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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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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 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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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hard problem
Cobedangiu   15
N 13 minutes ago by arqady
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$
15 replies
2 viewing
Cobedangiu
Apr 21, 2025
arqady
13 minutes ago
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One on reals
Rushil   30
N 17 minutes ago by Maximilian113
Source: INMO 2001 Problem 3
If $a,b,c$ are positive real numbers such that $abc= 1$, Prove that \[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]
30 replies
+1 w
Rushil
Oct 10, 2005
Maximilian113
17 minutes ago
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Bounding is hard
whatshisbucket   20
N 44 minutes ago by torch
Source: ELMO 2018 #5, 2018 ELMO SL A2
Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$holds for all integers $n>N.$

Proposed by Carl Schildkraut
20 replies
whatshisbucket
Jun 28, 2018
torch
44 minutes ago
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Yet another domino problem
juckter   14
N an hour ago by math-olympiad-clown
Source: EGMO 2019 Problem 2
Let $n$ be a positive integer. Dominoes are placed on a $2n \times 2n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way.
(A domino is a tile of size $2 \times 1$ or $1 \times 2$. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)
14 replies
juckter
Apr 9, 2019
math-olympiad-clown
an hour ago
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Amc 10 mock
Mathsboy100   4
N 3 hours ago by pooh123
let \[\lfloor x \rfloor\]denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which \[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]
4 replies
Mathsboy100
Oct 9, 2024
pooh123
3 hours ago
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Inequalities
sqing   0
4 hours ago
Let $ a,b,c>0 . $ Prove that
$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq 4\left(\frac{a+b}{b+c}+ \frac{b+c}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{32}{9}\left(\frac{a+b}{b+c}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{8}{3}\left( \frac{a+b}{b+c}+ \frac{b+c}{c+a}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a^2}{b^2}\right)\left(1+\frac{b^2}{c^2}\right)\left(1+\frac{c^2}{a^2}\right )\geq \frac{8}{3}\left( \frac{a^2+bc}{b^2+ca}+\frac{b^2+ca}{c^2+ab}+\frac{c^2+ab}{a^2+bc}\right)$$
0 replies
sqing
4 hours ago
0 replies
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Purple Comet High School Math Meet 2024 P1
franklin2013   3
N 4 hours ago by codegirl2013
Joe ate one half of a fifth of a pizza. Gale ate one third of a quarter of that pizza. The difference in the amounts that the two ate was $\frac{1}{n}$ of the pizza, where $n$ is a positive integer. Find $n$.
3 replies
franklin2013
5 hours ago
codegirl2013
4 hours ago
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Cool vieta sum
Kempu33334   0
4 hours ago
Let the roots of \[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]be $r_1, r_2, \dots, r_{108}$. Find \[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]without Newton Sums.
0 replies
Kempu33334
4 hours ago
0 replies
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Inequality, tougher than it looks
tom-nowy   2
N 4 hours ago by pooh123
Prove that for $a,b \in \mathbb{R}$
$$ 2(a^2+1)(b^2+1) \geq 3(a+b). $$Is there an elegant way to prove this?
2 replies
tom-nowy
Tuesday at 3:51 AM
pooh123
4 hours ago
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Stylish Numbers
pedronis   3
N 5 hours ago by pedronis
A positive even integer $n$ is called stylish if the set $\{1, 2, \ldots, n\}$ can be partitioned into $\frac{n}{2}$ pairs such that the sum of the elements in each pair is a power of $3$. For example, $6$ is stylish because the set $\{1, 2, 3, 4, 5, 6\}$ can be partitioned as $\{1,2\}, \{3,6\}, \{4,5\}$, with sums $3$, $9$, and $9$ respectively. Determine the number of stylish numbers less than $3^{2025}$.
3 replies
pedronis
Apr 13, 2025
pedronis
5 hours ago
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Transformation of a cross product when multiplied by matrix A
Math-lover1   2
N Yesterday at 9:23 PM by Math-lover1
I was working through AoPS Volume 2 and this statement from Chapter 11: Cross Products/Determinants confused me.
