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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Funny easy transcendental geo
qwerty123456asdfgzxcvb   0
a few seconds ago
Let $\mathcal{S}$ be a logarithmic spiral centered at the origin. Let $\mathcal{H}$ be a rectangular hyperbola centered at the origin, tangent to the logarithmic spiral.

Prove that for a point $P$ on the spiral, the polar of $P$ wrt. $\mathcal{H}$ is tangent to the spiral.
0 replies
1 viewing
qwerty123456asdfgzxcvb
a few seconds ago
0 replies
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Nice problem about a trapezoid
manlio   1
N 13 minutes ago by kiyoras_2001
Have you a nice solution for this problem?
Thank you very much
1 reply
manlio
Apr 19, 2025
kiyoras_2001
13 minutes ago
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product of the first n terms
FireBreathers   5
N 28 minutes ago by ihategeo_1969
Does there exist an infinite sequence of positive integers $a_i$ such that every positive integer appears exactly once and the product of the first $n$ terms is a perfect $n - th$ power ?
5 replies
1 viewing
FireBreathers
Dec 16, 2024
ihategeo_1969
28 minutes ago
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Inequality with three conditions
oVlad   1
N 33 minutes ago by Haris1
Source: Romania EGMO TST 2019 Day 1 P3
Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$
1 reply
oVlad
6 hours ago
Haris1
33 minutes ago
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geometry problem
kjhgyuio   1
N 5 hours ago by vanstraelen
.........
1 reply
kjhgyuio
Today at 8:27 AM
vanstraelen
5 hours ago
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Inequalities
sqing   4
N Today at 1:16 PM by sqing
Let $x,y\ge 0$ such that $ 13(x^3+y^3) \leq 125(1+xy)$. Prove that
$$ k(x+y)-xy\leq 5(2k-5)$$Where $k\geq 5.6797. $
$$ 6(x+y)-xy\leq 35$$
4 replies
sqing
Yesterday at 1:04 PM
sqing
Today at 1:16 PM
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Inscribed Semi-Circle!!!
ehz2701   2
N Today at 10:53 AM by mathafou
A right triangle $ABC$ with legs $AB = a$ and $BC = b$ is drawn with a semicircle inscribed into the triangle. What is the smallest possible radius of the semi-circle and the largest possible radius?

2 replies
ehz2701
Sep 11, 2022
mathafou
Today at 10:53 AM
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geometry
carvaan   1
N Today at 10:52 AM by vanstraelen
OABC is a trapezium with OC // AB and ∠AOB = 37°. Furthermore, A, B, C all lie on the circumference of a circle centred at O. The perpendicular bisector of OC meets AC at D. If ∠ABD = x°, find last 2 digit of 100x.
1 reply
carvaan
Yesterday at 5:48 PM
vanstraelen
Today at 10:52 AM
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Inequalities
nhathhuyyp5c   1
N Today at 9:09 AM by Mathzeus1024
Let $a, b, c$ be non-negative real numbers such that $a^2 + b^2 + c^2 = 3$. Find the maximum and minimum values of the expression
\[ P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}. \]
1 reply
nhathhuyyp5c
Yesterday at 6:35 AM
Mathzeus1024
Today at 9:09 AM
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In a school of $800$ students, $224$ students play cricket, $240$ students play
Vulch   2
N Today at 8:12 AM by MathBot101101
Hello everyone,
In a school of $800$ students, $224$ students play cricket, $240$ students play hockey and $336$ students play basketball. $64$ students play both basketball and hockey, $80$ students play both cricket and basketball, $40$ students play both cricket and hockey, and $24$ students play all three: basketball, hockey, and cricket. Find the number of students who do not play any game.

Edit:
In the above problem,I just want to know that why the number of students who don't play any game shouldn't be 0, because,if we add 224,240 and 336 it comes out to be 800.I have solution,but I just want to know how to explain it without theoretically.Thank you!
2 replies
Vulch
Yesterday at 11:41 PM
MathBot101101
Today at 8:12 AM
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Inequalities
sqing   25
N Today at 3:58 AM by sqing
Let $ a,b $ be reals such that $ a^2+ab+b^2 =3$ . Prove that
$$ \frac{4}{ 3}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{11}{4 }$$$$ \frac{13}{ 4}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{2}{3 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq -\frac{17}{6 }$$$$ \frac{19}{ 6}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2}$$Let $ a,b $ be reals such that $ a^2-ab+b^2 =1 $ . Prove that
$$ \frac{3}{ 2}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }+ab \geq \frac{4}{15 }$$$$ \frac{14}{ 15}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }-ab \geq -\frac{1}{2 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq \frac{13}{42 }$$$$ \frac{41}{ 42}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2 }$$
25 replies
sqing
Apr 16, 2025
sqing
Today at 3:58 AM
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Three variables inequality
Headhunter   4
N Today at 3:18 AM by lbh_qys
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
Prove that at least one of $(a-b)^{2}$, $(b-c)^{2}$, $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$.

