G
Topic
First Poster
Last Poster
24th PMO, Qualifying Stage #7
elpianista227 1
N
by elpianista227
Suppose 
are the roots of the polynomial 
. Let 
be the unique monic polynomial whose roots are 
. Find 
.
are the roots of the polynomial 
. Let 
be the unique monic polynomial whose roots are 
. Find 
.
1 reply
27th Philippine Mathematical Olympiad Area Stage #5
Siopao_Enjoyer 1
N
by Siopao_Enjoyer
Find the sum of the cubes of the roots of the polynomial p(x)=x^3-x^2+2x-3.
1 reply
Help me please
dssdgeww 1
N
by whwlqkd
Prove that there exists a positive integer n with 2024 prime divisors such that n| 2^n + 1
1 reply
[PMO20 Qualifying I.13] Log raised to Log
LilKirb 1
N
by LilKirb
Find the sum of the solutions to the logarithmic equation:
![\[ x^{\log{x}} = 10^{2 - 3\log{x} + 2(\log{x})^2}\]](//latex.artofproblemsolving.com/b/f/b/bfb8185992867a8926b5c62198a799c935771d53.png)
where 
is the logarithm of 
to the base 
![\[ x^{\log{x}} = 10^{2 - 3\log{x} + 2(\log{x})^2}\]](http://latex.artofproblemsolving.com/b/f/b/bfb8185992867a8926b5c62198a799c935771d53.png)
where 
is the logarithm of 
to the base 
1 reply
.
Prove that
Algebraic Manipulation
Darealzolt 4
N
by jasperE3
It is known that 
that satisfies
![\[
a^3+b^3=1957
\]](//latex.artofproblemsolving.com/5/d/8/5d8d2a40734ec6b0a9149f9a4ae3a5828d5055f3.png)
![\[
(a+b)(a+1)(b+1)=2014
\]](//latex.artofproblemsolving.com/3/0/0/300f2d16e669a1d8d9c54b357a6eaf7a8eb5d826.png)
Hence, find the value of 
that satisfies![\[
a^3+b^3=1957
\]](http://latex.artofproblemsolving.com/5/d/8/5d8d2a40734ec6b0a9149f9a4ae3a5828d5055f3.png)
![\[
(a+b)(a+1)(b+1)=2014
\]](http://latex.artofproblemsolving.com/3/0/0/300f2d16e669a1d8d9c54b357a6eaf7a8eb5d826.png)
Hence, find the value of 
4 replies
not obvious trig identity!
mathmax001 1
N
by joeym2011
Problem ( trigonometry )
Let
and n a positive integer 
, Show that : 
Here is my take in this video: https://youtu.be/DBPyHNqk0GI?si=9r-YDuwv794AGe1p
Let
and n a positive integer 
, Show that : 
Here is my take in this video: https://youtu.be/DBPyHNqk0GI?si=9r-YDuwv794AGe1p
1 reply
confused
greenplanet2050 3
N
by mathprodigy2011
um something weird happened today
I was doing the 2002 aime ii and i tried #9
I used PIE with
so for like 1 same element i did
cause there are 10 ways to choose 1 element that will be repeated. Similarly for 2 same elements it would be 
So if
the answer would be ![$(2^{10}-1)-([A_1+A_3+A_5+A_7+A_9]-[A_2+A_4+A_6+A_8+A_{10}].$](//latex.artofproblemsolving.com/7/3/1/73130cab8af6ffb5c2638548fdf6af3ecc70a6ed.png)
But this number turned out to be 
Later when looking at the solution, i found out that the correct number was
But I realized that 
So I was really confused of why i got the right answer somehow in my calculations.
Can someone explain why this happened? Thanks! :)
I was doing the 2002 aime ii and i tried #9
I used PIE with
so for like 1 same element i did
cause there are 10 ways to choose 1 element that will be repeated. Similarly for 2 same elements it would be 
So if
the answer would be ![$(2^{10}-1)-([A_1+A_3+A_5+A_7+A_9]-[A_2+A_4+A_6+A_8+A_{10}].$](http://latex.artofproblemsolving.com/7/3/1/73130cab8af6ffb5c2638548fdf6af3ecc70a6ed.png)
But this number turned out to be 
Later when looking at the solution, i found out that the correct number was
But I realized that 
So I was really confused of why i got the right answer somehow in my calculations.Can someone explain why this happened? Thanks! :)
3 replies
Easy one
irregular22104 2
N
by trangbui
Given two positive integers 
written on the board. We apply the following rule: At each step, we will add all the numbers that are the sum of the two numbers on the board so that the sum does not appear on the board. For example, if the two initial numbers are 
; then the numbers on the board after step 1 are 
; after step 2 are 
1) With
; 
, prove that the number 2024 cannot appear on the board.
2) With
; 
, prove that the number 2024 can appear on the board.
written on the board. We apply the following rule: At each step, we will add all the numbers that are the sum of the two numbers on the board so that the sum does not appear on the board. For example, if the two initial numbers are 
; then the numbers on the board after step 1 are 
; after step 2 are 
1) With
; 
, prove that the number 2024 cannot appear on the board.2) With
; 
, prove that the number 2024 can appear on the board.
2 replies
