College Math
Topics in undergraduate and graduate studies
Topics in undergraduate and graduate studies
3
M
G
BBookmark
VNew Topic
kLocked
College Math
Topics in undergraduate and graduate studies
Topics in undergraduate and graduate studies
3
M
G
BBookmark
VNew Topic
kLocked
No tags match your search
Manalytic geometry
calculus
real analysis
linear algebra
superior algebra
complex analysis
advanced fields
probability and stats
number theory
topology
Putnam
college contests
articles
function
integration
calculus computations
real analysis unsolved
limit
algebra
trigonometry
matrix
logarithms
derivative
superior algebra unsolved
polynomial
abstract algebra
geometry
inequalities
vector
group theory
linear algebra unsolved
probability
advanced fields unsolved
analytic geometry
domain
induction
LaTeX
Ring Theory
3D geometry
complex analysis unsolved
complex numbers
Functional Analysis
geometric transformation
superior algebra solved
real analysis theorems
search
parameterization
quadratics
real analysis solved
limits
ratio
No tags match your search
MG
Topic
First Poster
Last Poster
Combinatorics from EGMO 2018
BarishNamazov 27
N
an hour ago
by HamstPan38825
Source: EGMO 2018 P3
The
contestant of EGMO are named
. After the competition, they queue in front of the restaurant according to the following rules.
[list]
[*]The Jury chooses the initial order of the contestants in the queue.
[*]Every minute, the Jury chooses an integer
with
.
[list]
[*]If contestant
has at least
other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly
positions.
[*]If contestant
has fewer than
other contestants in front of her, the restaurant opens and process ends.
[/list]
[/list]
[list=a]
[*]Prove that the process cannot continue indefinitely, regardless of the Jury’s choices.
[*]Determine for every
the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.
[/list]


[list]
[*]The Jury chooses the initial order of the contestants in the queue.
[*]Every minute, the Jury chooses an integer


[list]
[*]If contestant



[*]If contestant


[/list]
[/list]
[list=a]
[*]Prove that the process cannot continue indefinitely, regardless of the Jury’s choices.
[*]Determine for every

[/list]
27 replies
Do you have any idea why they all call their problems' characters "Mykhailo"???
mshtand1 1
N
an hour ago
by sarjinius
Source: Ukrainian Mathematical Olympiad 2025. Day 2, Problem 10.7
In a row,
numbers
and
numbers
are written in some order.
Mykhailo counted the number of groups of adjacent numbers, consisting of at least two numbers, whose sum equals
.
(a) Find the smallest possible value of this number.
(b) Find the largest possible value of this number.
Proposed by Anton Trygub




Mykhailo counted the number of groups of adjacent numbers, consisting of at least two numbers, whose sum equals

(a) Find the smallest possible value of this number.
(b) Find the largest possible value of this number.
Proposed by Anton Trygub
1 reply

Polynomial divisible by x^2+1
Miquel-point 2
N
2 hours ago
by lksb
Source: Romanian IMO TST 1981, P1 Day 1
Consider the polynomial
, where
is a prime number. Show that if
is an even number, then the polynomial
is divisible by
.
Mircea Becheanu



![\[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\]](http://latex.artofproblemsolving.com/5/d/f/5df80f885e63852e4f53a6145dafd18d63dc3526.png)

Mircea Becheanu
2 replies
D1030 : An inequalitie
Dattier 1
N
2 hours ago
by lbh_qys
Source: les dattes à Dattier
Let
reals, and
.
Is it true that
?


Is it true that

1 reply
IGO 2021 P1
SPHS1234 14
N
3 hours ago
by LeYohan
Source: igo 2021 intermediate p1
Let
be a triangle with
. Let
be the orthocenter of
. Point
is the midpoint of
and point
lies on the side
such that
. Prove that
.
Proposed by Tran Quang Hung - Vietnam










Proposed by Tran Quang Hung - Vietnam
14 replies
Nationalist Combo
blacksheep2003 16
N
3 hours ago
by Martin2001
Source: USEMO 2019 Problem 5
Let
be a regular polygon, and let
be its set of vertices. Each point in
is colored red, white, or blue. A subset of
is patriotic if it contains an equal number of points of each color, and a side of
is dazzling if its endpoints are of different colors.
Suppose that
is patriotic and the number of dazzling edges of
is even. Prove that there exists a line, not passing through any point in
, dividing
into two nonempty patriotic subsets.
Ankan Bhattacharya





Suppose that




Ankan Bhattacharya
16 replies
subsets of {1,2,...,mn}
N.T.TUAN 10
N
3 hours ago
by de-Kirschbaum
Source: USA TST 2005, Problem 1
Let
be an integer greater than
. For a positive integer
, let
. Suppose that there exists a
-element set
such that
(a) each element of
is an
-element subset of
;
(b) each pair of elements of
shares at most one common element;
and
(c) each element of
is contained in exactly two elements of
.
Determine the maximum possible value of
in terms of
.






(a) each element of



(b) each pair of elements of

and
(c) each element of


Determine the maximum possible value of


10 replies
Sum and product of digits
Sadigly 4
N
3 hours ago
by jasperE3
Source: Azerbaijan NMO 2018
For a positive integer
, define
and
, where
and
denote the product and sum of the digits of
, respectively. Find all solutions to







4 replies
Geometry
smartvong 0
3 hours ago
Source: UM Mathematical Olympiad 2024
Let
be a point inside a triangle
. Let
meet
at
, let
meet
at
, and let
meet
at
. Let
be the point such that
is the midpoint of
, let
be the point such that
is the midpoint of
, and let
be the point such that
is the midpoint of
. Prove that points
cannot all lie strictly inside the circumcircle of triangle
.






















0 replies
angles in triangle
AndrewTom 34
N
4 hours ago
by happypi31415
Source: BrMO 2012/13 Round 2
The point
lies inside triangle
so that
. The point
is such that
is a parallelogram. Prove that
.






34 replies
