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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 2025
0 replies
well-known NT
Tuleuchina   9
N 27 minutes ago by Blackbeam999
Source: Kazakhstan mo 2019, P6, grade 9
Find all integer triples $(a,b,c)$ and natural $k$ such that $a^2+b^2+c^2=3k(ab+bc+ac)$
9 replies
Tuleuchina
Mar 20, 2019
Blackbeam999
27 minutes ago
Inequality involving square root cube root and 8th root
bamboozled   0
29 minutes ago
If $a,b,c,d,e,f,g,h,k\in R^+$ and $a+b+c=d+e+f=g+h+k=8$, then find the minimum value of $\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$
0 replies
bamboozled
29 minutes ago
0 replies
Old problem
kwin   2
N an hour ago by kwin
Let $a, b, c \ge 0$ and $ ab+bc+ca>0$. Prove that:
$$ \frac{1}{(a+b)^2} + \frac{1}{(b+c)^2} + \frac{1}{(c+a)^2} + \frac{15}{(a+b+c)^2} \ge \frac{6}{ab+bc+ca}$$Is there any generalizations?
2 replies
kwin
Sunday at 1:12 PM
kwin
an hour ago
functional equation
henderson   4
N an hour ago by megarnie
Source: unknown
Find all functions $f :\mathbb{R^+}\to\mathbb{R^+}$, satisfying the condition

$f(1+xf(y))=yf(x+y)$

for any positive reals $x$ and $y$.
4 replies
1 viewing
henderson
Oct 8, 2015
megarnie
an hour ago
Inequalities
sqing   8
N 2 hours ago by sqing
Let $ a,b,c>0 $ and $ a+b\leq 16abc. $ Prove that
$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$Where $ k>0. $
$$ a+b+3c^3\geq\sqrt{\frac{2} {3}}$$$$ a+b+2c^4\geq \frac{5} {8}$$
8 replies
sqing
Sunday at 12:46 PM
sqing
2 hours ago
A Collection of Good Problems from my end
SomeonecoolLovesMaths   10
N 3 hours ago by ReticulatedPython
This is a collection of good problems and my respective attempts to solve them. I would like to encourage everyone to post their solutions to these problems, if any. This will not only help others verify theirs but also perhaps bring forward a different approach to the problem. I will constantly try to update the pool of questions.

The difficulty level of these questions vary from AMC 10 to AIME. (Although the main pool of questions were prepared as a mock test for IOQM over the years)

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5
10 replies
SomeonecoolLovesMaths
Sunday at 8:16 AM
ReticulatedPython
3 hours ago
trapezoid
Darealzolt   0
3 hours ago
Let \(ABCD\) be a trapezoid such that \(A, B, C, D\) lie on a circle with center \(O\), and side \(AB\) is parallel to side \(CD\). Diagonals \(AC\) and \(BD\) intersect at point \(M\), and \(\angle AMD = 60^\circ\). It is given that \(MO = 10\). It is also known that the difference in length between \(AB\) and \(CD\) can be expressed in the form \(m\sqrt{n}\), where \(m\) and \(n\) are positive integers and \(n\) is square-free. Compute the value of \(m + n\).
0 replies
Darealzolt
3 hours ago
0 replies
Inequalities
sqing   2
N 3 hours ago by sqing
Let $ a,b,c\geq 0 ,   2a +ab + 12a bc \geq 8. $ Prove that
$$  a+  (b+c)(a+1)+\frac{4}{5}  bc \geq 4$$$$  a+  (b+c)(a+0.9996)+ 0.77  bc \geq 4$$
2 replies
sqing
Sunday at 5:23 AM
sqing
3 hours ago
anyone who can help me this 2 problems?
auroracliang   2
N 5 hours ago by ReticulatedPython
1. Let r be the radius of the largest circle which is tangent to the parabola y=x^2 at x=0 and which lies entirely on or inside (that is, above) the parabola, find r.

