and 
Prove that
Prove that
![\[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]](http://latex.artofproblemsolving.com/e/b/2/eb28918359848dd07784a3c2c6a34b934743c016.png)
be 
.![\[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]](http://latex.artofproblemsolving.com/3/1/5/315341a7adf35bc04e40ee0685a011c9f7b76115.png)
without Newton Sums.
Y by
Prove that the number 
is divisible by 2024.
is divisible by 2024.G
Topic
First Poster
Last Poster
Frankenstein FE
NamelyOrange 3
N
by jasperE3
[quote = My own problem]Solve the FE 
over 
. Ignore "pathological" solutions.[/quote]
How do I solve this? I made this while messing around, and I have no clue as to what to do...
over 
. Ignore "pathological" solutions.[/quote]How do I solve this? I made this while messing around, and I have no clue as to what to do...
3 replies
Find the domain and range of $f(x)=\frac{1}{1-2\cos x}.$
Vulch 1
N
by aidan0626
Find the domain and range of 
1 reply
Find the domain and range of $f(x)=\frac{4-x}{x-4}.$
Vulch 1
N
by aidan0626
Find the domain and range of 
1 reply
Amc 10 mock
Mathsboy100 4
N
by pooh123
let ![\[\lfloor x \rfloor\]](//latex.artofproblemsolving.com/d/3/f/d3f8facf62413e1f17c89b9c183a8735c2defe8b.png)
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which ![\[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]](//latex.artofproblemsolving.com/f/a/9/fa96f2d6217a16029e5f03499e0a83deffdfa28a.png)
![\[\lfloor x \rfloor\]](http://latex.artofproblemsolving.com/d/3/f/d3f8facf62413e1f17c89b9c183a8735c2defe8b.png)
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which ![\[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]](http://latex.artofproblemsolving.com/f/a/9/fa96f2d6217a16029e5f03499e0a83deffdfa28a.png)
4 replies
Purple Comet High School Math Meet 2024 P1
franklin2013 3
N
by codegirl2013
Joe ate one half of a fifth of a pizza. Gale ate one third of a quarter of that pizza. The difference in the amounts that the two ate was 
of the pizza, where 
is a positive integer. Find .
of the pizza, where 
is a positive integer. Find .
3 replies
Inequality, tougher than it looks
tom-nowy 2
N
by pooh123
Prove that for 
Is there an elegant way to prove this?
Is there an elegant way to prove this?
2 replies
2024 8's
G
H
G
H
Source: Moldova JTST 2024 P2
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Y by megarnie
We basically want to prove 
, or just 
.
Now since
we have 
, now since 
we must prove 
. Now from here we will split the division in prime powers which makes sense due to crt.
all hold trivially, the last one from fermat's theorem, therefore we have 
for all positive integers 
and ofc when 
we indeed get 
which implies the original problem thus we are done 
.
, or just 
.Now since
we have 
, now since 
we must prove 
. Now from here we will split the division in prime powers which makes sense due to crt.
all hold trivially, the last one from fermat's theorem, therefore we have 
for all positive integers 
and ofc when 
we indeed get 
which implies the original problem thus we are done .
Z
K
Y
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Y by
We have 
. Clearly, the number is divisible by 8, and we can see that it is also divisible by 11 from the divisibility rule. Now, all we have to show is that it is divisible by 23 now, or that 
. Since both 8 and 9 are relatively prime to 23, this is the same as showing that 
This is equivalent to 
, which is true by Fermat's.
. Clearly, the number is divisible by 8, and we can see that it is also divisible by 11 from the divisibility rule. Now, all we have to show is that it is divisible by 23 now, or that 
. Since both 8 and 9 are relatively prime to 23, this is the same as showing that 
This is equivalent to 
, which is true by Fermat's.
Z
K
Y
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x
x
it has a valid inverse allowing thisN Quick Reply
G
H
