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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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0 replies
jlacosta
May 1, 2025
0 replies
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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
J
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
J
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Convergent series with weight becomes divergent
P_Fazioli   1
N 14 minutes ago by solyaris
Initially, my problem was : is it true that if we fix $(b_n)$ positive such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$, then there exists $(a_n)$ positive such that $\displaystyle\sum_{n\geq 0}a_n$ converges and $\displaystyle\sum_{n\geq 0}a_nb_n$ diverges ?

Thinking about the continuous case : if $g:\mathbb{R}_+\longrightarrow\mathbb{R}$ is continuous, positive with $g(x)\underset{{x}\longrightarrow{+\infty}}\longrightarrow +\infty$, does $f$ continuous and positive exist on $\mathbb{R}_+$ such that $\displaystyle\int_0^{+\infty}f(x)\text{d}x$ converges and $\displaystyle\int_0^{+\infty}f(x)g(x)\text{d}x$ diverges ?

To the last question, the answer seems to be yes if $g$ is in the $\mathcal{C}^1$ class, increasing : I chose $f=\dfrac{g'}{g^2}$. With this idea, I had the idea to define $a_n=\dfrac{b_{n+1}-b_n}{b_n^2}$ but it is not clear that it is ok, even if $(b_n)$ is increasing.

Now I have some questions !

1) The main problem : is it true that if we fix $(b_n)$ positive such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$, then there exists $(a_n)$ positive such that $\displaystyle\sum_{n\geq 0}a_n$ converges and $\displaystyle\sum_{n\geq 0}a_nb_n$ diverges ? And if $(b_n)$ is increasing ?
2) is it true that if we fix $(b_n)$ positive increasing such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$
and $\frac{b_{n+1}}{b_n}\underset{{n}\longrightarrow{+\infty}}\longrightarrow 1$, then $\displaystyle\sum_{n\geq 0}\left(\frac{b_{n+1}}{b_n}-1\right)$ diverges ?
3) is it true that if we fix $(b_n)$ positive increasing such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$
and $\frac{b_{n+1}}{b_n}\underset{{n}\longrightarrow{+\infty}}\longrightarrow 1$, then $\displaystyle\sum_{n\geq 0}\frac{b_{n+1}-b_n}{b_n^2}$ converges ?
4) if $(b_n)$ is positive increasing and such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$
and $\frac{b_{n+1}}{b_n}$ does not converge to $1$, can $\displaystyle\sum_{n\geq 0}\frac{b_{n+1}-b_n}{b_n^2}$ diverge ?
5) for the continuous case, is it true if we suppose $g$ only to be continuous ?

1 reply
P_Fazioli
5 hours ago
solyaris
14 minutes ago
J
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2-var inequality
sqing   8
N 21 minutes ago by sqing
Source: Own
Let $ a,b\geq 0 $. Prove that
$$ \frac{a }{a^2+2b^2+1}+ \frac{b }{b^2+2a^2+1}\leq \frac{1}{\sqrt{3}} $$$$ \frac{a }{2a^2+ b^2+2ab+1}+ \frac{b }{2b^2+ a^2+2ab+1} \leq \frac{1}{\sqrt{5}} $$$$ \frac{a }{2a^2+ b^2+ ab+1}+ \frac{b }{2b^2+ a^2+ ab+1} \leq \frac{1}{2} $$$$\frac{a }{a^2+2b^2+2ab+1}+ \frac{b }{b^2+2a^2+2ab+1}\leq \frac{1}{2} $$
8 replies
sqing
Today at 1:39 AM
sqing
21 minutes ago
J
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Solution
KTYC   0
25 minutes ago





(a + b)/(a + c) = (b + c)/(b + a)


Cross-multiply,

(a + b)² = (a + c)(b + c)

(a + b)² = ab + bc + ac + c²

ab + bc + ac = (a + b)² - c²

ab + bc + ac = (a + b - c)(a + b + c)
—-Eqn(1)

—————————-

Since a, b, c ∈ ℕ,

(a + b + c) ∈ ℕ

(ab + bc + ac) ∈ ℕ


From Eqn(1),

(a + b - c) ∈ ℕ

—————————-

Since (ab + bc + ac) is prime,

The only positive factors of
(ab + bc + ac) are
1 and (ab + bc + ac)


Since c ∈ ℕ,

(a + b - c) < (a + b + c)


Hence,


a + b - c = 1
—-Eqn(2)

a + b + c = ab + bc + ac
—-Eqn(3)


From Eqn(2),

a + b = c + 1
—-Eqn(4)


From Eqn(3),

a + b + c = c(a + b) + ac
—-Eqn(5)


