High School Olympiads
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High School Olympiads
Regional, national, and international math olympiads
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MG
Topic
First Poster
Last Poster
Putnam 1968 A6
sqrtX 11
N
Yesterday at 11:47 PM
by ohiorizzler1434
Source: Putnam 1968
Find all polynomials whose coefficients are all
and whose roots are all real.

11 replies
Affine variety
YamoSky 1
N
Yesterday at 9:01 PM
by amplreneo
Let
. Is it possible to equip
with a finitely generated k-algebra with one generator such that make
be an affine variety?



1 reply
Reducing the exponents for good
RobertRogo 0
Yesterday at 6:38 PM
Source: The national Algebra contest (Romania), 2025, Problem 3/Abstract Algebra (a bit generalized)
Let
be a ring with unity such that for every
there exist
such that
. Prove that
a) If
then 
b) If there is an
such that
then the result from a) may no longer hold.
Authors: Laurențiu Panaitopol, Dorel Miheț, Mihai Opincariu, me, Filip Munteanu




a) If


b) If there is an


Authors: Laurențiu Panaitopol, Dorel Miheț, Mihai Opincariu, me, Filip Munteanu
0 replies
Differential equations , Matrix theory
c00lb0y 3
N
Yesterday at 12:26 PM
by loup blanc
Source: RUDN MATH OLYMP 2024 problem 4
Any idea?? Diff equational system combined with Matrix theory.
Consider the equation dX/dt=X^2, where X(t) is an n×n matrix satisfying the condition detX=0. It is known that there are no solutions of this equation defined on a bounded interval, but there exist non-continuable solutions defined on unbounded intervals of the form (t ,+∞) and (−∞,t). Find n.
Consider the equation dX/dt=X^2, where X(t) is an n×n matrix satisfying the condition detX=0. It is known that there are no solutions of this equation defined on a bounded interval, but there exist non-continuable solutions defined on unbounded intervals of the form (t ,+∞) and (−∞,t). Find n.
3 replies
The matrix in some degree is a scalar
FFA21 4
N
Yesterday at 12:06 PM
by FFA21
Source: MSU algebra olympiad 2025 P2



Is it true that


4 replies
hard number theory problem
danilorj 4
N
Yesterday at 9:01 AM
by c00lb0y
Let
and
be positive integers. Prove that
is not a perfect square.


![\[
a^2 + \left\lceil \frac{4a^2}{b} \right\rceil
\]](http://latex.artofproblemsolving.com/5/6/2/562d003cf06ca46031cec9e4e968f8b67784225f.png)
4 replies
1 viewing
maximum dimention of non-singular subspace
FFA21 1
N
Yesterday at 8:27 AM
by alexheinis
Source: MSU algebra olympiad 2025 P1
We call a linear subspace in the space of square matrices non-singular if all matrices contained in it, except for the zero one, are non-singular. Find the maximum dimension of a non-singular subspace in the space of
a) complex
matrices
b) real
matrices
c) rational
matrices
a) complex

b) real

c) rational

1 reply
functional equation
pratyush 4
N
Yesterday at 8:00 AM
by Mathzeus1024
For the functional equation
, if f ' (0)=p and f ' (5)=q, then prove f ' (-5) = q

4 replies
