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Cyclic ine
m4thbl3nd3r   5
N an hour ago by sqing
Let $a,b,c>0$ such that $a+b+c=3$. Prove that $$a^3b+b^3c+c^3a+9abc\le 12$$
5 replies
m4thbl3nd3r
Yesterday at 3:17 PM
sqing
an hour ago
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Interesting inequality
sqing   1
N 2 hours ago by sqing
Source: Own
Let $ a,b,c\geq 0,(ab+c)(ac+b)\neq 0 $ and $ a+b+c=3 . $ Prove that
$$ \frac{1}{ab+kc}+\frac{1}{ac+kb} \geq\frac{4}{3k} $$Where $ k\geq 3. $
$$ \frac{1}{ab+2c}+\frac{1}{ac+2b} \geq\frac{16}{25} $$$$ \frac{1}{ab+3c}+\frac{1}{ac+3b} \geq\frac{4}{9} $$$$ \frac{1}{ab+4c}+\frac{1}{ac+4b} \geq\frac{1}{3} $$

1 reply
sqing
2 hours ago
sqing
2 hours ago
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Problem 2
delegat   145
N 2 hours ago by Marcus_Zhang
Source: 0
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]

Proposed by Angelo Di Pasquale, Australia
145 replies
delegat
Jul 10, 2012
Marcus_Zhang
2 hours ago
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CMJ 1284 (Crazy Concyclic Circumcenter Circus)
kgator   0
2 hours ago
Source: College Mathematics Journal Volume 55 (2024), Issue 4: https://doi.org/10.1080/07468342.2024.2373015
1284. Proposed by Tran Quang Hung, High School for Gifted Students, Vietnam National University, Hanoi, Vietnam. Let quadrilateral $ABCD$ not be a trapezoid such that there is a circle centered at $I$ that is tangent to the four sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$. Let $X$, $Y$, $Z$, and $W$ be the circumcenters of the triangles $IAB$, $IBC$, $ICD$, and $IDA$, respectively. Prove that there is a circle containing the circumcenters of the triangles $XAB$, $YBC$, $ZCD$, and $WDA$.
0 replies
kgator
2 hours ago
0 replies
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euler-totient function
Laan   3
N 2 hours ago by top1vien
Proof that there are infinitely many positive integers $n$ such that
$\varphi(n)<\varphi(n+1)<\varphi(n+2)$
3 replies
Laan
Friday at 7:13 AM
top1vien
2 hours ago
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Loop of Logarithms
scls140511   12
N 2 hours ago by yyhloveu1314
Source: 2024 China Round 1 (Gao Lian)
Round 1

1 Real number $m>1$ satisfies $\log_9 \log_8 m =2024$. Find the value of $\log_3 \log_2 m$.
12 replies
scls140511
Sep 8, 2024
yyhloveu1314
2 hours ago
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A cyclic inequality
JK1603JK   1
N 2 hours ago by jokehim
Source: unknown
Let a,b,c be real numbers. Prove that a^6+b^6+c^6\ge 2(a+b+c)(ab+bc+ca)(a-b)(b-c)(c-a).
1 reply
JK1603JK
4 hours ago
jokehim
2 hours ago
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bank accounts
cloventeen   1
N 2 hours ago by jkim0656
edgar has three bank accounts, each with an integer amount of dollars in it. He is only allowed to transfer money from one account to another if, by doing so, the latter ends up with double the money it had previously. Prove that edgar can always transfer all of his money into two accounts. Will he always be able to transfer all of his money into a single account?
1 reply
cloventeen
3 hours ago
jkim0656
2 hours ago
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Divisibility
RenheMiResembleRice   0
2 hours ago
Source: Byer
Prove that for all n ∈ ℕ, 133|($11^{\left(n+2\right)}+12^{\left(2n+1\right)}$).
0 replies
RenheMiResembleRice
2 hours ago
0 replies
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2 var inquality
sqing   3
N 2 hours ago by sqing
Source: Own
Let $ a,b>0 $ and $ 3a+4b=a^3b^2. $ Prove that
$$2a+b+\dfrac{2}{a}+\dfrac{3}{b}\geq \frac{11}{\sqrt2}$$$$a+\dfrac{2}{a}+\dfrac{3}{b}\geq 4\sqrt[4]{\frac23}$$$$\dfrac{2}{a}+\dfrac{3}{b}\geq 2\sqrt[4]3$$$$3a+\dfrac{2}{a}+\dfrac{3}{b}\geq \sqrt[4]{354+66\sqrt{33}}$$
3 replies
sqing
Mar 4, 2025
sqing
2 hours ago
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Tangent.
steven_zhang123   1
N 2 hours ago by ehuseyinyigit
Source: China TST 2001 Quiz 6 P1
In \( \triangle ABC \) with \( AB > BC \), a tangent to the circumcircle of \( \triangle ABC \) at point \( B \) intersects the extension of \( AC \) at point \( D \). \( E \) is the midpoint of \( BD \), and \( AE \) intersects the circumcircle of \( \triangle ABC \) at \( F \). Prove that \( \angle CBF = \angle BDF \).
1 reply
steven_zhang123
3 hours ago
ehuseyinyigit
2 hours ago
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Find min
hunghd8   8
N Yesterday at 9:14 AM by imnotgoodatmathsorry
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$
8 replies
hunghd8
Friday at 12:10 PM
imnotgoodatmathsorry
Yesterday at 9:14 AM
J
Find min
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hunghd8
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hunghd8
#1 h  Friday at 12:10 PM
Y by
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$
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Mathzeus1024
742 posts
Mathzeus1024
#2 h  Friday at 2:06 PM
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Withdrawn
This post has been edited 3 times. Last edited by Mathzeus1024, Yesterday at 8:22 AM
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ehuseyinyigit
784 posts
ehuseyinyigit
#4 h  Friday at 3:14 PM
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You're wrong of course.
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sqing
41181 posts
sqing
#5 h  Friday at 3:20 PM
Y by
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Prove that
$$a^2+b^2+c^2\geq 2$$*
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imnotgoodatmathsorry
60 posts
imnotgoodatmathsorry
#6 h  Friday at 4:19 PM
Y by
hunghd8 wrote:
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$

