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No More than √㏑x㏑㏑x Digits
EthanWYX2009   4
N an hour ago by tom-nowy
Source: 2024 April 谜之竞赛-3
Let $f(x)\in\mathbb Z[x]$ have positive integer leading coefficient. Show that there exists infinte positive integer $x,$ such that the number of digit that doesn'r equal to $9$ is no more than $\mathcal O(\sqrt{\ln x\ln\ln x}).$

Created by Chunji Wang, Zhenyu Dong
4 replies
EthanWYX2009
Mar 24, 2025
tom-nowy
an hour ago
J
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Inequality with 4 variables
bel.jad5   2
N an hour ago by mihaig
Source: Own
Let $a$,$b$,$c$ $d$ positive real numbers. Prove that:
\[ \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} \geq 4+\frac{8(a-d)^2}{(a+b+c+d)^2}\]
2 replies
bel.jad5
Sep 5, 2018
mihaig
an hour ago
J
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Queue geo
vincentwant   3
N an hour ago by Ilikeminecraft
Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $Y, Z$ be the feet of the altitudes from $B, C$ to $AC, AB$ respectively. Let $D$ be the midpoint of $BC$. Let $\omega_1$ be the circle with diameter $AD$. Let $Q\neq A$ be the intersection of $(ABC)$ and $\omega$. Let $H$ be the orthocenter of $ABC$. Let $K$ be the intersection of $AQ$ and $BC$. Let $l_1,l_2$ be the lines through $Q$ tangent to $\omega,(AYZ)$ respectively. Let $I$ be the intersection of $l_1$ and $KH$. Let $P$ be the intersection of $l_2$ and $YZ$. Let $l$ be the line through $I$ parallel to $HD$ and let $O'$ be the reflection of $O$ across $l$. Prove that $O'P$ is tangent to $(KPQ)$.
3 replies
vincentwant
Wednesday at 3:54 PM
Ilikeminecraft
an hour ago
J
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Old hard problem
ItzsleepyXD   1
N an hour ago by ItzsleepyXD
Source: IDK
Let $ABC$ be a triangle and let $O$ be its circumcenter and $I$ its incenter.
Let $P$ be the radical center of its three mixtilinears and let $Q$ be the isogonal conjugate of $P$.
Let $G$ be the Gergonne point of the triangle $ABC$.
Prove that line $QG$ is parallel with line $OI$ .
1 reply
ItzsleepyXD
Apr 25, 2025
ItzsleepyXD
an hour ago
J
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Existence of a solution of a diophantine equation
syk0526   5
N an hour ago by cursed_tangent1434
Source: North Korea Team Selection Test 2013 #6
Show that $ x^3 + x+ a^2 = y^2 $ has at least one pair of positive integer solution $ (x,y) $ for each positive integer $ a $.
5 replies
syk0526
May 17, 2014
cursed_tangent1434
an hour ago
J
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Inequality with 3 variables
sqing   0
an hour ago
Source: Own
Let $ a,b,c\geq 0 ,a^3b^3+b^3c^3+c^3a^3+2abc\geq 1 . $ Prove that$$a+b+c\geq 2 $$Let $ a,b,c\geq 0 ,a^3b^3+b^3c^3+c^3a^3+6abc\geq 9 . $ Prove that$$a+b+c\geq 2\sqrt 3 $$Let $ a,b,c\geq 0 ,a^3b+b^3c+c^3a+6abc\geq 9 . $ Prove that$$a+b+c\geq 3 $$Let $ a,b,c\geq 0 ,a^3b+b^3c+c^3a+3abc\geq 3 . $ Prove that$$a+b+c\geq \frac{4}{\sqrt 3} $$
0 replies
sqing
an hour ago
0 replies
J
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Inequality with 3 variables and a special condition
Nuran2010   5
N 2 hours ago by sqing
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
For positive real numbers $a,b,c$ we have $3abc \geq ab+bc+ca$.
Prove that:

$\frac{1}{a^3+b^3+c}+\frac{1}{b^3+c^3+a}+\frac{1}{c^3+a^3+b} \leq \frac{3}{a+b+c}$.

Determine the equality case.
5 replies
Nuran2010
Apr 29, 2025
sqing
2 hours ago
J
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Chain of floors
Assassino9931   0
2 hours ago
Source: Vojtech Jarnik IMC 2025, Category I, P2
Determine all real numbers $x>1$ such that
\[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \]for any positive integer $n$.
0 replies
Assassino9931
2 hours ago
0 replies
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a^n + b is divisible by p but not by p^2
Assassino9931   0
3 hours ago
Source: Vojtech Jarnik IMC 2025, Category I, P1
Let $a\geq 2$ be an integer. Prove that there exists a positive integer $b$ with the following property: For each positive integer $n$, there is a prime number $p$ (possibly depending on $a,b,n$) such that $a^n + b$ is divisible by $p$, but not divisible by $p^2$.
0 replies
Assassino9931
3 hours ago
0 replies
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Do not try to bash on beautiful geometry
ItzsleepyXD   8
N 3 hours ago by Ianis
Source: Own , Mock Thailand Mathematic Olympiad P9
Let $ABC$be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$
8 replies
ItzsleepyXD
Wednesday at 9:30 AM
Ianis
3 hours ago
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a