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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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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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EGMO magic square
Lukaluce   15
N an hour ago by Assassino9931
Source: EGMO 2025 P6
In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$, and the sum of the numbers in each column is equal to $1$. Define $r_i$ to be the largest value in row $i$, and let $R = r_1 + r_2 + ... + r_{2025}$. Similarly, define $c_i$ to be the largest value in column $i$, and let $C = c_1 + c_2 + ... + c_{2025}$.
What is the largest possible value of $\frac{R}{C}$?

Proposed by Paulius Aleknavičius, Lithuania
15 replies
Lukaluce
Monday at 11:03 AM
Assassino9931
an hour ago
J
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Ant wanna come to A
Rohit-2006   2
N an hour ago by zhaoli
An insect starts from $A$ and in $10$ steps and has to reach $A$ again. But in between one of the s steps and can't go $A$. Find probability. For example $ABCDCDEDEA$ is valid but $ABCDEDEDEA$ is not valid.
2 replies
1 viewing
Rohit-2006
Yesterday at 6:47 PM
zhaoli
an hour ago
J
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Sum of squared areas of polyhedron's faces...
Miquel-point   2
N an hour ago by buratinogigle
Source: KoMaL B. 5453
The faces of a convex polyhedron are quadrilaterals $ABCD$, $ABFE$, $CDHG$, $ADHE$ and $EFGH$ according to the diagram. The edges from points $A$ and $G$, respectively are pairwise perpendicular. Prove that \[[ABCD]^2+[ABFE]^2+[ADHE]^2=[BCGF]^2+[CDHG]^2+[EFGH]^2,\]where $[XYZW]$ denotes the area of quadrilateral $XYZW$.

Proposed by Géza Kós, Budapest
2 replies
Miquel-point
Monday at 5:44 PM
buratinogigle
an hour ago
J
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IMO 2014 Problem 4
ipaper   168
N 2 hours ago by Bonime
Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Let $M$ and $N$ be the points on $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$ and $Q$ is the midpoint of $AN$. Prove that the intersection of $BM$ and $CN$ is on the circumference of triangle $ABC$.

Proposed by Giorgi Arabidze, Georgia.
168 replies
ipaper
Jul 9, 2014
Bonime
2 hours ago
J
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Angle oriented geometry
Problems_eater   0
4 hours ago
Let $A, B, C,D$ be four distinct points in the plane.
Which of the following statements, expressed using oriented angles, are always true?

1.If lines $AB$ and $CD$ are distinct and parallel, then
the oriented angle $ABC$ is equal to the oriented angle DCB.

2.If $B$ lies on the segment $AC$, then
the oriented angle $DBA$ plus the oriented angle $DBC $equals $180°$.

3.If the oriented angle$ ABC$ plus the oriented angle $BCD$ equals 0°, then
lines $AB $and $CD$ are parallel.

4.If the oriented angle $ABC$ plus the oriented angle $BCD$ equals $180°$, then
lines $AB$ and $CD$are parallel.
0 replies
Problems_eater
4 hours ago
0 replies
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how many quadrilaterals ?
Ecrin_eren   6
N Yesterday at 5:31 PM by mathprodigy2011
"All the diagonals of an 11-gon are drawn. How many quadrilaterals can be formed using these diagonals as sides? (The vertices of the quadrilaterals are selected from the vertices of the 11-gon.)"
6 replies
Ecrin_eren
Apr 13, 2025
mathprodigy2011
Yesterday at 5:31 PM
J
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Plane geometry problem with inequalities
ReticulatedPython   3
N Yesterday at 2:48 PM by vanstraelen
Let $A$ and $B$ be points on a plane such that $AB=1.$ Let $P$ be a point on that plane such that $$\frac{AP^2+BP^2}{(AP)(BP)}=3.$$Prove that $$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$
Source: Own
3 replies
ReticulatedPython
Apr 10, 2025
vanstraelen
Yesterday at 2:48 PM
J
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Inequalities
sqing   1
N Yesterday at 1:55 PM by sqing
Let $ a,b $ be reals such that $ a^2-ab+b^2 =3$ . Prove that
$$ \frac{13}{ 10 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq \frac{1}{ 2 }$$$$ \frac{6}{ 5 }>\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq \frac{1}{ 5 }$$$$ \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq \frac{1}{ 14 }$$
1 reply
sqing
Yesterday at 8:59 AM
sqing
Yesterday at 1:55 PM
J
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idk12345678 Math Contest
idk12345678   21
N Yesterday at 1:25 PM by idk12345678
Welcome to the 1st idk12345678 Math Contest.
You have 4 hours. You do not have to prove your answers.
Post \signup username to sign up. Post your answers in a hide tag and I will tell you your score.*


