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inequalities
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inequality
mathematical-forest   0
13 minutes ago
non-negative real numbers $x_{1} ,x_{2} , \cdots, x_{n} $, satisfied $\sum_{i=1}^{n} x_{i} =n$
Find maximum of $$\sum_{i=1}^{n-1} x_{i} x_{i+1}+x_{1} +x_{n} $$
0 replies
mathematical-forest
13 minutes ago
0 replies
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Non-linear Recursive Sequence
amogususususus   1
N 16 minutes ago by alpha31415
Given $a_1=1$ and the recursive relation
$$a_{i+1}=a_i+\frac{1}{a_i}$$for all natural number $i$. Find the general form of $a_n$.

Is there any way to solve this problem and similar ones?
1 reply
amogususususus
Jan 24, 2025
alpha31415
16 minutes ago
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Weird algebra with combinatorial flavour
a_507_bc   5
N 38 minutes ago by BreezeCrowd
Source: Kazakhstan National MO 2023 (10-11).2
Let $n>100$ be an integer. The numbers $1,2 \ldots, 4n$ are split into $n$ groups of $4$. Prove that there are at least $\frac{(n-6)^2}{2}$ quadruples $(a, b, c, d)$ such that they are all in different groups, $a<b<c<d$ and $c-b \leq |ad-bc|\leq d-a$.
5 replies
a_507_bc
Mar 21, 2023
BreezeCrowd
38 minutes ago
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Prove the inequality
Butterfly   1
N 39 minutes ago by Royal_mhyasd
Prove that for any positive numbers $x,y$, it holds that
$$x^2+y^2+7\ge 3(x+y)+\frac{9}{xy+2}.$$
1 reply
Butterfly
an hour ago
Royal_mhyasd
39 minutes ago
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old and easy imo inequality
Valentin Vornicu   216
N an hour ago by alexanderchew
Source: IMO 2000, Problem 2, IMO Shortlist 2000, A1
Let $ a, b, c$ be positive real numbers so that $ abc = 1$. Prove that
\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1. \]
216 replies
Valentin Vornicu
Oct 24, 2005
alexanderchew
an hour ago
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Tangent to incircles.
dendimon18   7
N an hour ago by Gggvds1
Source: ISR 2021 TST1 p.3
Let $ABC$ be an acute triangle with orthocenter $H$. Prove that there is a line $l$ which is parallel to $BC$ and tangent to the incircles of $ABH$ and $ACH$.
7 replies
dendimon18
May 4, 2022
Gggvds1
an hour ago
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Problem 3 of RMO 2006 (Regional Mathematical Olympiad-India)
makar   36
N an hour ago by SomeonecoolLovesMaths
Source: Elementry inequality
If $ a,b,c$ are three positive real numbers, prove that $ \frac {a^{2}+1}{b+c}+\frac {b^{2}+1}{c+a}+\frac {c^{2}+1}{a+b}\ge 3$
36 replies
makar
Sep 13, 2009
SomeonecoolLovesMaths
an hour ago
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Three numbers cannot be squares simultaneously
WakeUp   40
N an hour ago by Adywastaken
Source: APMO 2011
Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.
40 replies
WakeUp
May 18, 2011
Adywastaken
an hour ago
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Kaprekar Number
CSJL   5
N an hour ago by Adywastaken
Source: 2025 Taiwan TST Round 1 Independent Study 2-N
Let $k$ be a positive integer. A positive integer $n$ is called a $k$-good number if it satisfies
the following two conditions:

1. $n$ has exactly $2k$ digits in decimal representation (it cannot have leading zeros).

2. If the first $k$ digits and the last $k$ digits of $n$ are considered as two separate $k$-digit
numbers (which may have leading zeros), the square of their sum is equal to $n$.

