Vieta's Polynomial x^20-7x^3+1=0
by Goblik, Apr 2, 2025, 7:46 PM
If 
are roots of 
, then find 
are roots of 
, then find 
Number theory
by Maaaaaaath, Apr 2, 2025, 5:10 PM
Let 
be a positive integer . Prove that there exists infinitely many pairs of positive integers 
such that 
and :
be a positive integer . Prove that there exists infinitely many pairs of positive integers 
such that 
and :
inequalities hard
by Cobedangiu, Mar 31, 2025, 11:45 AM
Attachments:
This post has been edited 1 time. Last edited by Cobedangiu, Mar 31, 2025, 2:50 PM
and 
.
Prove 
Find maximum 
2025 Caucasus MO Seniors P8
by BR1F1SZ, Mar 26, 2025, 12:52 AM
Determine for which integers 
the cells of a 
table can be filled with the numbers 
such that the following conditions are satisfied:
the cells of a 
table can be filled with the numbers 
such that the following conditions are satisfied:
- Each of the numbers appears exactly once.
- In any
rectangle, one of the numbers is the arithmetic mean of the other two. - The number
is located in the middle cell of the table.
rectangle, one of the numbers is the arithmetic mean of the other two.
is located in the middle cell of the table.Geo Final but hard to solve with Conics...
by Seungjun_Lee, Jan 18, 2025, 7:13 AM
Let 
be the circumcircle of triangle 
with center 
, and the 
inmixtilinear circle is tangent to 
at 
respectively. 
is the intersection of 
and 
and 
is the intersection of 
and . Prove that the isogonal conjugate of lies on the line passing through the midpoint of 
and .
be the circumcircle of triangle 
with center 
, and the 
inmixtilinear circle is tangent to 
at 
respectively. 
is the intersection of 
and 
and 
is the intersection of 
and . Prove that the isogonal conjugate of lies on the line passing through the midpoint of 
and .This post has been edited 1 time. Last edited by Seungjun_Lee, Jan 18, 2025, 12:44 PM
Polynomial
by EtacticToe, Dec 14, 2024, 6:43 PM
Let 
be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer 
such that 
.
Find all
such that 
be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer 
such that 
.Find all
such that 
This post has been edited 1 time. Last edited by EtacticToe, Dec 14, 2024, 6:44 PM
Reason: The row
Reason: The row
Unlimited candy in PAGMO
by JuanDelPan, Oct 6, 2021, 10:28 PM
Celeste has an unlimited amount of each type of 
types of candy, numerated type 1, type 2, ... type n. Initially she takes 
candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:
She eats a candy of type , and in its position in the row she places one candy type 
followed by one candy type 
(we consider type 
to be type 1, and type 0 to be type ).
She chooses two consecutive candies which are the same type, and eats them.
Find all positive integers for which Celeste can leave the table empty for any value of and any configuration of candies on the table.
types of candy, numerated type 1, type 2, ... type n. Initially she takes 
candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:
She eats a candy of type , and in its position in the row she places one candy type 
followed by one candy type 
(we consider type 
to be type 1, and type 0 to be type ).
She chooses two consecutive candies which are the same type, and eats them.Find all positive integers
for which Celeste can leave the table empty for any value of and any configuration of candies on the table.
This post has been edited 4 times. Last edited by JuanDelPan, Oct 7, 2021, 12:35 AM
Ways to Place Counters on 2mx2n board
by EpicParadox, Mar 28, 2019, 7:23 PM
You have a 
by 
grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place 
counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.
by 
grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place 
counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.Problem 4 from IMO 1997
by iandrei, Jul 28, 2003, 1:33 PM
An 
matrix whose entries come from the set 
is called a silver matrix if, for each 
, the 
-th row and the -th column together contain all elements of 
. Show that:
(a) there is no silver matrix for
;
(b) silver matrices exist for infinitely many values of
.
matrix whose entries come from the set 
is called a silver matrix if, for each 
, the 
-th row and the -th column together contain all elements of 
. Show that:(a) there is no silver matrix for
;(b) silver matrices exist for infinitely many values of
.The oldest, shortest words — "yes" and "no" — are those which require the most thought.
April 2021
February 2021
March 2020
February 2018
January 2018
November 2017
September 2017
July 2017
June 2017
May 2017
April 2017
March 2017
February 2017
December 2016
April 2016