Inspired by old results
by sqing, Apr 27, 2025, 2:18 AM
Let 
and 
Prove that
and 
Prove that
This post has been edited 1 time. Last edited by sqing, an hour ago
On a conditon for Hamilton Graphs
by flower417477, Apr 27, 2025, 1:55 AM
Prove that: for a graph 
,if for any vertex 
for which 
and 
there's 
,then is a Hamiltonian graph
,if for any vertex 
for which 
and 
there's 
,then is a Hamiltonian graphWeird ninja points collinearity
by americancheeseburger4281, Apr 26, 2025, 10:12 PM
For some triangle, define its Ninja Point as the point on its circumcircle such that its Steiner line coincides with the Euler line of the triangle. For an triangle 
, define:
Points 
, 
and 
are collinear, that is is the Feuerbach point.
Points 
, and 
are collinear
, define:
as its circumcentre, as its orthocentre and as its nine-point centre.
, 
and 
to be the midpoint of the smaller arcs.- as the isogonal conjugate of the Nagel point (i.e. the exsimillicenter of the incircle and circumcircle)
- as the ninja point of
- as the ninja point of the contact triangle
as its circumcentre, 
,
and 
to be the midpoint of the smaller arcs.
Points 
, 
and 
are collinear, that is is the Feuerbach point.
Points 
, and 
are collinear
and 
Prove that
easy functional
by B1t, Apr 26, 2025, 6:45 AM
Denote the set of real numbers by 
. Find all functions 
such that for all 
,
![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](//latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
. Find all functions 
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
This post has been edited 2 times. Last edited by B1t, Yesterday at 7:01 AM
Dou Fang Geometry in Taiwan TST
by Li4, Apr 26, 2025, 5:03 AM
Let 
and 
be the incircle and circumcircle of the acute triangle , respectively. Draw a square 
so that all of its sides are tangent to , and 
, 
are both on 
. Extend 
and 
, intersecting at 
and 
, respectively. Prove that 
and 
intersects on .
Proposed by kyou46, Li4, Revolilol.
and 
be the incircle and circumcircle of the acute triangle , respectively. Draw a square 
so that all of its sides are tangent to , and 
, 
are both on 
. Extend 
and 
, intersecting at 
and 
, respectively. Prove that 
and 
intersects on .Proposed by kyou46, Li4, Revolilol.
Inequalities
by Scientist10, Apr 23, 2025, 6:36 PM
If , then prove that the following inequality holds:
![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](//latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
, then prove that the following inequality holds:![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
2025 Caucasus MO Juniors P7
by BR1F1SZ, Mar 26, 2025, 1:03 AM
It is known that from segments of lengths 
, 
and 
, a triangle can be formed. Could it happen that from segments of lengths 
a right-angled triangle can be formed?
, 
and 
, a triangle can be formed. Could it happen that from segments of lengths 
a right-angled triangle can be formed?To Mixtilinear or not to mixtilinear
by ihategeo_1969, Jan 8, 2025, 3:21 AM
Let be a triangle and let 
be the contact point of the 
-mixtilinear incircle with the circumcircle, and let 
be the reflection of over . Prove that the nine-point circle of 
is tangent to the incircle.
be a triangle and let 
be the contact point of the 
-mixtilinear incircle with the circumcircle, and let 
be the reflection of over . Prove that the nine-point circle of 
is tangent to the incircle.Continued fraction
by tapir1729, Jun 24, 2024, 6:36 PM
Let 
be an odd prime number. Suppose and are polynomials with integer coefficients such that 
, there is no nonconstant polynomial dividing both and , and
![\[
1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 +
(p-1)x}}}=\frac{P(x)}{Q(x)}.
\]](//latex.artofproblemsolving.com/b/c/a/bca9034e2affa1d80b167451d77434fbcd8bd88a.png)
Show that all coefficients of except for the constant coefficient are divisible by , and all coefficients of are not divisible by .
Andrew Gu
be an odd prime number. Suppose and are polynomials with integer coefficients such that 
, there is no nonconstant polynomial dividing both and , and![\[
1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 +
(p-1)x}}}=\frac{P(x)}{Q(x)}.
\]](http://latex.artofproblemsolving.com/b/c/a/bca9034e2affa1d80b167451d77434fbcd8bd88a.png)
Show that all coefficients of except for the constant coefficient are divisible by , and all coefficients of are not divisible by .Andrew Gu
This post has been edited 2 times. Last edited by tapir1729, Jan 6, 2025, 2:37 AM
Reason: fixed italics
Reason: fixed italics
April 2025
March 2025