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Find the range of a symmetric function

by DinDean, Apr 10, 2025, 6:38 AM

For $a,b,c\in[0,1]$ , assume $a+b+c=1$ , compute the range of
$f(a,b,c)=(1-a^2)^2+(1-b^2)^2+(1-c^2)^2.$

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Funky and nice NT FE

by yaybanana, Apr 10, 2025, 5:48 AM

find all functions $f: \mathbb{Z}^* \rightarrow \mathbb{Z}^*$ ,s.t :

$f(n)+f(m) \mid n^2+nf(m)$

for all $m,n \in \mathbb{Z}^*$

functional equation

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F=(F^3+F^3)/9-2F^3

by Yiyj1, Apr 10, 2025, 3:12 AM

Define the Fibonacci sequence $F_n$ as $F_1=F_2=1, F_{n+1}+F_n=F_{n-1}$ fir $n \in \mathbb{N}$ . Prove that $F_{2n}=\dfrac{F_{2n+2}^3+F_{2n-2}^3}{9}-2F_{2n}^3$ for all $n \ge 2$ .

Fibonacci Numbers

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Inspired by Ruji2018252

by sqing, Apr 10, 2025, 2:00 AM

Let $a,b,c$ be reals such that $a^2+b^2+c^2-2a-4b-4c=7.$ Prove that
$-4\leq 2a+b+2c\leq 20$ $5-4\sqrt 3\leq a+b+c\leq 5+4\sqrt 3$ $11-4\sqrt {14}\leq a+2b+3c\leq 11+4\sqrt {14}$

inequalities proposed

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Linear recurrence fits with factorial finitely often

by Assassino9931, Apr 9, 2025, 10:25 PM

Let $k\geq 3$ be an integer. The sequence $(a_n)_{n\geq 1}$ is defined via $a_1 = 1$ , $a_2 = k$ and
$a_{n+2} = ka_{n+1} + a_n$ for any positive integer $n$ . Prove that there are finitely many pairs $(m, \ell)$ of positive integers such that $a_m = \ell!$ .

Linear Recurrences

Diophantine equation

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Thanks u!

by Ruji2018252, Apr 9, 2025, 5:52 PM

Let $a^2+b^2+c^2-2a-4b-4c=7(a,b,c\in\mathbb{R})$
Find minimum $T=2a+3b+6c$

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4 lines concurrent

by Zavyk09, Apr 9, 2025, 11:51 AM

Let $ABC$ be triangle with circumcenter $(O)$ and orthocenter $H$ . $BH, CH$ intersect $(O)$ again at $K, L$ respectively. Lines through $H$ parallel to $AB, AC$ intersects $AC, AB$ at $E, F$ respectively. Point $D$ such that $HKDL$ is a parallelogram. Prove that lines $KE, LF$ and $AD$ are concurrent at a point on $OH$ .

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Beautiful problem

by luutrongphuc, Apr 4, 2025, 5:35 AM

(Phan Quang Tri) Let triangle $ABC$ be circumscribed about circle $(I)$ , and let $H$ be the orthocenter of $\triangle ABC$ . The circle $(I)$ touches line $BC$ at $D$ . The tangent to the circle $(BHC)$ at $H$ meets $BC$ at $S$ . Let $J$ be the midpoint of $HI$ , and let the line $DJ$ meet $(I)$ again at $X$ . The tangent to $(I)$ parallel to $BC$ meets the line $AX$ at $T$ . Prove that $ST$ is tangent to $(I)$ .

This post has been edited 1 time. Last edited by luutrongphuc, Apr 7, 2025, 1:49 AM

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Easy Number Theory

by VicKmath7, Mar 16, 2020, 5:14 PM

If $n$ is positive integer and $p, q, r$ are primes solve the system: $pqr=n$ and $(p+1)(q+1)r=n+138$

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Quadratic residues in a given interval

by cyshine, Nov 2, 2007, 1:50 AM

Find the number of integers $c$ such that $-2007 \leq c \leq 2007$ and there exists an integer $x$ such that $x^2 + c$ is a multiple of $2^{2007}$ .

modular arithmetic

number theory unsolved

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Submit

how do you have so many posts
by krithikrokcs, Jul 14, 2023, 6:20 PM
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by math_explorer, Jan 20, 2021, 8:43 AM
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by suvamkonar, Jan 20, 2021, 4:14 AM
https://artofproblemsolving.com/community/c47h361466
by math_explorer, Jun 10, 2020, 1:20 AM
when did the first greed control game start?
by piphi, May 30, 2020, 1:08 AM
ok..........
by asdf334, Sep 10, 2019, 3:48 PM
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by math_explorer, Sep 10, 2019, 2:03 PM
SO MANY VIEWS!!!
PLEASE CONTRIB

by asdf334, Sep 10, 2019, 1:58 PM
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Hullo bye
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by Dsquared, Dec 16, 2010, 3:14 PM
$\sum_{n=0}^{\infty}2^n=-1$
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Delete the shouts...

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Enjoy. (no offense)
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Changed your avatar?
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by math_explorer, Sep 12, 2010, 9:20 AM
Hello. Shout?
by asf, Sep 11, 2010, 5:18 PM

91 shouts

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