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Isogonal Conjugates of Nagel and Gergonne Point
SerdarBozdag 6
N
by ohiorizzler1434
Source: http://math.fau.edu/yiu/Oldwebsites/Geometry2013Fall/Geometry2013Chapter12.pdf
Proposition 12.1.
(a) The isogonal conjugate of the Gergonne point is the insimilicenter of
the circumcircle and the incircle.
(b) The isogonal conjugate of the Nagel point is the exsimilicenter of the circumcircle and
the incircle.
Note: I need synthetic solution.
(a) The isogonal conjugate of the Gergonne point is the insimilicenter of
the circumcircle and the incircle.
(b) The isogonal conjugate of the Nagel point is the exsimilicenter of the circumcircle and
the incircle.
Note: I need synthetic solution.
6 replies
Looking for someone to work with
midacer 1
N
by wipid98
I’m looking for a motivated study partner (or small group) to collaborate on college-level competition math problems, particularly from contests like the Putnam, IMO Shortlist, IMC, and similar. My goal is to improve problem-solving skills, explore advanced topics (e.g., combinatorics, NT, analysis), and prepare for upcoming competitions. I’m new to contests but have a strong general math background(CPGE in Morocco). If interested, reply here or DM me to discuss
1 reply
USAMO 1983 Problem 2 - Roots of Quintic
Binomial-theorem 33
N
by SomeonecoolLovesMaths
Source: USAMO 1983 Problem 2
Prove that the roots of![\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\]](//latex.artofproblemsolving.com/f/d/0/fd0ab11635ee44ad272a54c0f20a2bc47a3596de.png)
cannot all be real if 
.
![\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\]](http://latex.artofproblemsolving.com/f/d/0/fd0ab11635ee44ad272a54c0f20a2bc47a3596de.png)
cannot all be real if 
.
33 replies
Compact powers of 2
NO_SQUARES 1
N
by Isolemma
Source: 239 MO 2025 8-9 p3 = 10-11 p2
Let's call a power of two compact if it can be represented as the sum of no more than 
not necessarily distinct factorials of positive integer numbers. Prove that the set of compact powers of two is finite.
not necessarily distinct factorials of positive integer numbers. Prove that the set of compact powers of two is finite.
1 reply
Cute NT Problem
M11100111001Y1R 4
N
by RANDOM__USER
Source: Iran TST 2025 Test 4 Problem 1
A number 
is called lucky if it has at least two distinct prime divisors and can be written in the form:
![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](//latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)
where 
are distinct prime numbers that divide . (Note: it is possible that has other prime divisors not among .) Prove that for every prime number 
, there exists a lucky number such that 
.
is called lucky if it has at least two distinct prime divisors and can be written in the form:![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](http://latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)
where 
are distinct prime numbers that divide . (Note: it is possible that has other prime divisors not among .) Prove that for every prime number 
, there exists a lucky number such that 
.
4 replies
USAMO 2003 Problem 4
MithsApprentice 72
N
by endless_abyss
Let 
be a triangle. A circle passing through 
and 
intersects segments 
and 
at 
and 
, respectively. Lines 
and 
intersect at 
, while lines 
and 
intersect at 
. Prove that 
if and only if 
.
be a triangle. A circle passing through 
and 
intersects segments 
and 
at 
and 
, respectively. Lines 
and 
intersect at 
, while lines 
and 
intersect at 
. Prove that 
if and only if 
.
72 replies
Easy but unusual junior ineq
Maths_VC 1
N
by blug
Source: Serbia JBMO TST 2025, Problem 2
Real numbers 
satisfy 
. Determine the minimal and the maximal value of the expression 
satisfy 
. Determine the minimal and the maximal value of the expression 
1 reply
Bosnia and Herzegovina JBMO TST 2009 Problem 1
gobathegreat 1
N
by FishkoBiH
Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2009
Lengths of sides of triangle are positive integers, and smallest side is equal to 
. Determine the area of triangle 
if 
, where 
, 
and 
are lengths of altitudes in triangle from vertices , and 
, respectively.
are positive integers, and smallest side is equal to 
. Determine the area of triangle 
if 
, where 
, 
and 
are lengths of altitudes in triangle from vertices , and 
, respectively.
1 reply
USAMO 2001 Problem 2
MithsApprentice 53
N
by lksb
Let be a triangle and let 
be its incircle. Denote by 
and 
the points where is tangent to sides and , respectively. Denote by 
and 
the points on sides and , respectively, such that 
and 
, and denote by the point of intersection of segments 
and 
. Circle intersects segment at two points, the closer of which to the vertex is denoted by 
. Prove that 
.
be a triangle and let 
be its incircle. Denote by 
and 
the points where is tangent to sides and , respectively. Denote by 
and 
the points on sides and , respectively, such that 
and 
, and denote by the point of intersection of segments 
and 
. Circle intersects segment at two points, the closer of which to the vertex is denoted by 
. Prove that 
.
53 replies
A=b
k2c901_1 89
N
by reni_wee
Source: Taiwan 1st TST 2006, 1st day, problem 3
Let 
, 
be positive integers such that 
is a multiple of 
for all positive integers 
. Prove that 
.
Proposed by Mohsen Jamali, Iran
, 
be positive integers such that 
is a multiple of 
for all positive integers 
. Prove that 
.Proposed by Mohsen Jamali, Iran
89 replies
