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9 Three concurrent chords
v_Enhance   1
N an hour ago by YaoAOPS
Three distinct circles $\Omega_1$, $\Omega_2$, $\Omega_3$ cut three common chords concurrent at $X$. Consider two distinct circles $\Gamma_1$, $\Gamma_2$ which are internally tangent to all $\Omega_i$. Determine, with proof, which of the following two statements is true.

(1) $X$ is the insimilicenter of $\Gamma_1$ and $\Gamma_2$
(2) $X$ is the exsimilicenter of $\Gamma_1$ and $\Gamma_2$.
1 reply
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v_Enhance
an hour ago
YaoAOPS
an hour ago
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Simple vector geometry existence
AndreiVila   3
N 2 hours ago by Ianis
Source: Romanian District Olympiad 2025 9.1
Let $ABCD$ be a parallelogram of center $O$. Prove that for any point $M\in (AB)$, there exist unique points $N\in (OC)$ and $P\in (OD)$ such that $O$ is the center of mass of $\triangle MNP$.
3 replies
AndreiVila
Mar 8, 2025
Ianis
2 hours ago
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An inequality
jokehim   2
N 2 hours ago by anduran
Let $a,b,c \in \mathbb{R}: a+b+c=3$ then prove $$\color{black}{\frac{a^2}{a^{2}-2a+3}+\frac{b^2}{b^{2}-2b+3}+\frac{c^2}{c^{2}-2c+3}\ge \frac{3}{2}.}$$
2 replies
jokehim
Today at 3:05 PM
anduran
2 hours ago
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BD tangent to (MDE) , rhombus ABCD with <DCB=60^o
parmenides51   1
N 3 hours ago by vanstraelen
Source: 2021 Germany R4 10.6 https://artofproblemsolving.com/community/c3208025_
Let a rhombus $ABCD$ with $|\angle DCB| = 60^o$ be given . On the extension of the segment $\overline{CD}$ beyond $D$, a point $E$ is chosen arbitrarily. Let the line through $E$ and $A$ intersect the line $BC$ at the point $F$. Let $M$ be the intersection of the lines $BE$ and $DF$. Prove that the line $BD$ is tangent to the circumcircle of the triangle $MDE$.
1 reply
parmenides51
Oct 6, 2024
vanstraelen
3 hours ago
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Geometry Problem #42
vankhea   2
N 3 hours ago by kaede_Arcadia
Source: Van Khea
Let $P$ be any point. Let $D, E, F$ be projection point from $P$ to $BC, CA, AB$. Circumcircle $(ABC)$ cuts circumcircle $(AEF), (BFD), (CDE)$ at $A_1, B_1, C_1$. Let $A_2, B_2, C_2$ be antipode of $A_1, B_1, C_1$ wrt $(AEF), (BFD), (CDE)$. Prove that $A_2, B_2, C_2, P$ are cyclic.
2 replies
vankhea
Sep 6, 2023
kaede_Arcadia
3 hours ago
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divisibility
srnjbr   3
N 3 hours ago by srnjbr
Find all natural numbers n such that there exists a natural number l such that for every m members of the natural numbers the number m+m^2+...m^l is divisible by n.
3 replies
srnjbr
6 hours ago
srnjbr
3 hours ago
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Very easy inequality
pggp   5
N 3 hours ago by ionbursuc
Source: Polish Junior MO Second Round 2019
Let $x$, $y$ be real numbers, such that $x^2 + x \leq y$. Prove that $y^2 + y \geq x$.
5 replies
pggp
Oct 26, 2020
ionbursuc
3 hours ago
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Solve in gaussian integers
CHESSR1DER   0
3 hours ago
Solve in gaussian integers.
$ \sin\left(\ln\left(x^{x^{x^2}}\right)\right) = x^4 $
0 replies
CHESSR1DER
3 hours ago
0 replies
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Inequality and function
srnjbr   4
N 4 hours ago by srnjbr
Find all f:R--R such that for all x,y, yf(x)+f(y)>=f(xy)
4 replies
srnjbr
6 hours ago
srnjbr
4 hours ago
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Problem 4
blug   3
N 4 hours ago by sunken rock
Source: Polish Junior Math Olympiad Finals 2025
In a rhombus $ABCD$, angle $\angle ABC=100^{\circ}$. Point $P$ lies on $CD$ such that $\angle PBC=20^{\circ}$. Line parallel to $AD$ passing trough $P$ intersects $AC$ at $Q$. Prove that $BP=AQ$.
3 replies
blug
Mar 15, 2025
sunken rock
4 hours ago
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CMI Entrance 19#6
bubu_2001   5
N 5 hours ago by quasar_lord
$(a)$ Compute -
\begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg] \end{align*}$(b)$ For $x > 0 $ define $F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t . $

$1.$ Determine the open interval(s) (if any) where $F ( x )$ is decreasing and all the open interval(s) (if any) where $F ( x )$ is increasing.

