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USAMO 2002 Problem 2
MithsApprentice   35
N 40 minutes ago by sami1618
Let $ABC$ be a triangle such that
\[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \]
where $s$ and $r$ denote its semiperimeter and its inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisors and determine these integers.
35 replies
MithsApprentice
Sep 30, 2005
sami1618
40 minutes ago
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annoying algebra with sequence :/
tabel   1
N an hour ago by L_.
Source: random 9th grade text book (section meant for contests)
Let \( a_1 = 1 \) and \( a_{n+1} = 1 + \frac{n}{a_n} \) for \( n \geq 1 \). Prove that the sequence \( (a_n)_{n \geq 1} \) is increasing.
1 reply
tabel
Today at 4:55 PM
L_.
an hour ago
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The Return of Triangle Geometry
peace09   16
N an hour ago by NO_SQUARES
Source: 2023 ISL A7
Let $N$ be a positive integer. Prove that there exist three permutations $a_1,\dots,a_N$, $b_1,\dots,b_N$, and $c_1,\dots,c_N$ of $1,\dots,N$ such that \[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\]for every $k=1,2,\dots,N$.
16 replies
peace09
Jul 17, 2024
NO_SQUARES
an hour ago
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f(1)f(2)...f(n) has at most n prime factors
MarkBcc168   40
N an hour ago by shendrew7
Source: 2020 Cyberspace Mathematical Competition P2
Let $f(x) = 3x^2 + 1$. Prove that for any given positive integer $n$, the product
$$f(1)\cdot f(2)\cdot\dots\cdot f(n)$$has at most $n$ distinct prime divisors.

Proposed by Géza Kós
40 replies
MarkBcc168
Jul 15, 2020
shendrew7
an hour ago
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smallest a so that S(n)-S(n+a) = 2018, where S(n)=sum of digits
parmenides51   3
N an hour ago by TheBaiano
Source: Lusophon 2018 CPLP P3
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$.
3 replies
parmenides51
Sep 13, 2018
TheBaiano
an hour ago
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ABC is similar to XYZ
Amir Hossein   55
N 2 hours ago by Mr.Sharkman
Source: China TST 2011 - Quiz 2 - D2 - P1
Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.
55 replies
Amir Hossein
May 20, 2011
Mr.Sharkman
2 hours ago
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Russia 2001
sisioyus   25
N 3 hours ago by cubres
Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a+b-1$ divides $ n$.
25 replies
sisioyus
Aug 18, 2007
cubres
3 hours ago
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conditional sequence
MithsApprentice   16
N 3 hours ago by shendrew7
Source: USAMO 1995
Suppose $\, q_{0}, \, q_{1}, \, q_{2}, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions:

(i) $\, m-n \,$ divides $\, q_{m}-q_{n}\,$ for $\, m > n \geq 0,$
(ii) there is a polynomial $\, P \,$ such that $\, |q_{n}| < P(n) \,$ for all $\, n$

Prove that there is a polynomial $\, Q \,$ such that $\, q_{n}= Q(n) \,$ for all $\, n$.
16 replies
MithsApprentice
Oct 23, 2005
shendrew7
3 hours ago
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P(z) and P(z)-1 have roots of magnitude 1
anser   16
N 3 hours ago by monval
Source: USA TSTST 2020 Problem 7
Find all nonconstant polynomials $P(z)$ with complex coefficients for which all complex roots of the polynomials $P(z)$ and $P(z) - 1$ have absolute value 1.