I assume that all are greater than it, but can't go more.
4 replies
Headhunter
Yesterday at 6:58 AM
lbh_qys
Today at 3:18 AM
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Indonesia Regional MO 2019 Part A
parmenides51   23
N Today at 2:08 AM by chinawgp
Indonesia Regional MO
Year 2019 Part A

Time: 90 minutes Rules


p1. In the bag there are $7$ red balls and $8$ white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...


p2. Given a regular hexagon with a side length of $1$ unit. The area of the hexagon is ...


p3. It is known that $r, s$ and $1$ are the roots of the cubic equation $x^3 - 2x + c = 0$. The value of $(r-s)^2$ is ...


p4. The number of pairs of natural numbers $(m, n)$ so that $GCD(n,m) = 2$ and $LCM(m,n) = 1000$ is ...


p5. A data with four real numbers $2n-4$, $2n-6$, $n^2-8$, $3n^2-6$ has an average of $0$ and a median of $9/2$. The largest number of such data is ...


p6. Suppose $a, b, c, d$ are integers greater than $2019$ which are four consecutive quarters of an arithmetic row with $a <b <c <d$. If $a$ and $d$ are squares of two consecutive natural numbers, then the smallest value of $c-b$ is ...


p7. Given a triangle $ABC$, with $AB = 6$, $AC = 8$ and $BC = 10$. The points $D$ and $E$ lies on the line segment $BC$. with $BD = 2$ and $CE = 4$. The measure of the angle $\angle DAE$ is ...


p8. Sequqnce of real numbers $a_1,a_2,a_3,...$ meet $\frac{na_1+(n-1)a_2+...+2a_{n-1}+a_n}{n^2}=1$ for each natural number $n$. The value of $a_1a_2a_3...a_{2019}$ is ....


p9. The number of ways to select four numbers from $\{1,2,3, ..., 15\}$ provided that the difference of any two numbers at least $3$ is ...


p10. Pairs of natural numbers $(m , n)$ which satisfies $$m^2n+mn^2 +m^2+2mn = 2018m + 2019n + 2019$$are as many as ...


p11. Given a triangle $ABC$ with $\angle ABC =135^o$ and $BC> AB$. Point $D$ lies on the side $BC$ so that $AB=CD$. Suppose $F$ is a point on the side extension $AB$ so that $DF$ is perpendicular to $AB$. The point $E$ lies on the ray $DF$ such that $DE> DF$ and $\angle ACE = 45^o$. The large angle $\angle AEC$ is ...


p12. The set of $S$ consists of $n$ integers with the following properties: For every three different members of $S$ there are two of them whose sum is a member of $S$. The largest value of $n$ is ....


p13. The minimum value of $\frac{a^2+2b^2+\sqrt2}{\sqrt{ab}}$ with $a, b$ positive reals is ....


p14. The polynomial P satisfies the equation $P (x^2) = x^{2019} (x+ 1) P (x)$ with $P (1/2)= -1$ is ....


p15. Look at a chessboard measuring $19 \times 19$ square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of $k$ coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of $k$ so that the game never ends for any initial square selection is ....
23 replies
parmenides51
Nov 11, 2021
chinawgp
Today at 2:08 AM
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k VOLUNTEERING OPPORTUNITIES OPEN TO HIGH/MIDDLE SCHOOLERS
im_space_cadet   13
N Today at 12:30 AM by im_space_cadet
Hi everyone!
Do you specialize in contest math? Do you have a passion for teaching? Do you want to help leverage those college apps? Well, I have something for all of you.

I am im_space_cadet, and during the fall of last year, I opened my non-profit DeltaMathPrep which teaches students preparing for contest math the problem-solving skills they need in order to succeed at these competitions. Currently, we are very much understaffed and would greatly appreciate the help of more tutors on our platform.

Each week on Saturday and Wednesday, we meet once for each competition: Wednesday for AMC 8 and Saturday for AMC 10 and we go over a past year paper for the entire class. On both of these days, we meet at 9PM EST in the night.

This is a great opportunity for anyone who is looking to have a solid activity to add to their college resumes that requires low effort from tutors and is very flexible with regards to time.

This is the link to our non-profit for anyone who would like to view our initiative:
https://www.deltamathprep.org/

If you are interested in this opportunity, please send me a DM on AoPS or respond to this post expressing your interest. I look forward to having you all on the team!

Thanks,
im_space_cadet
13 replies
im_space_cadet
Yesterday at 2:27 PM
im_space_cadet
Today at 12:30 AM
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Induction
Mathlover_1   2
N Apr 4, 2025 by GreekIdiot
Hello, can you share links of same interesting induction problems in algebra
2 replies
Mathlover_1
Mar 24, 2025
GreekIdiot
Apr 4, 2025
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Induction
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Mathlover_1
295 posts
Mathlover_1
#1 h  Mar 24, 2025, 4:52 PM
Y by
Hello, can you share links of same interesting induction problems in algebra
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Primeniyazidayi
73 posts
Primeniyazidayi
#2 h  Mar 26, 2025, 6:56 PM
Y by
Well,you can try to prove AM-GM inequality with Cauchy induction.
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GreekIdiot
174 posts
GreekIdiot
#3 h  Apr 4, 2025, 3:42 PM
Y by
Algebra and induction huh. Try this year's HMO p3... HMO means Hellenic Mathematical Olympiad.
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