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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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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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Tricky inequality
Orestis_Lignos   28
N 13 minutes ago by MR.1
Source: JBMO 2023 Problem 2
Prove that for all non-negative real numbers $x,y,z$, not all equal to $0$, the following inequality holds

$\displaystyle \dfrac{2x^2-x+y+z}{x+y^2+z^2}+\dfrac{2y^2+x-y+z}{x^2+y+z^2}+\dfrac{2z^2+x+y-z}{x^2+y^2+z}\geq 3.$

Determine all the triples $(x,y,z)$ for which the equality holds.

Milan Mitreski, Serbia
28 replies
Orestis_Lignos
Jun 26, 2023
MR.1
13 minutes ago
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JBMO Shortlist 2023 A1
Orestis_Lignos   5
N 21 minutes ago by MR.1
Source: JBMO Shortlist 2023, A1
Prove that for all positive real numbers $a,b,c,d$,

$$\frac{2}{(a+b)(c+d)+(b+c)(a+d)} \leq \frac{1}{(a+c)(b+d)+4ac}+\frac{1}{(a+c)(b+d)+4bd}$$
and determine when equality occurs.
5 replies
Orestis_Lignos
Jun 28, 2024
MR.1
21 minutes ago
J
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square root problem
kjhgyuio   6
N 23 minutes ago by kjhgyuio
........
6 replies
kjhgyuio
May 3, 2025
kjhgyuio
23 minutes ago
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60 posts!(and a question )
kjhgyuio   0
33 minutes ago
Finally 60 posts :D
0 replies
kjhgyuio
33 minutes ago
0 replies
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2023 Official Mock NAIME #15 f(f(f(x))) = f(f(x))
parmenides51   1
N 6 hours ago by jasperE3
How many non-bijective functions $f$ exist that satisfy $f(f(f(x))) = f(f(x))$ for all real $x$ and the domain of f is strictly within the set of $\{1,2,3,5,6,7,9\}$, the range being $\{1,2,4,6,7,8,9\}$?

Even though this is an AIME problem, a proof is mandatory for full credit. Constants must be ignored as we dont want an infinite number of solutions.
1 reply
parmenides51
Dec 4, 2023
jasperE3
6 hours ago
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Inequalities
sqing   3
N Today at 3:29 AM by sqing
Let $ a,b>0 $ and $\frac{a}{a^2+3}+ \frac{b}{b^2+ 3} \geq \frac{1}{2} . $ Prove that
$$a^2+ab+b^2\geq 3$$$$a^2-ab+b^2 \geq 1 $$Let $ a,b>0 $ and $\frac{a}{a^3+3}+ \frac{b}{b^3+ 3}\geq \frac{1}{2} . $ Prove that
$$a^3+ab+b^3 \geq 3$$$$ a^3-ab+b^3\geq 1 $$
3 replies
sqing
Wednesday at 12:59 PM
sqing
Today at 3:29 AM
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exist solutions?
teomihai   6
N Today at 12:05 AM by iwastedmyusername
Find how many perfect squares of five different digits there are, with elements from the set ${0,1,4,6,9}$.
6 replies
teomihai
Yesterday at 5:04 PM
iwastedmyusername
Today at 12:05 AM
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A pentagon inscribed in a circle of radius √2
tom-nowy   6
N Yesterday at 11:55 PM by anticodon
Can a pentagon with all rational side lengths be inscribed in a circle of radius $\sqrt{2}$ ?
6 replies
tom-nowy
May 6, 2025
anticodon
Yesterday at 11:55 PM
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Menelau's theorem
noneofyou34   6
N Yesterday at 11:10 PM by Shan3t
Please can someone help me prove that orthocenter of a triangle exists by using Menelau's Theorem!
6 replies
noneofyou34
Yesterday at 5:52 PM
Shan3t
Yesterday at 11:10 PM
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Math analytical geometry
JDog22   2
N Yesterday at 8:37 PM by Alex-131
Could you please tell me if this is correct:
If you take the position vector of an arbitrary point P, which is known to lie on the plane E, then the dot product of P and the normal vector n always results in the same number, regardless of the coordinates of point P. As long as the point lies on the plane, this calculation always gives the same number. Is that true?
2 replies
JDog22
Yesterday at 8:24 PM
Alex-131
Yesterday at 8:37 PM
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Hard Inequality
William_Mai   14
N Yesterday at 5:34 PM by hi2024IMOp14
Given $a, b, c \in \mathbb{R}$ such that $a^2 + b^2 + c^2 = 1$.
Find the minimum value of $P = ab + 2bc + 3ca$.

