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USAMO 2002 Problem 2
MithsApprentice 34
N
by Giant_PT
Let 
be a triangle such that
![\[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \]](//latex.artofproblemsolving.com/6/9/b/69b1964be5f92eb44a36b0b8604bf473fe27e210.png)
where
and 
denote its semiperimeter and its inradius, respectively. Prove that triangle is similar to a triangle 
whose side lengths are all positive integers with no common divisors and determine these integers.
be a triangle such that![\[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \]](http://latex.artofproblemsolving.com/6/9/b/69b1964be5f92eb44a36b0b8604bf473fe27e210.png)
where
and 
denote its semiperimeter and its inradius, respectively. Prove that triangle is similar to a triangle 
whose side lengths are all positive integers with no common divisors and determine these integers.
34 replies
Another config geo with concurrent lines
a_507_bc 17
N
by Rayvhs
Source: BMO SL 2023 G5
Let be a triangle with circumcenter 
. Point 
is the intersection of the parallel line from to 
with the perpendicular line to 
from 
. Let 
be the point where the external bisector of 
intersects with . Let 
be the projection of onto 
. Prove that the lines 
have a common point.
be a triangle with circumcenter 
. Point 
is the intersection of the parallel line from to 
with the perpendicular line to 
from 
. Let 
be the point where the external bisector of 
intersects with . Let 
be the projection of onto 
. Prove that the lines 
have a common point.
17 replies
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
