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Ducks can play games now apparently
MortemEtInteritum 35
N
by pi271828
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.
, 
, 
be fixed positive integers. There are 
ducks sitting in acircle, one behind the other. Each duck picks either rock, paper, or scissors, with
duckspicking 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 takeplace, over all possible initial configurations.
35 replies
1 viewing
2017 IGO Advanced P3
bgn 18
N
by Circumcircle
Source: 4th Iranian Geometry Olympiad (Advanced) P3
Let 
be the circumcenter of triangle 
. Line 
intersects the altitude from 
at point 
. Let 
be the midpoints of 
, 
respectively. If 
intersects 
at 
, and the circumcircle of triangle 
meets 
at 
, prove that 
is cyclic.
Proposed by Ali Daeinabi - Hamid Pardazi
be the circumcenter of triangle 
. Line 
intersects the altitude from 
at point 
. Let 
be the midpoints of 
, 
respectively. If 
intersects 
at 
, and the circumcircle of triangle 
meets 
at 
, prove that 
is cyclic.Proposed by Ali Daeinabi - Hamid Pardazi
18 replies
![\[
\text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\
f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right)
\]](http://latex.artofproblemsolving.com/2/a/7/2a73a5e58eab4a994fbd83160fb79a0b00152951.png)
Euler line of incircle touching points /Reposted/
Eagle116 6
N
by pigeon123
Let be a triangle with incentre 
and circumcentre . Let 
be the touchpoints of the incircle with , 
, respectively. Prove that 
is the Euler line of 
.
be a triangle with incentre 
and circumcentre . Let 
be the touchpoints of the incircle with , 
, respectively. Prove that 
is the Euler line of 
.
6 replies
Parallel lines on a rhombus
buratinogigle 1
N
by Giabach298
Source: Own, Entrance Exam for Grade 10 Admission, HSGS 2025
Given the rhombus 
with its incircle 
. Let 
and 
be the points of tangency of with and respectively. On the edges 
and 
, take points 
and 
such that 
is tangent to at 
. Suppose 
is the intersection point of the lines 
and 
. Prove that two lines 
and 
are parallel or coincide.
with its incircle 
. Let 
and 
be the points of tangency of with and respectively. On the edges 
and 
, take points 
and 
such that 
is tangent to at 
. Suppose 
is the intersection point of the lines 
and 
. Prove that two lines 
and 
are parallel or coincide.
1 reply
Orthocenter lies on circumcircle
whatshisbucket 90
N
by bjump
Source: 2017 ELMO #2
Let be a triangle with orthocenter 
and let 
be the midpoint of 
Suppose that and are distinct points on the circle with diameter 
different from 
such that lies on line 
Prove that the orthocenter of 
lies on the circumcircle of 
Proposed by Michael Ren
be a triangle with orthocenter 
and let 
be the midpoint of 
Suppose that and are distinct points on the circle with diameter 
different from 
such that lies on line 
Prove that the orthocenter of 
lies on the circumcircle of 
Proposed by Michael Ren
90 replies
Polish MO Finals 2014, Problem 4
j___d 3
N
by ariopro1387
Source: Polish MO Finals 2014
Denote the set of positive rational numbers by 
. Find all functions 
that satisfy
for all integers 
and rational numbers 
.
. Find all functions 
that satisfy
for all integers 
and rational numbers 
.
3 replies
S(an) greater than S(n)
ilovemath0402 1
N
by ilovemath0402
Source: Inspired by an old result
Find all positive integer 
such that 
(
is sum of digit of in base 10)
P/s:
such that 
(
is sum of digit of in base 10)P/s:
are one of such 1 reply
Hagge-like circles, Jerabek hyperbola, Lemoine cubic
kosmonauten3114 0
Source: My own
Let 
be a scalene oblique triangle with circumcenter and orthocenter , and (
, 
) a point in the plane.
Let
, 
be the circumcevian triangles of , , respectively.
Let
be the pedal triangle of with respect to .
Let
be the reflection in 
of 
. Define 
, 
cyclically.
Let
be the reflection in of 
. Define 
, 
cyclically.
Let
, 
be the circumcenters of 
, 
, respectively.
Prove that:
1), , are collinear if and only if lies on the Jerabek hyperbola of .
2), , are collinear if and only if lies on the Lemoine cubic (= 
) of .
be a scalene oblique triangle with circumcenter and orthocenter , and (
, 
) a point in the plane.Let
, 
be the circumcevian triangles of , , respectively.Let
be the pedal triangle of with respect to .Let
be the reflection in 
of 
. Define 
, 
cyclically.Let
be the reflection in of 
. Define 
, 
cyclically.Let
, 
be the circumcenters of 
, 
, respectively.Prove that:
1)
, , are collinear if and only if lies on the Jerabek hyperbola of .2)
, , are collinear if and only if lies on the Lemoine cubic (= 
) of .
0 replies
Incenter perpendiculars and angle congruences
math154 84
N
by zuat.e
Source: ELMO Shortlist 2012, G3
is a triangle with incenter . The foot of the perpendicular from to is 
, and the foot of the perpendicular from to 
is . Prove that 
.Alex Zhu.
84 replies