For example, $2025$ is a $2$-good number because
\[(20 + 25)^2 = 2025.\]Find all $3$-good numbers.
5 replies
CSJL
Mar 6, 2025
Adywastaken
an hour ago
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Projective geometry
definite_denny   0
an hour ago
Source: IDK
Let ABC be a triangle and let DEF be the tangency point of incircirle with sides BC,CA,AB. Points P,Q are chosen on sides AB,AC such that PQ is parallel to BC and PQ is tangent to the incircle. Let M denote the midpoint of PQ. Let EF intersect BC at T. Prove that TM is tangent to the incircle
0 replies
definite_denny
an hour ago
0 replies
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Problem 7 of RMO 2006 (Regional Mathematical Olympiad-India)
makar   11
N 2 hours ago by SomeonecoolLovesMaths
Source: Functional Equation
Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x+y)=f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)=9$, determine $ f(9) .$
11 replies
makar
Sep 13, 2009
SomeonecoolLovesMaths
2 hours ago
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Symmetric inequalities under two constraints
ChrP   5
N Apr 15, 2025 by ChrP
Let $a+b+c=0$ such that $a^2+b^2+c^2=1$. Prove that $$ \sqrt{2-3a^2}+\sqrt{2-3b^2}+\sqrt{2-3c^2} \leq 2\sqrt{2} $$
and

$$ a\sqrt{2-3a^2}+b\sqrt{2-3b^2}+c\sqrt{2-3c^2} \geq 0 $$
What about the lower bound in the first case and the upper bound in the second?
5 replies
ChrP
Apr 7, 2025
ChrP
Apr 15, 2025
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Symmetric inequalities under two constraints
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inequalities
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ChrP
127 posts
ChrP
#1 h  Apr 7, 2025, 9:03 PM • 1 Y
Y by cubres
Let $a+b+c=0$ such that $a^2+b^2+c^2=1$. Prove that $$ \sqrt{2-3a^2}+\sqrt{2-3b^2}+\sqrt{2-3c^2} \leq 2\sqrt{2} $$
and

$$ a\sqrt{2-3a^2}+b\sqrt{2-3b^2}+c\sqrt{2-3c^2} \geq 0 $$
What about the lower bound in the first case and the upper bound in the second?
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arqady
30258 posts
arqady
#2 h  Apr 7, 2025, 9:38 PM • 2 Y
Y by ChrP, cubres
ChrP wrote:
Let $a+b+c=0$ such that $a^2+b^2+c^2=1$. Prove that $$ \sqrt{2-3a^2}+\sqrt{2-3b^2}+\sqrt{2-3c^2} \leq 2\sqrt{2} $$
We need to prove that: $$|x+2|+|2x+1|+|x-1|\leq4\sqrt{x^2+x+1},$$which is trivial.
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arqady
30258 posts
arqady
#3 h  Apr 7, 2025, 9:59 PM • 2 Y
Y by ChrP, cubres
ChrP wrote:
Let $a+b+c=0$ such that $a^2+b^2+c^2=1$. Prove that

$$ a\sqrt{2-3a^2}+b\sqrt{2-3b^2}+c\sqrt{2-3c^2} \geq 0 $$
Try $a=b=-\frac{1}{\sqrt6}$ and $c=\frac{2}{\sqrt6}.$
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ChrP
127 posts
ChrP
#4 h  Apr 10, 2025, 1:15 PM • 1 Y
Y by cubres
Does a generalization of this hold for $a,b,c,d$ such that $a+b+c+d=0$ and $a^2+b^2+c^2+d^2=1$ ? For example,


$$ \sqrt{3-4a^2}+ \sqrt{3-4b^2}+\sqrt{3-4c^2} +\sqrt{3-4d^2} \leq 2\sqrt{2} $$
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ChrP
127 posts
ChrP
#5 h  Apr 13, 2025, 10:39 AM • 1 Y
Y by cubres
Bump! Bump!
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ChrP
127 posts
ChrP
#6 h  Apr 15, 2025, 12:42 PM • 1 Y
Y by yyhloveu1314
I'm sorry— the correct generalization is as follows: $a,b,c,d$ such that $a+b+c+d=0$ and $a^2+b^2+c^2+d^2=1$, then


$$ \sqrt{3-4a^2}+ \sqrt{3-4b^2}+\sqrt{3-4c^2} +\sqrt{3-4d^2} \leq 2+2 \sqrt{3} $$
This post has been edited 1 time. Last edited by ChrP, Apr 15, 2025, 12:42 PM
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