Ankan Bhattacharya
16 replies
anser
Jan 25, 2021
monval
3 hours ago
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Sums of n mod k
EthanWYX2009   3
N 3 hours ago by Safal
Source: 2025 May 谜之竞赛-3
Given $0<\varepsilon <1.$ Show that there exists a constant $c>0,$ such that for all positive integer $n,$
\[\sum_{k\le n^{\varepsilon}}(n\text{ mod } k)>cn^{2\varepsilon}.\]Proposed by Cheng Jiang
3 replies
EthanWYX2009
May 26, 2025
Safal
3 hours ago
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diophantine with factorials and exponents
skellyrah   11
N 3 hours ago by maromex
find all positive integers $a,b,c$ such that $$ a! + 5^b = c^3 $$
11 replies
skellyrah
May 30, 2025
maromex
3 hours ago
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J
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inequality ( 4 var
SunnyEvan   11
N Apr 10, 2025 by SunnyEvan
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$$$ a^4+b^4+c^4+d^4+ \frac{76}{25} \geq \frac{44}{25}(a^3+b^3+c^3+d^3) $$
11 replies
SunnyEvan
Apr 4, 2025
SunnyEvan
Apr 10, 2025
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inequality ( 4 var
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inequalities
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G H BBookmark kLocked kLocked NReply
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SunnyEvan
146 posts
SunnyEvan
#1 h  Apr 4, 2025, 5:19 AM
Y by
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$$$ a^4+b^4+c^4+d^4+ \frac{76}{25} \geq \frac{44}{25}(a^3+b^3+c^3+d^3) $$
This post has been edited 6 times. Last edited by SunnyEvan, Apr 8, 2025, 10:45 AM
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SunnyEvan
146 posts
SunnyEvan
#5 h  Apr 4, 2025, 9:55 AM
Y by
no one ? :(
This post has been edited 1 time. Last edited by SunnyEvan, Apr 4, 2025, 9:55 AM
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ektorasmiliotis
113 posts
ektorasmiliotis
#6 h  Apr 4, 2025, 9:59 AM
Y by
i havent try it because i am not home rn,but i think lagrange multipliers work
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arqady
30263 posts
arqady
#9 h  Apr 5, 2025, 2:47 PM
Y by
ektorasmiliotis wrote:
i havent try it because i am not home rn,but i think lagrange multipliers work
For the first LM does not help, I think, but I am wrong, of course. BW helps. Also, the Vasc's RCF Theorem helps.
This post has been edited 2 times. Last edited by arqady, Apr 7, 2025, 6:36 PM
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SunnyEvan
146 posts
SunnyEvan
#10 h  Apr 6, 2025, 11:00 AM
Y by
What are "LM" and "BW"?
@arqady.
This post has been edited 1 time. Last edited by SunnyEvan, Apr 6, 2025, 11:00 AM
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sqing
42574 posts
sqing
#11 h  Apr 6, 2025, 12:53 PM • 1 Y
Y by SunnyEvan
SunnyEvan wrote:
What are "LM" and "BW"?
@arqady.
LM
BW
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SunnyEvan
146 posts
SunnyEvan
#12 h  Apr 7, 2025, 9:42 AM • 1 Y
Y by teomihai
Thank you very much @ Sqing :-D :-D
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teomihai
2966 posts
teomihai
#14 h  Apr 7, 2025, 10:05 AM
Y by
it is good second?
This post has been edited 2 times. Last edited by teomihai, Apr 8, 2025, 4:57 AM
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arqady
30263 posts
arqady
#15 h  Apr 7, 2025, 6:40 PM • 1 Y
Y by teomihai
SunnyEvan wrote:
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :

$$ a^4+b^4+c^4+d^4+ \frac{252}{25} \geq \frac{88}{25}(a^3+b^3+c^3+d^3) $$equality cases : ?
It's wrong. Try $a=b=c=1.1$.
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SunnyEvan
146 posts
SunnyEvan
#16 h  Apr 8, 2025, 9:57 AM
Y by
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+ 2\sqrt{273} \geq \frac{\sqrt{273}-13}{2}(a^3+b^3+c^3+d^3)+30 $$
This post has been edited 1 time. Last edited by SunnyEvan, Apr 8, 2025, 10:44 AM
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GeoMorocco
44 posts
GeoMorocco
#18 h  Apr 8, 2025, 12:04 PM
Y by
SunnyEvan wrote:
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$

Let $f(x) = 4x^4+3-7x^3$. $f$ has two inflexion points and is right convex for $x \geq 1$. Therefore it is enough to check the inquality for $a \leq 1 \leq b=c=d$. In this case, we get:
$$\frac{1}{27}(a-1)^2(27a^2+(a-1)^2) \geq 0$$Equality for $a=b=c=d=1$.
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SunnyEvan
146 posts
SunnyEvan
#20 h  Apr 10, 2025, 2:07 PM
Y by
GeoMorocco wrote:
SunnyEvan wrote:
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$

Let $f(x) = 4x^4+3-7x^3$. $f$ has two inflexion points and is right convex for $x \geq 1$. Therefore it is enough to check the inquality for $a \leq 1 \leq b=c=d$. In this case, we get:
$$\frac{1}{27}(a-1)^2(27a^2+(a-1)^2) \geq 0$$Equality for $a=b=c=d=1$.

Thanks for your help.
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