such that 
.
and 
Prove that
Where 
)
and 
Prove that
![$$a+\dfrac{2}{a}+\dfrac{3}{b}\geq 4\sqrt[4]{\frac23}$$](http://latex.artofproblemsolving.com/b/c/1/bc1ceb6a41d5fc0598cdd0cc455875087f9d4585.png)
![$$\dfrac{2}{a}+\dfrac{3}{b}\geq 2\sqrt[4]3$$](http://latex.artofproblemsolving.com/f/2/2/f22009d039177d6343c55023754310e0e59c1744.png)
![$$3a+\dfrac{2}{a}+\dfrac{3}{b}\geq \sqrt[4]{354+66\sqrt{33}}$$](http://latex.artofproblemsolving.com/e/c/9/ec9133ad1679ba58cefdb5de88de5550e6903d83.png)
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a^2+b^2+c^2+x^2+y^2=1. Find the maximum value of the expression (ax+by)^2+(bx+cy)^2
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Problem 2
delegat 145
N
by Marcus_Zhang
Source: 0
Let 
be an integer, and let 
be positive real numbers such that 
. Prove that
![\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]](//latex.artofproblemsolving.com/7/7/f/77fb719e975225b59c64f1f3aa102c98a6c25f50.png)
Proposed by Angelo Di Pasquale, Australia
be an integer, and let 
be positive real numbers such that 
. Prove that![\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]](http://latex.artofproblemsolving.com/7/7/f/77fb719e975225b59c64f1f3aa102c98a6c25f50.png)
Proposed by Angelo Di Pasquale, Australia
145 replies
CMJ 1284 (Crazy Concyclic Circumcenter Circus)
kgator 0
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 
not be a trapezoid such that there is a circle centered at 
that is tangent to the four sides 
, 
, 
, and 
. Let 
, 
, 
, and 
be the circumcenters of the triangles 
, 
, 
, and 
, respectively. Prove that there is a circle containing the circumcenters of the triangles 
, 
, 
, and 
.
not be a trapezoid such that there is a circle centered at 
that is tangent to the four sides 
, 
, 
, and 
. Let 
, 
, 
, and 
be the circumcenters of the triangles 
, 
, 
, and 
, respectively. Prove that there is a circle containing the circumcenters of the triangles 
, 
, 
, and 
.
0 replies
euler-totient function
Laan 3
N
by top1vien
Proof that there are infinitely many positive integers 
such that
such that
3 replies
Loop of Logarithms
scls140511 12
N
by yyhloveu1314
Source: 2024 China Round 1 (Gao Lian)
Round 1
1 Real number
satisfies 
. Find the value of 
.
1 Real number
satisfies 
. Find the value of 
.
12 replies
A cyclic inequality
JK1603JK 1
N
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
bank accounts
cloventeen 1
N
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
Tangent.
steven_zhang123 1
N
by ehuseyinyigit
Source: China TST 2001 Quiz 6 P1
In 
with 
, a tangent to the circumcircle of at point 
intersects the extension of 
at point 
. 
is the midpoint of 
, and 
intersects the circumcircle of at 
. Prove that 
.
with 
, a tangent to the circumcircle of at point 
intersects the extension of 
at point 
. 
is the midpoint of 
, and 
intersects the circumcircle of at 
. Prove that 
.
1 reply
Inequality
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srnjbr wrote:
a^2+b^2+c^2+x^2+y^2=1. Find the maximum value of the expression (ax+by)^2+(bx+cy)^2
This post has been edited 1 time. Last edited by sqing, Yesterday at 2:59 AM
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