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Any nice way to do this?
NamelyOrange 1
N
by HockeyMaster85
Source: Taichung P.S.1 math program tryouts
How many ordered pairs
are there such that 
and 
?
How many ordered pairs
are there such that 
and 
?
1 reply
Polynomial optimization problem
ReticulatedPython 2
N
by Mathzeus1024
Let 
, where 
is a real number. Prove that for all positive , 
, where 
is a real number. Prove that for all positive , 
2 replies
Complex + Radical Evaluation
Saucepan_man02 4
N
by Mathzeus1024
Evaluate: (without calculators)
4 replies
Inequality
JK1603JK 1
N
by lbh_qys
Let 
then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
1 reply
Inequality
JK1603JK 1
N
by lbh_qys
Prove that 9ab\left(a-b+c\right)+9bc\left(b-c+a\right)+9ca\left(c-a+b\right)\ge \left(a+b+c\right)^{3},\ \ a\ge 0\ge b\ge c: a+b+c\le 0.
1 reply
Inequalities
sqing 2
N
by sqing
Let 
and 
Prove that
Let and 
Prove that
Let and 
Prove that
Let and 
Prove that
and 
Prove that
Let and 
Prove that
Let and 
Prove that
Let and 
Prove that
2 replies
geometry
Tony_stark0094 1
N
by Tony_stark0094
Consider 
let 
and 
be the circles passing through 
and 
respectively such that 
is tangent to and define 
to be a point such that it lies on both the circles and prove that 
and 
are perpendicular.
let 
and 
be the circles passing through 
and 
respectively such that 
is tangent to and define 
to be a point such that it lies on both the circles and prove that 
and 
are perpendicular.
1 reply
one very nice!
MihaiT 1
N
by MihaiT
Given 
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
. What will be displayed after three successive runs: 5,2,5,1,4 | 5,2,5,1,11 | 5,2,5,1,20?
One answer is possible:
a)10;5;0;
b)20;7;0;
c)20;7;1;
d)10;10;0;
e)10;7;0;
f)20;5;0,
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
. What will be displayed after three successive runs: 5,2,5,1,4 | 5,2,5,1,11 | 5,2,5,1,20?One answer is possible:
a)10;5;0;
b)20;7;0;
c)20;7;1;
d)10;10;0;
e)10;7;0;
f)20;5;0,
1 reply
