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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 3:18 PM
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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0 replies
jlacosta
Yesterday at 3:18 PM
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Similar triangles and parallelism
KAME06   2
N 24 minutes ago by jordiejoh
Source: OMEC Ecuador National Olympiad Final Round 2022 N3 P5 day 2
Let $ABC$ be a 90-degree triangle with hypotenuse $BC$. Let $D$ and $E$ distinct points on segment $BC$ and $P, Q$ be the foot of the perpendicular from $D$ to $AB$ and $E$ to $AC$, respectively. $DP$ and $EQ$ intersect at $R$.
Lines $CR$ and $AB$ intersect at $M$ and lines $BR$ and $AC$ intersect at $N$.
Prove that $MN \parallel BC$ if and only if $BD=CE$.
2 replies
KAME06
Nov 4, 2024
jordiejoh
24 minutes ago
J
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Mmo 9-10 graders P5
Bet667   2
N 29 minutes ago by Bet667
Let $a,b,c,d$ be real numbers less than 2.Then prove that $\frac{a^3}{b^2+4}+\frac{b^3}{c^2+4}+\frac{c^3}{d^2+4}+\frac{d^3}{a^2+4}\le4$
2 replies
Bet667
an hour ago
Bet667
29 minutes ago
J
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Modular Matching Pairs
steven_zhang123   2
N 42 minutes ago by CHN_Lucas
Source: China TST 2025 P20
Let \( n \) be an odd integer, \( m = \frac{n+1}{2} \). Consider \( 2m \) integers \( a_1, a_2, \ldots, a_m, b_1, b_2, \ldots, b_m \) such that for any \( 1 \leq i < j \leq m \), \( a_i \not\equiv a_j \pmod{n} \) and \( b_i \not\equiv b_j \pmod{n} \). Prove that the number of \( k \in \{0, 1, \ldots, n-1\} \) for which satisfy \( a_i + b_j \equiv k \pmod{n} \) for some \( i \neq j \), $i, j \in \left \{ 1,2,\cdots,m \right \} $ is greater than \( n - \sqrt{n} - \frac{1}{2} \).
2 replies
steven_zhang123
Mar 29, 2025
CHN_Lucas
42 minutes ago
J
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Turkish MO 1994 P5
xeroxia   9
N 44 minutes ago by Primeniyazidayi
Source: Turkish Mathematical Olympiad 2nd Round 1994
Find the set of all ordered pairs $(s,t)$ of positive integers such that \[t^{2}+1=s(s+1).\]
9 replies
xeroxia
Sep 27, 2006
Primeniyazidayi
44 minutes ago
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When is this well known sequence periodic?
Assassino9931   2
N Mar 30, 2025 by Assassino9931
Source: Bulgaria Spring Mathematical Competition 2025 12.2
Determine all values of $a_0$ for which the sequence of real numbers with $a_{n+1}=3a_n - 4a_n^3$ for all $n\geq 0$ is periodic from the beginning.
2 replies
Assassino9931
Mar 30, 2025
Assassino9931
Mar 30, 2025
J
When is this well known sequence periodic?
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periodic
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Source: Bulgaria Spring Mathematical Competition 2025 12.2
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Assassino9931
1220 posts
Assassino9931
#1 h  Mar 30, 2025, 1:17 PM
Y by
Determine all values of $a_0$ for which the sequence of real numbers with $a_{n+1}=3a_n - 4a_n^3$ for all $n\geq 0$ is periodic from the beginning.
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RagvaloD
4894 posts
RagvaloD
#2 h  Mar 30, 2025, 2:28 PM • 1 Y
Y by MihaiT
Let $a_0>1$
Then $a_1 =3a_0-4a_0^3<-a_0$
And $a_2=3a_1-4a_1^3>-a_1>a_0>1$ so $a_0<a_2<...<a_{2k}$ and so sequence is not periodic

Same logic for $a_0<-1$

So $a_0 \in [-1,1]$ and so $a_0= \sin t$
then $a_1=\sin{3t} \in [-1,1]$ so $a_n=\sin {3^n t}$

$a_n=a_0 \to \sin{3^nt}=\sin {t} \to \sin {\frac{3^n+1}{2}t}\cos{\frac{3^n-1}{2}t}=0 \to t = \frac{2\pi m}{3^n+1}$ or $ t =\frac{\pi (2m-1)}{3^n-1}$
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Assassino9931
1220 posts
Assassino9931
#3 h  Mar 30, 2025, 4:56 PM
Y by
For $|a_0| > 1$ one simply has $|a_{n+1}| > |a_n|$ by induction, as $|3 - 4a_n^2| = 4|a_n|^2 - 3 > 1$.
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