The contest is attached to the post

Clarifications

*I mightve done them wrong feel free to ask about an answer
21 replies
idk12345678
Apr 10, 2025
idk12345678
Yesterday at 1:25 PM
J
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purple comet math competition question
AVY2024   4
N Yesterday at 1:02 PM by K1mchi_
Given that (1 + tan 1)(1 + tan 2). . .(1 + tan 45) = 2n, find n
4 replies
AVY2024
Yesterday at 11:00 AM
K1mchi_
Yesterday at 1:02 PM
J
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Inequalities
sqing   25
N Yesterday at 12:06 PM by sqing
Let $ a,b,c,d>0 $ and $(a+c)(b+d)=ac+\frac{3}{2}bd.$ Prove that
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq \frac{20-\sqrt{10}}{3}$$Let $ a,b,c,d>0 $ and $(a+c)(b+d)=ac+\frac{4}{3}bd.$ Prove that
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq \frac{21-\sqrt{6}}{3}$$
25 replies
sqing
Dec 3, 2024
sqing
Yesterday at 12:06 PM
J
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Polynomials
CuriousBabu   3
N Yesterday at 11:40 AM by osszhangbanvan
\[ \frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0 \]
Find the number of real solutions.
3 replies
CuriousBabu
Monday at 4:09 PM
osszhangbanvan
Yesterday at 11:40 AM
J
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Combination
aria123   0
Yesterday at 10:59 AM
Prove that three squares of side length $4$ cannot completely cover a square of side length $5$, if the three smaller squares do not overlap in their interiors (i.e., they may touch at edges or corners, but no part of one lies over another).
0 replies
aria123
Yesterday at 10:59 AM
0 replies
J
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Geo Mock #6
Bluesoul   3
N Yesterday at 3:16 AM by dudade
Consider triangle $ABC$ with $AB=5, BC=8, AC=7$, denote the incenter of the triangle as $I$. Extend $BI$ to meet the circumcircle of $\triangle{AIC}$ at $Q\neq I$, find the length of $QC$.
3 replies
Bluesoul
Apr 1, 2025
dudade
Yesterday at 3:16 AM
J
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J
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<ACE=? for tangential pentagon ABCDE and <BAE =<DCB =<AED = 90^o
parmenides51   0
Jul 11, 2021
Source: 2003 Ukraine NMO 8.2
A convex pentagon $ABCDE$ has a circle that touches all its sides. It is known that $\angle BAE = \angle DCB = \angle AED = 90^o$. Find the measure of the angle $ACE$ .
0 replies
parmenides51
Jul 11, 2021
0 replies
J
<ACE=? for tangential pentagon ABCDE and <BAE =<DCB =<AED = 90^o
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Reveal topic tags
geometry
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Source: 2003 Ukraine NMO 8.2
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parmenides51
30630 posts
parmenides51
#1 h  Jul 11, 2021, 7:17 PM
Y by
A convex pentagon $ABCDE$ has a circle that touches all its sides. It is known that $\angle BAE = \angle DCB = \angle AED = 90^o$. Find the measure of the angle $ACE$ .
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