High School Olympiads
Regional, national, and international math olympiads
Regional, national, and international math olympiads
3
M
G
BBookmark
VNew Topic
kLocked
High School Olympiads
Regional, national, and international math olympiads
Regional, national, and international math olympiads
3
M
G
BBookmark
VNew Topic
kLocked
No tags match your search
Mfloor function
algebra
combinatorics
geometry
inequalities
number theory
IMO
articles
inequalities proposed
function
algebra unsolved
circumcircle
trigonometry
number theory unsolved
inequalities unsolved
polynomial
geometry unsolved
geometry proposed
combinatorics unsolved
number theory proposed
functional equation
algebra proposed
modular arithmetic
induction
geometric transformation
incenter
calculus
3D geometry
combinatorics proposed
quadratics
Inequality
reflection
ratio
logarithms
prime numbers
analytic geometry
floor function
angle bisector
search
parallelogram
integration
Diophantine equation
rectangle
LaTeX
limit
complex numbers
probability
graph theory
conics
Euler
cyclic quadrilateral
No tags match your search
MG
Topic
First Poster
Last Poster
Geometry with orthocenter config
thdnder 6
N
2 hours ago
by ohhh
Source: Own
Let
be a triangle, and let
be its altitudes. Let
be its orthocenter, and let
and
be the circumcenters of triangles
and
. Let
be the second intersection of the circumcircles of triangles
and
. Prove that the lines
,
, and
-median of
are concurrent.














6 replies
1 viewing
Strange Inequality
anantmudgal09 40
N
2 hours ago
by starchan
Source: INMO 2020 P4
Let
be an integer and let
be
real numbers such that
. Prove that
Proposed by Kapil Pause





Proposed by Kapil Pause
40 replies
Finding Solutions
MathStudent2002 22
N
2 hours ago
by ihategeo_1969
Source: Shortlist 2016, Number Theory 5
Let
be a positive integer which is not a perfect square, and consider the equation
Let
be the set of positive integers
for which the equation admits a solution in
with
, and let
be the set of positive integers for which the equation admits a solution in
with
. Show that
.

![\[k = \frac{x^2-a}{x^2-y^2}.\]](http://latex.artofproblemsolving.com/c/2/2/c223ccbc272d07c19632c5b8883571023e47a395.png)








22 replies

USAMO 2000 Problem 3
MithsApprentice 10
N
2 hours ago
by HamstPan38825
A game of solitaire is played with
red cards,
white cards, and
blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of
and
the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.





10 replies
Hard limits
Snoop76 7
N
2 hours ago
by MihaiT












7 replies
Additive combinatorics (re Cauchy-Davenport)
mavropnevma 3
N
2 hours ago
by Orzify
Source: Romania TST 3 2010, Problem 4
Let
and
be two finite subsets of the half-open interval
such that
and
for no
and no
. Prove that the set
has at least
elements.
***









***
3 replies
Ducks can play games now apparently
MortemEtInteritum 34
N
3 hours ago
by HamstPan38825
Source: USA TST(ST) 2020 #1
Let
,
,
be fixed positive integers. There are
ducks sitting in a
circle, one behind the other. Each duck picks either rock, paper, or scissors, with
ducks
picking rock,
ducks picking paper, and
ducks picking scissors.
A move consists of an operation of one of the following three forms:
[list]
[*] If a duck picking rock sits behind a duck picking scissors, they switch places.
[*] If a duck picking paper sits behind a duck picking rock, they switch places.
[*] If a duck picking scissors sits behind a duck picking paper, they switch places.
[/list]
Determine, in terms of
,
, and
, the maximum number of moves which could take
place, over all possible initial configurations.




circle, one behind the other. Each duck picks either rock, paper, or scissors, with

picking rock,


A move consists of an operation of one of the following three forms:
[list]
[*] If a duck picking rock sits behind a duck picking scissors, they switch places.
[*] If a duck picking paper sits behind a duck picking rock, they switch places.
[*] If a duck picking scissors sits behind a duck picking paper, they switch places.
[/list]
Determine, in terms of



place, over all possible initial configurations.
34 replies
Floor sequence
va2010 87
N
3 hours ago
by Mathgloggers
Source: 2015 ISL N1
Determine all positive integers
such that the sequence
defined by
contains at least one integer term.


![\[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \]](http://latex.artofproblemsolving.com/2/1/b/21b2a1f93b11a94b84e7b55f4b4f679aa20e36c6.png)
87 replies
INMO 2019 P3
div5252 45
N
3 hours ago
by anudeep
Let
be distinct positive integers. Prove that
Further, determine when equality holds.


45 replies
My unsolved problem
ZeltaQN2008 1
N
3 hours ago
by Adywastaken
Source: Belarus 2017
Find all funcition
such that for all any
:



1 reply
