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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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Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
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[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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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diophantine equation
m4thbl3nd3r   1
N 5 minutes ago by whwlqkd
Find all positive integers $n,k$ such that $$5^{2n+1}-5^n+1=k^2$$
1 reply
m4thbl3nd3r
6 hours ago
whwlqkd
5 minutes ago
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ISI UGB 2025 P2
SomeonecoolLovesMaths   11
N 10 minutes ago by ProMaskedVictor
Source: ISI UGB 2025 P2
If the interior angles of a triangle $ABC$ satisfy the equality, $$\sin ^2 A + \sin ^2 B + \sin^2 C = 2 \left( \cos ^2 A + \cos ^2 B + \cos ^2 C \right),$$prove that the triangle must have a right angle.
11 replies
SomeonecoolLovesMaths
May 11, 2025
ProMaskedVictor
10 minutes ago
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Functional Inequaility
ariopro1387   2
N 15 minutes ago by Triborg-V
Source: Own
Find all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that for any real numbers \(x\) and \(y\), the following inequality holds:
\[ f\left(x^2+2y f(x)\right) + (f(y))^2 \leq (f(x+y))^2 \]
2 replies
ariopro1387
Apr 9, 2025
Triborg-V
15 minutes ago
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Problem 7
SlovEcience   4
N 16 minutes ago by SlovEcience
Consider the sequence \((u_n)\) defined by \(u_0 = 5\) and
\[ u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}. \]a) Prove that there exist infinitely many positive integers \(n\) such that \(u_n > 2020n\).

b) Compute
\[ \lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}. \]
4 replies
SlovEcience
May 14, 2025
SlovEcience
16 minutes ago
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22nd PMO Qualifying Stage #11
pensive   2
N 26 minutes ago by pensive
Let $x$ and $y$ be positive real numbers such that
\[ \log_x 64 + \log_{y^2} 16 = \frac{5}{3} \quad \text{and} \quad \log_y 64 + \log_{x^2} 16 = 1 \]What is the value of $\log_2 (xy)$?

Answer
2 replies
+1 w
pensive
an hour ago
pensive
26 minutes ago
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quadratic discriminant probability problem
elpianista227   2
N an hour ago by elpianista227
Consider the quadratic equation $x^2 - \sqrt{a}x + 5b = 0$. Find the probability of selecting two real numbers $a, b \in [0, 10]$ such that the equation has at least one real root.
2 replies
elpianista227
2 hours ago
elpianista227
an hour ago
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Infinite polynomial's value
Kempu33334   1
N an hour ago by Aaronjudgeisgoat
Let us define $P(x)$ to be an infinite polynomial where \[P(x) = \lim_{n\to \infty}\prod_{k=0}^n \left(x-\left(\cos\left(\dfrac{2\pi k}{n}\right)+i\sin\left(\dfrac{2\pi k}{n}\right)\right)\right).\]
a) Find $P(0)$, $P(\pi)$. Hint
b) Find the coefficient of the $x^2$, and $x^4$ terms. Hint

@below, yep thank you!
1 reply
Kempu33334
2 hours ago
Aaronjudgeisgoat
an hour ago
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[PMO25 Areas, I.13] log + cubic equation
jdcuber13   1
N 2 hours ago by jdcuber13
Let $a,b,c$ be real numbers with $1 < a < b < c$ that satisfy the equations
\begin{align*} \log_{a} b + \log_{b} c + \log_{c} a &= 6.5 \\\\ \log_{b} a + \log_{c} b + \log_{a} c &= 5 \end{align*}
Then max{$\log_{a} b, \log_{b} c, \log_{c} a$} can be written in the form $\sqrt{x} + \sqrt{y}$, where $x$, and $y$ are positive integers. What is $x+y$?