Sub. Eqn(4) into Eqn(5):

c + 1 + c = c(c + 1) + ac

2c + 1 = c² + c + ac

c² + c + ac - 2c - 1 = 0

c² + ac - c - 1 = 0

c² + c(a - 1) - 1 = 0
—-Eqn(6)

——————————

Eqn(6) is a quadratic equation about c


Hence,

Discriminant of Eqn(6)
= (a - 1)² - 4(1)(-1)
= (a - 1)² + 4


Since c ∈ ℕ,
Discriminant of Eqn(6) is a perfect square


Hence,

[(a - 1)² + 4] is a perfect square


Let
(a - 1)² + 4 = k²
where k ∈ ℕ


Hence,

k² - (a - 1)² = 4

[k - (a - 1)][k + (a - 1)] = 4

(k - a + 1)(k + a - 1) = 4
—-Eqn(7)

——————————

Since a, k ∈ ℕ,

a ≥ 1

k ≥ 1


Hence,

(k + a - 1) ≥ (1 + 1 - 1)

(k + a - 1) ≥ 1


Hence,

(k + a - 1) ∈ ℕ


4 ∈ ℕ


Hence,

From Eqn(7),

(k - a + 1) ∈ ℕ


Hence,

4 is a factor of two positive integers in Eqn(7)

——————————

(k - a + 1) + (k + a - 1)
= 2k,
which is even


Hence,

(k - a + 1) and (k + a - 1) have the same parity


Since 4 is even,
(k - a + 1) and (k + a - 1) are even

——————————

Express 4 as a product of two even positive integers,

4 = 2 x 2


Hence,

k - a + 1 = 2
—-Eqn(8)

k + a - 1 = 2
—-Eqn(9)


From Eqn(8) and Eqn(9),

k - a + 1 = k + a - 1

-a + 1 = a - 1

-a - a = -1 - 1

-2a = -2

a = 1
—-Eqn(10)


Sub. Eqn(10) into Eqn(6):

c² + c(1 - 1) - 1 = 0

c² = 1


Hence,

c = 1 —-Eqn(11)
Or
c = -1 (rej. as c > 0)


Sub. Eqn(10) and Eqn(11) into Eqn(4):

1 + b = 1 + 1

b = 1

——————————————————

Therefore,


Ans:

(a, b, c)
= (1, 1, 1)
0 replies
KTYC
25 minutes ago
0 replies
J
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Inequality
MathsII-enjoy   4
N 30 minutes ago by ehuseyinyigit
A interesting problem generalized :-D
4 replies
MathsII-enjoy
Saturday at 1:59 PM
ehuseyinyigit
30 minutes ago
J
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old one but good one
Sunjee   0
35 minutes ago
If $x_1,x_2,...,x_n $ are positive numbers, then prove that
$$\frac{x_1}{1+x_1^2}+\frac{x_2}{1+x_1^2+x_2^2}+\cdots+ \frac{x_n}{1+x_1^2+\cdots+x_n^2}\geq \sqrt{n}$$
0 replies
Sunjee
35 minutes ago
0 replies
J
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Insects walk
Giahuytls2326   1
N an hour ago by removablesingularity
Source: somewhere in the internet
A 100 × 100 chessboard is divided into unit squares. Each square has an arrow pointing up, down, left, or right. The board square is surrounded by a wall, except to the right of the top right corner square. An insect is placed in one of the squares.

Every second, the insect moves one unit in the direction of the arrow in its square. As the insect moves, the arrow of the square it just left rotates 90° clockwise.

If the specified movement cannot be performed, then the insect will not move for that second, but the arrow in the square it is standing on will still rotate. Is it possible that the insect never leaves the board?
1 reply
Giahuytls2326
May 3, 2025
removablesingularity
an hour ago
J
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something...
SunnyEvan   1
N an hour ago by SunnyEvan
Source: unknown
Try to prove : $$ \sum csc^{20} \frac{2^{i} \pi}{7} csc^{23} \frac{2^{j}\pi }{7} csc^{2023} \frac{2^{k} \pi}{7} $$is a rational number.
Where $ (i,j,k)=(1,2,3) $ and other permutations.
1 reply
SunnyEvan
Today at 1:13 AM
SunnyEvan
an hour ago
J
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something interesting...
SunnyEvan   1
N an hour ago by SunnyEvan
Source: old result
Let $x$, $y$, $z$ be non-negative real numbers, no two of which are zero. Such that $ x+y+z=3.$ Prove that :
$$ \sum \frac{16(9-xyz)}{9(z+x)^2(y+3)^2} \geq \frac{2xyz}{\sum x^2} + \frac{\sum (z^2-x^2)(z^2-y^2)(z+x)(z+y)}{\sum(x+y)^3(y+z)^3} $$
1 reply
SunnyEvan
3 hours ago
SunnyEvan
an hour ago
J
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we can find one pair of a boy and a girl
orl   17
N an hour ago by raghu7
Source: Vietnam TST 2001 for the 42th IMO, problem 3
Some club has 42 members. It’s known that among 31 arbitrary club members, we can find one pair of a boy and a girl that they know each other. Show that from club members we can choose 12 pairs of knowing each other boys and girls.
17 replies
orl
Jun 26, 2005
raghu7
an hour ago
J
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IMO Genre Predictions
ohiorizzler1434   37
N an hour ago by flower417477
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
37 replies
ohiorizzler1434
May 3, 2025
flower417477
an hour ago
J
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Inequality
lgx57   6
N 2 hours ago by MihaiT
Source: Own
$a,b,c>0,ab+bc+ca=1$. Prove that