2. Counting number n has the following property,: if we take any 50 different numbers from 1,2,3, ... n, there always are two numbers with the difference of 7. what is the largest value among all value of n?


thanks a lot
2 replies
auroracliang
Nov 3, 2024
ReticulatedPython
5 hours ago
What conic section is this? Is this even a conic section?
invincibleee   2
N 5 hours ago by ReticulatedPython
IMAGE

The points in this are given by
P = (sin2A, sin4A)∀A [0,2π]
Is this a conic section? what is this?
2 replies
invincibleee
Nov 15, 2024
ReticulatedPython
5 hours ago
Spheres, ellipses, and cones
ReticulatedPython   0
6 hours ago
A sphere is inscribed inside a cone with base radius $1$ and height $2.$ Another sphere of radius $r$ is internally tangent to the lateral surface of the cone, but does not intersect the larger inscribed sphere. A plane is tangent to both of these spheres, and passes through the inside of the cone. The intersection of the plane and the cone forms an ellipse. Find the maximum area of this ellipse.
0 replies
ReticulatedPython
6 hours ago
0 replies
Looking for users and developers
derekli   13
N 6 hours ago by DreamineYT
Guys I've been working on a web app that lets you grind high school lvl math. There's AMCs, AIME, BMT, HMMT, SMT etc. Also, it's infinite practice so you can keep grinding without worrying about finding new problems. Please consider helping me out by testing and also consider joining our developer team! :P :blush:

Link: https://stellarlearning.app/competitive
13 replies
derekli
May 4, 2025
DreamineYT
6 hours ago
trigonometric functions
VivaanKam   12
N Yesterday at 11:06 PM by aok
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
12 replies
VivaanKam
Apr 29, 2025
aok
Yesterday at 11:06 PM
find number of elements in H
Darealzolt   1
N Yesterday at 6:47 PM by alexheinis
If \( H \) is the set of positive real solutions to the system
\[
x^3 + y^3 + z^3 = x + y + z
\]\[
x^2 + y^2 + z^2 = xyz
\]then find the number of elements in \( H \).
1 reply
Darealzolt
Yesterday at 1:50 AM
alexheinis
Yesterday at 6:47 PM
Geo with unnecessary condition
egxa   8
N Apr 4, 2025 by ErTeeEs06
Source: Turkey Olympic Revenge 2024 P4
Let the circumcircle of a triangle $ABC$ be $\Gamma$. The tangents to $\Gamma$ at $B,C$ meet at point $E$. For a point $F$ on line $BC$ which is not on the segment $BC$, let the midpoint of $EF$ be $G$. Lines $GB,GC$ meet $\Gamma$ again at points $I,H$ respectively. Let $M$ be the midpoint of $BC$. Prove that the points $F,I,H,M$ lie on a circle.

Proposed by Mehmet Can Baştemir
8 replies
egxa
Aug 6, 2024
ErTeeEs06
Apr 4, 2025
Geo with unnecessary condition
G H J
G H BBookmark kLocked kLocked NReply
Source: Turkey Olympic Revenge 2024 P4
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egxa
210 posts
#1 • 2 Y
Y by SerdarBozdag, Rounak_iitr
Let the circumcircle of a triangle $ABC$ be $\Gamma$. The tangents to $\Gamma$ at $B,C$ meet at point $E$. For a point $F$ on line $BC$ which is not on the segment $BC$, let the midpoint of $EF$ be $G$. Lines $GB,GC$ meet $\Gamma$ again at points $I,H$ respectively. Let $M$ be the midpoint of $BC$. Prove that the points $F,I,H,M$ lie on a circle.