Answer Confirmation
1 reply
jdcuber13
3 hours ago
jdcuber13
2 hours ago
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Minimize
lgx57   7
N 2 hours ago by lgx57
Minimize $\sqrt{\cos^2 x+(2-\sin x)^2}+\dfrac{1}{2}\sqrt{(\sqrt 3-\cos x)^2+(\sin x+1)^2}$
7 replies
lgx57
May 23, 2025
lgx57
2 hours ago
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[Own problem] geometric sequence of logarithms
aops-g5-gethsemanea2   1
N 4 hours ago by aops-g5-gethsemanea2
A geometric sequence has the property where the third term is $\log32$ more than the first term, and the fourth term is $\log(128\sqrt2)$ more than the second term. Find the first term.
1 reply
aops-g5-gethsemanea2
4 hours ago
aops-g5-gethsemanea2
4 hours ago
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Functional Equations with DP
Magdalo   1
N 5 hours ago by Magdalo
Suppose a function $f:\mathbb Z_{\geq0}\to \mathbb Z_{\geq0}$ satisfies the following conditions:
\begin{align} &f(3x+1)=f(3x)=f(x)\\ &f(3x+2)=f(x)+2\\ &f(0)=f(1)=0\\ &f(2)=2 \end{align}Find the value of $f(1000)$.
1 reply
Magdalo
5 hours ago
Magdalo
5 hours ago
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[PMO18 Qualifying] III.3 Functional Equation
Magdalo   1
N 5 hours ago by Magdalo
Suppose a function $f:\mathbb R\to \mathbb R$ satisfies the following conditions:
\begin{align*} &f(4xy)=2y[f(x+y)+f(x-y)]\text{ for all }x,y\in\mathbb R\\ &f(5)=3 \end{align*}
Find the value of $f(2015)$.
1 reply
Magdalo
5 hours ago
Magdalo
5 hours ago
J
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Functional Equations
burntpizza001   0
Today at 9:46 AM
Let $f:R\rightarrow{}R$ is a function satisfying $f(2-x)=f(2+x)$ and $f(20-x)=f(x)$, $\forall x\in R$.

(i) If $f(0) = 5$ then minimum possible number of values of $x$ satisfying $f(x)=5$, for $x\in[0,170]$ is?

(ii) Graph of $y=f(x)$$ is symmetrical about which line?
0 replies
burntpizza001
Today at 9:46 AM
0 replies
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[Mathira 2018 T7-C2] Inverse of exponential spam
aops-g5-gethsemanea2   1
N Today at 6:50 AM by aops-g5-gethsemanea2
Find the inverse of the function $f(x)=\dfrac12\left(\dfrac{2^{-x}+2^x}{2^{-x}-2^x}+\dfrac{2^{-x}-2^x}{2^{-x}+2^x}\right)$.
1 reply
aops-g5-gethsemanea2
Today at 6:49 AM
aops-g5-gethsemanea2
Today at 6:50 AM
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angle bisectors and radii
rogue   2
N May 11, 2010 by moldovan
Let $O$ be the circumcenter and $I$ be the incenter of triangle $ABC.$ Prove that if $AI\perp OB$ and $BI\perp OC$ then $CI\parallel OA$.
2 replies
rogue
May 8, 2010
moldovan
May 11, 2010
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angle bisectors and radii
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Reveal topic tags
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Kyiv mathematical festival
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rogue
554 posts
rogue
#1 h  May 8, 2010, 11:37 PM • 2 Y
Y by Adventure10, Mango247
Let $O$ be the circumcenter and $I$ be the incenter of triangle $ABC.$ Prove that if $AI\perp OB$ and $BI\perp OC$ then $CI\parallel OA$.
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Batominovski
1608 posts
Batominovski
#2 h  May 9, 2010, 3:22 AM • 3 Y
Y by Adventure10, Mango247, and 1 other user
Angle chasing to get
\[\angle{A}=\frac{2\pi}{7}\,,\;\angle{B}=\frac{4\pi}{7}\,,\text{ and }\angle{C}=\frac{\pi}{7}\,.\]
The remaining is obvious.
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moldovan
1311 posts
moldovan
#3 h  May 11, 2010, 7:52 PM • 2 Y
Y by Adventure10, Mango247
$AI \perp OB \Rightarrow \frac{A}{2}+\frac{\pi-2C}{2}=\frac{\pi}{2} \Rightarrow C=\frac{A}{2}$

$BI \perp OC \Rightarrow \frac{B}{2}+\frac{\pi-2A}{2}=\frac{\pi}{2} \Rightarrow A=\frac{B}{2}$

$\Rightarrow A=\frac{2 \pi}{7}, B=\frac{4 \pi}{7} , C=\frac{\pi}{7}$
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