$2.$ Determine all the local minima of $F ( x )$ (if any) and all the local maxima of $F ( x )$ (if any) $.$
5 replies
bubu_2001
Nov 1, 2019
quasar_lord
5 hours ago
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Inequalites
Mario16   17
N Mar 17, 2025 by sqing
If a+b+c=3 ;a,b,c>=0 prove that 1/(5+a^2)+1/(5+b^2)+1/(5+c^2)<=1/2
17 replies
Mario16
Feb 1, 2021
sqing
Mar 17, 2025
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Inequalites
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Mario16
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#1 h  Feb 1, 2021, 10:08 PM • 1 Y
Y by Mango247
If a+b+c=3 ;a,b,c>=0 prove that 1/(5+a^2)+1/(5+b^2)+1/(5+c^2)<=1/2
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IAmTheHazard
5000 posts
IAmTheHazard
#2 h  Feb 1, 2021, 10:46 PM
Y by
You literally posted this 2 hours ago.
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Mario16
100 posts
Mario16
#3 h  Feb 1, 2021, 10:50 PM
Y by
Yes but i forgot to write Something
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IAmTheHazard
5000 posts
IAmTheHazard
#4 h  Feb 1, 2021, 10:52 PM
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It seems to be the exact same to me.
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Wildabandon
506 posts
Wildabandon
#5 h  Feb 2, 2021, 12:22 AM
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Mario16 wrote:
Yes but i forgot to write Something

You can edit your post.

If $a,b,c\ge 0$ and $a+b+c=3$, prove that
\[\frac{1}{5+a^2} + \frac{1}{5+b^2} + \frac{1}{5+c^2}\le \frac{1}{2}\]
This post has been edited 1 time. Last edited by Wildabandon, Feb 2, 2021, 12:23 AM
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Quantum_fluctuations
1282 posts
Quantum_fluctuations
#6 h  Feb 2, 2021, 12:23 AM • 1 Y
Y by Mango247
This is where you should go.
https://artofproblemsolving.com/community/c6h2387664
This post has been edited 1 time. Last edited by Quantum_fluctuations, Feb 2, 2021, 12:23 AM
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KP9
27 posts
KP9
#7 h  Feb 2, 2021, 8:38 AM
Y by
Another solution :

Easy to prove : $\frac{1}{5+a^2} \leq \frac{1}{5}(1-\frac{1}{18}a^3 - \frac{1}{9}a)$

So we have : $\sum \frac{1}{5+a^2} \leq \frac{3}{5} - \frac{1}{90}(a^3+b^3+c^3) - \frac{1}{45}(a+b+c) = \frac{3}{5} -\frac{3}{45} - \frac{1}{90}(a^3+b^3+c^3) = \frac{8}{15} - \frac{a^3+b^3+c^3}{90}$ (1)

We also have : $(1+1+1)(1+1+1)(a^3+b^3+c^3)\geq (a+b+c)^3$

$\Rightarrow a^3 + b^3 + c^3 \geq 3$ (2)

(1) and (2) $\Rightarrow \sum \frac{1}{5+a^2} \leq \frac{8}{15} - \frac{3}{90} = \frac{1}{2}$ ( Q.E.D)
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Quantum_fluctuations
1282 posts
Quantum_fluctuations
#8 h  Feb 2, 2021, 8:43 AM
Y by
KP9 wrote:
Another solution :

Easy to prove : $\frac{1}{5+a^2} \leq \frac{1}{5}(1-\frac{1}{18}a^3 - \frac{1}{9}a)$