Proposed by Mehmet Can Baştemir
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sami1618
904 posts
#3
Y by
Consider the linear function $f(X)=\mathbb{P}(X,\Gamma)-\mathbb{P}(X,G)$. Then $$\mathbb{P}(G,\Gamma)=f(G)=\frac{f(E)+f(F)}{2}=\frac{BE^2+BF \cdot CF -EG^2-FG^2}{2}=$$$$\frac{(BM^2+BF\cdot CF+EM^2)-2EG^2}{2}=\frac{(FM^2+EM^2)-2EG^2}{2}=\frac{EF^2-2EG^2}{2}=EG$$Thus inverting about $G$ with radius $EG$ maps $F$, $I$, $H$, and $M$ all to line $BC$, finishing the problem.
Attachments:
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SerdarBozdag
892 posts
#4 • 4 Y
Y by alexanderhamilton124, bin_sherlo, egxa, ehuseyinyigit
$\angle FME = 90, GF = GE$ implies $GF = GM = GE$. Note that $G$ is on the radical axis of the point circle $(E)$ and $(ABC)$ because it lies on the $E$-midline of the triangle $EBC$. This gives
$GF^2 = GM^2 = GB \cdot GI = GH \cdot GC$ and because $B,C \in FM$; $F,I,M,H,G$ are cyclic.
This post has been edited 2 times. Last edited by SerdarBozdag, Aug 6, 2024, 2:23 PM
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hukilau17
288 posts
#5 • 1 Y
Y by ehuseyinyigit
We complex bash. Let $\Gamma$ be the unit circle (and let $j$, rather than $i$, denote the coordinate of $I$), so that
$$|a|=|b|=|c|=1$$$$e=\frac{2bc}{b+c}$$$$m=\frac{b+c}2$$$$\overline{f} = \frac{b+c-f}{bc}$$$$g = \frac{e+f}2 = \frac{2bc+bf+cf}{2(b+c)}$$$$\overline{g} = \frac{\frac2{bc}+\frac{b+c-f}{b^2c}+\frac{b+c-f}{bc^2}}{2\left(\frac1b+\frac1c\right)} = \frac{b^2+4bc+c^2-bf-cf}{2bc(b+c)}$$$$j = \frac{b-g}{b\overline{g}-1} = \frac{\frac{2b^2-bf-cf}{2(b+c)}}{\frac{b^2+2bc-c^2-bf-cf}{2c(b+c)}} = \frac{c(2b^2-bf-cf)}{b^2+2bc-c^2-bf-cf}$$$$h = \frac{c-g}{c\overline{g}-1} = \frac{\frac{2c^2-bf-cf}{2(b+c)}}{\frac{-b^2+2bc+c^2-bf-cf}{2b(b+c)}} = \frac{b(2c^2-bf-cf)}{-b^2+2bc+c^2-bf-cf}$$$$j-f = \frac{bf^2+cf^2-b^2f-3bcf+2b^2c}{b^2+2bc-c^2-bf-cf} = \frac{(b-f)(2bc-bf-cf)}{b^2+2bc-c^2-bf-cf}$$$$h-f = \frac{bf^2+cf^2-c^2f-3bcf+2bc^2}{-b^2+2bc+c^2-bf-cf} = \frac{(c-f)(2bc-bf-cf)}{-b^2+2bc+c^2-bf-cf}$$$$j-m = \frac{-b^3+b^2c-bc^2+c^3+b^2f-c^2f}{2(b^2+2bc-c^2-bf-cf)} = \frac{(c-b)(b^2+c^2-bf-cf)}{2(b^2+2bc-c^2-bf-cf)}$$$$h-m = \frac{b^3-b^2c+bc^2-c^3-b^2f+c^2f}{2(-b^2+2bc+c^2-bf-cf)} = \frac{(b-c)(b^2+c^2-bf-cf)}{2(-b^2+2bc+c^2-bf-cf)}$$Thus
$$\frac{(j-f)(h-m)}{(h-f)(j-m)} = -\frac{b-f}{c-f}$$which is real since $F$ lies on line $BC$. $\blacksquare$
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ohhh
48 posts
#6
Y by
Fix triangle $ABC$ and move the point $F$ along the line $BC$ (Yes, ignore the "which is not on the segment" condition).
$degF = 1 \implies degG  = 1 \implies degH = degI =1$
Notice that the condition $(IMGF)$ cyclic has degree at most $degGI + degMI + degMF +deg GF= 1 + 1 + 0 + 1= 3$.