WLOG, we have $(a-1)(b-1) \ge 0$ so we have: $a+b+c \ge 2 +abc \ge 2-c+c(a+b)$
so: $(a+b-2)(1-c)>=0$
Now you need to solve 2 cases: $a+b \ge 2$ and $c \le 1$ or $a+b \le 2$ and $c \ge 1$ which is kinda ez
So, you can prove $P \ge 2$ with equality occurs when $(a;b;c)=(2;0;0)$
This post has been edited 2 times. Last edited by imnotgoodatmathsorry, Friday at 4:20 PM
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hunghd8
40 posts
hunghd8
#7 h  Yesterday at 1:57 AM
Y by
imnotgoodatmathsorry wrote:
hunghd8 wrote:
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$

WLOG, we have $(a-1)(b-1) \ge 0$ so we have: $a+b+c \ge 2 +abc \ge 2-c+c(a+b)$
so: $(a+b-2)(1-c)>=0$
Now you need to solve 2 cases: $a+b \ge 2$ and $c \le 1$ or $a+b \le 2$ and $c \ge 1$ which is kinda ez
So, you can prove $P \ge 2$ with equality occurs when $(a;b;c)=(2;0;0)$
$a+b \le 2$ and $c \ge 1$ which is kinda ez??
Z K Y
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imnotgoodatmathsorry
60 posts
imnotgoodatmathsorry
#8 h  Yesterday at 4:10 AM
Y by
hunghd8 wrote:
imnotgoodatmathsorry wrote:
hunghd8 wrote:
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$

WLOG, we have $(a-1)(b-1) \ge 0$ so we have: $a+b+c \ge 2 +abc \ge 2-c+c(a+b)$
so: $(a+b-2)(1-c)>=0$
Now you need to solve 2 cases: $a+b \ge 2$ and $c \le 1$ or $a+b \le 2$ and $c \ge 1$ which is kinda ez
So, you can prove $P \ge 2$ with equality occurs when $(a;b;c)=(2;0;0)$
$a+b \le 2$ and $c \ge 1$ which is kinda ez??

Well, you should kind of present $P$ by $c$ and that's all right?
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hunghd8
40 posts
hunghd8
#9 h  Yesterday at 6:32 AM
Y by
imnotgoodatmathsorry wrote:
hunghd8 wrote:
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$

WLOG, we have $(a-1)(b-1) \ge 0$ so we have: $a+b+c \ge 2 +abc \ge 2-c+c(a+b)$
so: $(a+b-2)(1-c)>=0$
Now you need to solve 2 cases: $a+b \ge 2$ and $c \le 1$ or $a+b \le 2$ and $c \ge 1$ which is kinda ez
So, you can prove $P \ge 2$ with equality occurs when $(a;b;c)=(2;0;0)$
You're wrong of course
Z K Y
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imnotgoodatmathsorry
60 posts
imnotgoodatmathsorry
#10 h  Yesterday at 9:14 AM
Y by
hunghd8 wrote:
imnotgoodatmathsorry wrote:
hunghd8 wrote:
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$

WLOG, we have $(a-1)(b-1) \ge 0$ so we have: $a+b+c \ge 2 +abc \ge 2-c+c(a+b)$
so: $(a+b-2)(1-c)>=0$
Now you need to solve 2 cases: $a+b \ge 2$ and $c \le 1$ or $a+b \le 2$ and $c \ge 1$ which is kinda ez
So, you can prove $P \ge 2$ with equality occurs when $(a;b;c)=(2;0;0)$
You're wrong of course

oh, well, thanks for noticing btw
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