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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Japan Mathematical Olympiad Preliminary 2007 Problem 3
Kunihiko_Chikaya   1
N 9 minutes ago by Mathzeus1024
On a plane given the line segment with length 7. The distance of a point $P$ and the segment is 3. Find the possible minimum value of $AP*BP.$
1 reply
Kunihiko_Chikaya
Jan 18, 2007
Mathzeus1024
9 minutes ago
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Inspired by old results
sqing   5
N 13 minutes ago by Sunjee
Source: Own
Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that
$$ \frac{2}{a}+\frac {2}{ab}+\frac{1}{abc}\geq 4$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{2}{abc}\geq 2+\sqrt 3$$$$ \frac{3}{a}+\frac {3}{ab}+\frac{1}{abc}\geq\frac {7+\sqrt {13}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{3}{abc}\geq\frac {5+\sqrt {21}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{4}{abc}\geq 3+2\sqrt 2$$
5 replies
sqing
Apr 26, 2025
Sunjee
13 minutes ago
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Polynomial approximation and intersections
egxa   2
N 13 minutes ago by iliya8788
Source: All Russian 2025 10.6
What is the smallest value of \( k \) such that for any polynomial \( f(x) \) of degree $100$ with real coefficients, there exists a polynomial \( g(x) \) of degree at most \( k \) with real coefficients such that the graphs of \( y = f(x) \) and \( y = g(x) \) intersect at exactly $100$ points?
2 replies
+1 w
egxa
Apr 18, 2025
iliya8788
13 minutes ago
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Quadrilateral geo
a_507_bc   16
N 21 minutes ago by zuat.e
Source: Mexico 2023/3
Let $ABCD$ be a convex quadrilateral. If $M, N, K$ are the midpoints of the segments $AB, BC$, and $CD$, respectively, and there is also a point $P$ inside the quadrilateral $ABCD$ such that, $\angle BPN= \angle PAD$ and $\angle CPN=\angle PDA$. Show that $AB \cdot CD=4PM\cdot PK$.
16 replies
a_507_bc
Nov 8, 2023
zuat.e
21 minutes ago
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triangle inscribed in a triangle, similar with min area
parmenides51   1
N Dec 26, 2021 by gvole
Source: Australian 1989
Points $X, Y$ and $Z$ on sides $BC, CA$ and $AB$ respectively of triangle $ABC$ are such that triangles $ABC$ and $XYZ$ are similar, the angles at $X, Y$ and $Z$ being equal to those at $A, B$ and $C$ respectively. Find $X, Y$ and $Z$ so that triangle $XYZ$ has minimum area.
1 reply
parmenides51
Jul 13, 2019
gvole
Dec 26, 2021
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triangle inscribed in a triangle, similar with min area
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Reveal topic tags
geometry
similar triangles
area of a triangle
minimum
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Source: Australian 1989
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parmenides51
30650 posts
parmenides51
#1 h  Jul 13, 2019, 6:15 PM • 1 Y
Y by Adventure10
Points $X, Y$ and $Z$ on sides $BC, CA$ and $AB$ respectively of triangle $ABC$ are such that triangles $ABC$ and $XYZ$ are similar, the angles at $X, Y$ and $Z$ being equal to those at $A, B$ and $C$ respectively. Find $X, Y$ and $Z$ so that triangle $XYZ$ has minimum area.
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gvole
201 posts
gvole
#2 h  Dec 26, 2021, 8:45 PM
Y by
By Miquel's theorem, the circles $AYZ$, $BZX$ and $CXY$ concur at a point $M$. By trivial angle chasing $M$ is the orthocenter of $\triangle XYZ$ and then it is also the circumcenter of $\triangle ABC$. That means $MX$, $MY$ and $MZ$ are shortest when $X$, $Y$, and $Z$ are the midpoints of the sides of $\triangle ABC$, it is trivial to check that the medial triangle is similar to $\triangle ABC$
This post has been edited 2 times. Last edited by gvole, Dec 27, 2021, 12:17 PM
Reason: typo
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