How did you find that?
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KP9
27 posts
KP9
#10 h  Feb 2, 2021, 9:00 AM • 1 Y
Y by Mango247
oh , iam sorry , i have a problem when i prove it
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logrange
120 posts
logrange
#11 h  Feb 2, 2021, 9:00 AM • 1 Y
Y by Mango247
I have another similar problem in which sum is cyclic in one variable only. In these kind of problems we can use tangent line, but I want to know that whether it can be used in problems which have expression ≤ constant. I have used it only in cases like expression ≥ constant. If you can do this by tangent line then please post the solution of this by tangent line also.
Prove that cyclic sum $\frac{a}{2a^2+a+1}\leq \frac{3}{4}$
Given a+b+c=3 (Sorry, I missed that earlier)
This post has been edited 3 times. Last edited by logrange, Feb 2, 2021, 5:39 PM
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logrange
120 posts
logrange
#12 h  Feb 2, 2021, 5:40 PM
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Bump bump bump
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logrange
120 posts
logrange
#13 h  Feb 3, 2021, 5:07 AM
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Anyone?
Note - I want a solution without n-1 EV (Calculus)
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Wildabandon
506 posts
Wildabandon
#14 h  Feb 3, 2021, 5:21 AM
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I'm thinking Jensen but still using the calculus LOL
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logrange
120 posts
logrange
#15 h  Feb 3, 2021, 5:25 AM
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@below
Thanks
This post has been edited 1 time. Last edited by logrange, Feb 3, 2021, 11:13 AM
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starchan
1601 posts
starchan
#16 h  Feb 3, 2021, 5:36 AM
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I think Chebyshev's kills this one..
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sqing
41157 posts
sqing
#17 h  Mar 17, 2025, 2:31 PM
Y by
Wildabandon wrote:
If $a,b,c\ge 0$ and $a+b+c=3$, prove that
\[\frac{1}{5+a^2} + \frac{1}{5+b^2} + \frac{1}{5+c^2}\le \frac{1}{2}\]
https://artofproblemsolving.com/community/c4h2391785p19635446
https://artofproblemsolving.com/community/c6h2387664p20486022
Let $a,b,c$ be non-negative numbers such that $ab+bc+ca+abc=4.$ Prove that
$$\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}= 1$$$$\frac{1}{2}\leq \frac{1}{a^2+4}+\frac{1}{b^2+4}+\frac{1}{c^2+4}\leq \frac{3}{5}$$https://artofproblemsolving.com/community/c6h1510436p8962718
Let $ a,b,c>0 $ and $a^2+b^2+c^2+ab+bc+ca =6.$ Prove that
$$\frac{1}{a^2+5}+\frac{1}{b^2+5}+\frac{1}{c^2+5}\leq \frac{1}{2}$$( Vasile Cîrtoaje)
$$a^2b+b^2c+c^2a\leq \frac{368}{3}-\frac{176\sqrt{33}}{9}$$https://artofproblemsolving.com/community/c6h382474p2119615
This post has been edited 2 times. Last edited by sqing, Mar 17, 2025, 2:55 PM
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sqing
41157 posts
sqing
#18 h  Mar 17, 2025, 2:50 PM
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Let $ a,b,c>0 $ and $a^2+b^2+c^2+ab+bc+ca =6.$ Prove that
$$ ab+bc+ca- abc\leq 2$$$$\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\geq \frac{3}{2}$$$$\frac{1}{a^2+8}+\frac{1}{b^2+8}+\frac{1}{c^2+8}\leq \frac{1}{3}$$
This post has been edited 2 times. Last edited by sqing, Mar 17, 2025, 3:30 PM
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sqing
41157 posts
sqing
#19 h  Mar 17, 2025, 3:21 PM
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Let $ a,b\geq 0 $ and $a+b+a^2+ab+b^2 =5.$ Prove that
$$ \frac{1}{a^2+1}+ \frac{1}{b^2+1} \geq1$$$$ \frac{1}{a^2+2}+ \frac{1}{b^2+2} \geq \frac{2}{3}$$$$ \frac{1}{a^2+\frac{53}{20}}+ \frac{1}{b^2+\frac{53}{20}} \geq \frac{40}{73}$$$$ \frac{1}{a^2+\frac{1327}{500}}+ \frac{1}{b^2+ \frac{1327}{500}} \geq \frac{1000}{1827}$$$$ \frac{1}{a^2+3}+ \frac{1}{b^2+3} \geq \frac{185+3\sqrt{21}}{402}$$
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