Source: Pham Le Van
14 replies
William_Mai
May 3, 2025
hi2024IMOp14
Yesterday at 5:34 PM
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Geometry
MTA_2024   7
N Yesterday at 4:56 PM by hi2024IMOp14
Let $ABC$ be a triangle such that $AB=3$,$BC=5$ and $AC=6$.Let $D$ be a point on side $AC$ and $E$ one on side $BC$ so that the line $DE$ is tangent to the incircle of $\triangle ABC$ .
Evaluate the perimeter of triangle $\triangle CDE$.
7 replies
MTA_2024
Wednesday at 6:52 PM
hi2024IMOp14
Yesterday at 4:56 PM
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four point lie on circle
Kizaruno   0
Yesterday at 3:29 PM
Let triangle ABC be inscribed in a circle with center O. A line d intersects sides AB and AC at points E and D, respectively. Let M, N, and P be the midpoints of segments BD, CE, and DE, respectively. Let Q be the foot of the perpendicular from O to line DE. Prove that the points M, N, P, and Q lie on a circle.
0 replies
Kizaruno
Yesterday at 3:29 PM
0 replies
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Range if \omega for No Inscribed Right Triangle y = \sin(\omega x)
ThisIsJoe   0
Yesterday at 2:02 PM
For a positive number \omega , determine the range of \omega for which the curve y = \sin(\omega x) has no inscribed right triangle.
Could someone help me figure out how to approach this?
0 replies
ThisIsJoe
Yesterday at 2:02 PM
0 replies
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Is the geometric function injective?
Project_Donkey_into_M4   1
N Apr 20, 2025 by Funcshun840
Source: Mock RMO TDP and Kayak 2018, P3
A non-degenerate triangle $\Delta ABC$ is given in the plane, let $S$ be the set of points which lie strictly inside it. Also let $\mathfrak{C}$ be the set of circles in the plane. For a point $P \in S$, let $A_P, B_P, C_P$ be the reflection of $P$ in sides $\overline{BC}, \overline{CA}, \overline{AB}$ respectively. Define a function $\omega: S \rightarrow \mathfrak{C}$ such that $\omega(P)$ is the circumcircle of $A_PB_PC_P$. Is $\omega$ injective?

Note: The function $\omega$ is called injective if for any $P, Q \in S$, $\omega(P) = \omega(Q) \Leftrightarrow P = Q$
1 reply
Project_Donkey_into_M4
Apr 20, 2025
Funcshun840
Apr 20, 2025
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Is the geometric function injective?
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Reveal topic tags
algebra
geometry
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Source: Mock RMO TDP and Kayak 2018, P3
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Project_Donkey_into_M4
148 posts
Project_Donkey_into_M4
#1 h  Apr 20, 2025, 6:23 PM
Y by
A non-degenerate triangle $\Delta ABC$ is given in the plane, let $S$ be the set of points which lie strictly inside it. Also let $\mathfrak{C}$ be the set of circles in the plane. For a point $P \in S$, let $A_P, B_P, C_P$ be the reflection of $P$ in sides $\overline{BC}, \overline{CA}, \overline{AB}$ respectively. Define a function $\omega: S \rightarrow \mathfrak{C}$ such that $\omega(P)$ is the circumcircle of $A_PB_PC_P$. Is $\omega$ injective?

Note: The function $\omega$ is called injective if for any $P, Q \in S$, $\omega(P) = \omega(Q) \Leftrightarrow P = Q$
This post has been edited 1 time. Last edited by Project_Donkey_into_M4, Apr 20, 2025, 6:24 PM
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Funcshun840
22 posts
Funcshun840
#2 h  Apr 20, 2025, 9:04 PM
Y by
I’m confused, doesn’t the injectivity of the function follow from the fact that the center of $\omega(P)$ is the isogonal conjugate of $P$?
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