$$\sum \sqrt{8ab+1} \ge 5$$
(I don't know whether the equality holds)
6 replies
lgx57
Saturday at 3:14 PM
MihaiT
2 hours ago
J
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D1026 : An equivalent
Dattier   4
N 6 hours ago by 3ch03s
Source: les dattes à Dattier
Let $u_0=1$ and $\forall n \in \mathbb N, u_{2n+1}=\ln(1+u_{2n}), u_{2n+2}=\sin(u_{2n+1})$.

Find an equivalent of $u_n$.
4 replies
Dattier
Saturday at 1:39 PM
3ch03s
6 hours ago
J
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Rolles theorem
sasu1ke   4
N Yesterday at 6:13 PM by sasu1ke

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that
\[ f(0) = 2, \quad f'(0) = -2, \quad \text{and} \quad f(1) = 1. \]Prove that there exists a point \( \xi \in (0, 1) \) such that
\[ f(\xi) \cdot f'(\xi) + f''(\xi) = 0. \]

4 replies
sasu1ke
Saturday at 9:00 PM
sasu1ke
Yesterday at 6:13 PM
J
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Putnam 1947 B1
sqrtX   3
N Yesterday at 5:38 PM by Levieee
Source: Putnam 1947
Let $f(x)$ be a function such that $f(1)=1$ and for $x \geq 1$
$$f'(x)= \frac{1}{x^2 +f(x)^{2}}.$$Prove that
$$\lim_{x\to \infty} f(x)$$exists and is less than $1+ \frac{\pi}{4}.$
3 replies
sqrtX
Apr 3, 2022
Levieee
Yesterday at 5:38 PM
J
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J
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Matrix Row and column relation.
Schro   6
N May 1, 2025 by Schro
If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...
6 replies
Schro
Apr 28, 2025
Schro
May 1, 2025
J
Matrix Row and column relation.
G H J
Reveal topic tags
linear algebra
L
G H BBookmark kLocked kLocked NReply
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Schro
4 posts
Schro
#1 h  Apr 28, 2025, 2:54 PM
Y by
If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...
Z K Y
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Samujjal101
2799 posts
Samujjal101
#2 h  Apr 28, 2025, 3:06 PM
Y by
What do you mean by dependent ?
Z K Y
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rchokler
2975 posts
rchokler
#3 h  Apr 28, 2025, 8:40 PM
Y by
For square matrices, if the rows are linearly dependent, then so are the columns and vice-versa. This is because for every matrix of any size, the row rank and column rank are equal.
Z K Y
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Schro
4 posts
Schro
#4 h  Apr 29, 2025, 4:31 PM
Y by
Samujjal101 wrote:
What do you mean by dependent ?


I mean if ith row is a combination of other rows.....then ith column will also be a combination of other columns
Z K Y
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Schro
4 posts
Schro
#5 h  Apr 29, 2025, 4:32 PM
Y by
rchokler wrote:
For square matrices, if the rows are linearly dependent, then so are the columns and vice-versa. This is because for every matrix of any size, the row rank and column rank are equal.

I m not talking in general sense....
I m being specific.....if "i th" row is a combination of other rows.....then "i th " column will be a combination of other columns and vice versa
Z K Y
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Etkan
1563 posts
Etkan
#6 h  Apr 29, 2025, 5:01 PM
Y by
Schro wrote:
rchokler wrote:
For square matrices, if the rows are linearly dependent, then so are the columns and vice-versa. This is because for every matrix of any size, the row rank and column rank are equal.

I m not talking in general sense....
I m being specific.....if "i th" row is a combination of other rows.....then "i th " column will be a combination of other columns and vice versa

It's not true. For example, in the matrix$$\begin{pmatrix}0&0&0\\1&0&0\\-1&0&0\end{pmatrix}$$the first row is the sum of the second and third rows, but any linear combination of the second and third columns is $0$, so it cannot be equal to the first column.
Z K Y
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Schro
4 posts
Schro
#7 h  May 1, 2025, 6:20 AM
Y by
Thnks, i got it
Z K Y
N Quick Reply
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