it's true?
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giangtruong13
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giangtruong13
#3 h  Apr 19, 2025, 3:03 PM
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Oh sorry, i write wrongly, i will fix it here :oops_sign:
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giangtruong13
146 posts
giangtruong13
#4 h  Apr 20, 2025, 2:41 AM
Y by
Bummppppp
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giangtruong13
146 posts
giangtruong13
#5 h  Apr 20, 2025, 2:57 PM
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This inequality was from a book by an inactive user $toanmuonmau$
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arqady
30251 posts
arqady
#6 h  Apr 20, 2025, 7:57 PM • 1 Y
Y by kiyoras_2001
giangtruong13 wrote:
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{b+c}{a} \geq 2\sqrt{3(ab+bc+ca)}$$
Because by C-S we obtain:
$$\sum_{cyc}\frac{b+c}{a}=\frac{\sum\limits_{cyc}(a^2b+a^2c)}{abc}=\frac{\sum\limits_{cyc}a^2\sum\limits_{cyc}a-\sum\limits_{cyc}a^3}{abc}=$$$$=\frac{\sqrt{\left(\sum\limits_{cyc}a^2\right)^2\left(\sum\limits_{cyc}a\right)^2}-\sum\limits_{cyc}a^3}{abc}=\frac{\sqrt{\sum\limits_{cyc}(a^4+2a^2b^2)\sum\limits_{cyc}(a^2+2ab)}-\sum\limits_{cyc}a^3}{abc}\geq$$$$\geq\frac{\sqrt{\sum\limits_{cyc}a^4\sum\limits_{cyc}a^2}+2\sqrt{\sum\limits_{cyc}a^2b^2\sum\limits_{cyc}ab}-\sum\limits_{cyc}a^3}{abc}\geq\frac{2\sqrt{\sum\limits_{cyc}a^2b^2\sum\limits_{cyc}ab}}{abc}=2\sqrt{3(ab+ac+bc)}.$$
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