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Nice sequence problem.
mathlover1231 1
N
by vgtcross
Source: Own
Scientists found a new species of bird called “N-coloured rainbow”. They also found out 3 interesting facts about the bird’s life: 1) every day, N-coloured rainbow is coloured in one of N colors.
2) every day, the color is different from yesterday (not every previous day, just yesterday).
3) there are no four days i, j, k, l in the bird’s life such that i<j<k<l with colours a, b, c, d respectively for which a=c ≠ b=d.
Find the greatest possible age (in days) of this bird as a function of N.
2) every day, the color is different from yesterday (not every previous day, just yesterday).
3) there are no four days i, j, k, l in the bird’s life such that i<j<k<l with colours a, b, c, d respectively for which a=c ≠ b=d.
Find the greatest possible age (in days) of this bird as a function of N.
1 reply
Three circles are concurrent
Twoisaprime 23
N
by Curious_Droid
Source: RMM 2025 P5
Let triangle 
be an acute triangle with 
and let 
and 
be its orthocenter and circumcenter, respectively. Let 
be the circle 
. The line 
and the circle of radius centered at 
cross at 
and 
, respectively. Prove that , the circle on diameter 
and circle 
are concurrent.
Proposed by Romania, Radu-Andrew Lecoiu
be an acute triangle with 
and let 
and 
be its orthocenter and circumcenter, respectively. Let 
be the circle 
. The line 
and the circle of radius centered at 
cross at 
and 
, respectively. Prove that , the circle on diameter 
and circle 
are concurrent.Proposed by Romania, Radu-Andrew Lecoiu
23 replies
|a_i/a_j - a_k/a_l| <= C
mathwizard888 32
N
by ezpotd
Source: 2016 IMO Shortlist A2
Find the smallest constant 
for which the following statement holds: among any five positive real numbers 
(not necessarily distinct), one can always choose distinct subscripts 
such that
![\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]](//latex.artofproblemsolving.com/a/e/e/aeeab8e846634c7922e9312ef139739ad3fbe6eb.png)
for which the following statement holds: among any five positive real numbers 
(not necessarily distinct), one can always choose distinct subscripts 
such that![\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]](http://latex.artofproblemsolving.com/a/e/e/aeeab8e846634c7922e9312ef139739ad3fbe6eb.png)
32 replies
Two lines meeting on circumcircle
Zhero 54
N
by Ilikeminecraft
Source: ELMO Shortlist 2010, G4; also ELMO #6
Let be a triangle with circumcircle 
, incenter 
, and -excenter 
. Let the incircle and the -excircle hit 
at 
and 
, respectively, and let 
be the midpoint of arc without . Consider the circle tangent to at and arc 
at 
. If 
intersects again at 
, prove that 
and 
meet on .
Amol Aggarwal.
be a triangle with circumcircle 
, incenter 
, and -excenter 
. Let the incircle and the -excircle hit 
at 
and 
, respectively, and let 
be the midpoint of arc without . Consider the circle tangent to at and arc 
at 
. If 
intersects again at 
, prove that 
and 
meet on .Amol Aggarwal.
54 replies
Help me this problem. Thank you
illybest 3
N
by jasperE3
Find f: R->R such that
f( xy + f(z) ) = (( xf(y) + yf(x) )/2) + z
f( xy + f(z) ) = (( xf(y) + yf(x) )/2) + z
3 replies
line JK of intersection points of 2 lines passes through the midpoint of BC
parmenides51 4
N
by reni_wee
Source: Rioplatense Olympiad 2018 level 3 p4
Let be an acute triangle with 
. be the circumcircle circumscribed to the triangle and the midpoint of the smallest arc of this circle. Let and points of the segments 
and 
respectively such that 
. Let 
be the second intersection point of the circumcircle circumscribed to 
with . Let 
and be the intersections of lines 
and 
with other than 
, respectively. Let 
and 
be the intersection points of lines 
and 
with lines and respectively. Show that the 
line passes through the midpoint of
be an acute triangle with 
. be the circumcircle circumscribed to the triangle and the midpoint of the smallest arc of this circle. Let and points of the segments 
and 
respectively such that 
. Let 
be the second intersection point of the circumcircle circumscribed to 
with . Let 
and be the intersections of lines 
and 
with other than 
, respectively. Let 
and 
be the intersection points of lines 
and 
with lines and respectively. Show that the 
line passes through the midpoint of
4 replies
AGM Problem(Turkey JBMO TST 2025)
HeshTarg 3
N
by ehuseyinyigit
Source: Turkey JBMO TST Problem 6
Given that 
are real numbers, find the smallest possible value of the expression:
are real numbers, find the smallest possible value of the expression:
3 replies
Shortest number theory you might've seen in your life
AlperenINAN 8
N
by HeshTarg
Source: Turkey JBMO TST 2025 P4
Let 
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
8 replies
Squeezing Between Perfect Squares and Modular Arithmetic(JBMO TST Turkey 2025)
HeshTarg 3
N
by BrilliantScorpion85
Source: Turkey JBMO TST Problem 4
If and are primes and is a perfect square, prove that 
is a perfect square.
and are primes and is a perfect square, prove that 
is a perfect square.
3 replies
A bit too easy for P2(Turkey 2025 JBMO TST)
HeshTarg 0
Source: Turkey 2025 JBMO TST P2
Let 
be a positive integer. Aslı and Zehra are playing a game on an 
chessboard. Initially, 
stones are placed on the squares of the board. In each move, Aslı chooses a row or a column; Zehra chooses a row or a column. The number of stones in each square of the chosen row or column must change such that the difference between the number of stones in a square with the most stones and a square with the fewest stones in that same row or column is at most 1. For which values of can Aslı guarantee that after a finite number of moves, all squares on the board will have an equal number of stones, regardless of the initial distribution?
be a positive integer. Aslı and Zehra are playing a game on an 
chessboard. Initially, 
stones are placed on the squares of the board. In each move, Aslı chooses a row or a column; Zehra chooses a row or a column. The number of stones in each square of the chosen row or column must change such that the difference between the number of stones in a square with the most stones and a square with the fewest stones in that same row or column is at most 1. For which values of can Aslı guarantee that after a finite number of moves, all squares on the board will have an equal number of stones, regardless of the initial distribution?
0 replies