So $4$ points are enough!
$1: F = B$
$2: F = M$
$3: F$ such that $G = M_{bc}$
$4: F$ such that $G = M_a$
Okay! Similarly, we can prove $(HMGF)$ and consequently $(IMHGF)$, as desired.
This post has been edited 3 times. Last edited by ohhh, Aug 7, 2024, 12:41 AM
Reason: Typos
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Anancibedih
18 posts
#7 • 2 Y
Y by ehuseyinyigit, Ege_Saribass
Let $T$ be the point that satisfies $FBET$ parallelogram. $\angle{TFB}=\angle{EBC}=\angle{BCE},FC||TE\Rightarrow \hspace{1mm}\text{FTEC is cyclic}$ $\angle{TEC}+\angle{FCE}=180^\circ=\angle{TEC}+\angle{TIC}\Rightarrow \text{I lies on FTEC circle}$. So $FB\cdot BC=2FB\cdot BM=IB\cdot BT=2BG\cdot IB\Rightarrow \text{FGMI is cyclic}$. We know that $FG=GM$ so $\angle{FIG}=\angle {GIM}=\angle{FMG}$ so $\triangle {GMB}\sim \triangle {GIM}\Rightarrow GM^2=GB\cdot GI=GH\cdot GC$ so $\angle{GMF}=\angle{MHC}$ and we get $\angle{GIM}+\angle{GHM}=180^\circ$ so $H$ lies on $FGMI$ circle. We're done.
This post has been edited 3 times. Last edited by Anancibedih, Aug 21, 2024, 2:01 PM
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khanhnx
1618 posts
#8 • 1 Y
Y by MS_asdfgzxcvb
We have $BC$ is polar of $E$ WRT $\Gamma$ so $\bigodot(EF)$ is orthogonal to $\Gamma$. From this, we have $\mathcal{I}^{k = GM^2}_G: \Gamma \longleftrightarrow \Gamma$. Hence $\mathcal{I}^{k = GM^2}_G: F \longleftrightarrow F, M \longleftrightarrow M, B \longleftrightarrow I, C \longleftrightarrow H$. But $B, C, M, F$ are collinear then $G, I, H, F, M$ lie on a circle
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ehuseyinyigit
821 posts
#10
Y by
The most natural way is constructing parallelograms. Let $C'$ and $B'$ be reflection of the points $C$ and $B$ wrt to $G$ respectively. Then $\angle BHC=\angle BAC=\angle BCE=\angle BFC'$ implies $BHC'F$ is cyclic and thus $FHMG$ (Since $BC.CF=2MC.CF=C'C.CH=2GC.CH$). On the other hand, $BE=EC$ implies $CEB'F$ is cyclic-isosceles trapezoid. Furthermore $\angle BHC=\angle CIB'=\angle CEB'$ gives $B'EICF$ is cyclic. Then PoP gives $BC.BF=2BM.BF=BI.BB'=2BI.BG$ implies $FGMI$ is cyclic. The result follows from $(FGMI)$ and $(FHMG)$ as desired.
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ErTeeEs06
64 posts
#11
Y by
$G$ is on the radax of $(ABC)$ and pointcircle $E$, since that radax is the midparallel in triangle $EBC$. So inversion with radius $GE$ around $G$ sends $H$ to $C$ and $I$ to $B$. Now since $G$ is midpoint of $EF$ and $\angle EMF=90^\circ$ we see that $GE=GM=GF$ so $M$ and $F$ remain fixed. So the images of $F, I, H, M$ are collinear so $F, I, H, M$ are concyclic.
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