[quote=AoPS Volume 2]A quick comparison of $|\underline{A}|$ to the cross product $(\underline{A}\vec{i}) \times (\underline{A}\vec{j})$ reveals that a negative determinant [of $\underline{A}$] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
I understand that this is true for the unit vectors $\vec{i} = (1 \ 0)$ and $\vec{j} = (0 \ 1)$, but am confused on how to prove this statement for general vectors $\vec{v}$ and $\vec{w}$ although its supposed to be a quick comparison.

How do I prove this statement easily with any two 2D vectors?
2 replies
Math-lover1
Tuesday at 10:29 PM
Math-lover1
Yesterday at 9:23 PM
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unfair coin, points winning 2024 TMC AIME Mock #9
parmenides51   5
N Yesterday at 8:44 PM by Math-lover1
Krithik has an unfair coin with a $\frac13$ chance of landing heads when flipped. Krithik is playing a game where he starts with $1$ point. Every turn, he flips the coin, and if it lands heads, he gains $1$ point, and if it lands tails, he loses $1$ point. However, after the turn, if he has a negative number of points, his point counter resets to $1$. Krithik wins when he earns $8$ points. Find the expected number of turns until Krithik wins.
5 replies
parmenides51
Apr 26, 2025
Math-lover1
Yesterday at 8:44 PM
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BrUMO 2025 Team Round Problem 15
lpieleanu   1
N Yesterday at 8:01 PM by vanstraelen
Let $\triangle{ABC}$ be an isosceles triangle with $AB=AC.$ Let $D$ be a point on the circumcircle of $\triangle{ABC}$ on minor arc $AB.$ Let $\overline{AD}$ intersect the extension of $\overline{BC}$ at $E.$ Let $F$ be the midpoint of segment $AC,$ and let $G$ be the intersection of $\overline{EF}$ and $\overline{AB}.$ Let the extension of $\overline{DG}$ intersect $\overline{AC}$ and the circumcircle of $\triangle{ABC}$ at $H$ and $I,$ respectively. Given that $DG=3, GH=5,$ and $HI=1,$ compute the length of $\overline{AE}.$
1 reply
lpieleanu
Apr 27, 2025
vanstraelen
Yesterday at 8:01 PM
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trigonometric functions
VivaanKam   9
N Yesterday at 7:12 PM by aok
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
9 replies
VivaanKam
Tuesday at 8:29 PM
aok
Yesterday at 7:12 PM
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k congruent faces if centers coincide, circumsphere , insphere of tetrahedron
parmenides51   1
N Dec 22, 2022 by parmenides51
Source: 1998 Argentina IberoAmerican Training List p25 https://artofproblemsolving.com/community/c2434876_argentina_training_lists_ibero
The centers of the spheres circumscribed and inscribed to a given tetrahedron coincide. Prove that the faces are congruent triangles.
1 reply
parmenides51
Oct 16, 2021
parmenides51
Dec 22, 2022
J
congruent faces if centers coincide, circumsphere , insphere of tetrahedron
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Reveal topic tags
geometry
3D geometry
tetrahedron
circumsphere
insphere
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G H BBookmark kLocked kLocked NReply
Source: 1998 Argentina IberoAmerican Training List p25 https://artofproblemsolving.com/community/c2434876_argentina_training_lists_ibero
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parmenides51
30650 posts
parmenides51
#1 h  Oct 16, 2021, 10:26 PM
Y by
The centers of the spheres circumscribed and inscribed to a given tetrahedron coincide. Prove that the faces are congruent triangles.
This post has been edited 1 time. Last edited by parmenides51, Oct 16, 2021, 10:26 PM
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parmenides51
30650 posts
parmenides51
#2 h  Dec 22, 2022, 2:20 AM
Y by
solved in an older source
Austrian-Polish 1979
This post has been edited 1 time. Last edited by parmenides51, Dec 22, 2022, 2:24 AM
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