four variables inequality
by JK1603JK, Apr 4, 2025, 4:26 PM
Prove that 
a hard geometry problen
by Tuguldur, Apr 4, 2025, 3:56 PM
Let 
be a convex quadrilateral. Suppose that the circles with diameters 
and 
intersect at points 
and 
. Let 
and 
. Prove that the points 
, 
, and are concyclic.
( and are not the diagnols)
be a convex quadrilateral. Suppose that the circles with diameters 
and 
intersect at points 
and 
. Let 
and 
. Prove that the points 
, 
, and are concyclic.(
and are not the diagnols)Problem 2
by SlovEcience, Apr 4, 2025, 3:52 PM
Let 
be positive integers and 
be an odd prime such that:
![\[
a^p \equiv 1 \pmod{p^n}.
\]](//latex.artofproblemsolving.com/3/4/8/34815b8211ded2f37b0653525dc4b436356e6eee.png)
Prove that:
![\[
a \equiv 1 \pmod{p^{n-1}}.
\]](//latex.artofproblemsolving.com/0/7/a/07a4ed2ab3c94f41028080f7789cfb6e2973955c.png)
be positive integers and 
be an odd prime such that:![\[
a^p \equiv 1 \pmod{p^n}.
\]](http://latex.artofproblemsolving.com/3/4/8/34815b8211ded2f37b0653525dc4b436356e6eee.png)
Prove that:![\[
a \equiv 1 \pmod{p^{n-1}}.
\]](http://latex.artofproblemsolving.com/0/7/a/07a4ed2ab3c94f41028080f7789cfb6e2973955c.png)
Problem 1
by SlovEcience, Apr 4, 2025, 3:00 PM
Prove that
![\[
C(p-1, k-1) \equiv (-1)^{k-1} \pmod{p}
\]](//latex.artofproblemsolving.com/4/0/7/4072235afbaa1f2d8d4c9ca044615a113a4306ce.png)
for 
, where 
is the binomial coefficient 
choose 
.
![\[
C(p-1, k-1) \equiv (-1)^{k-1} \pmod{p}
\]](http://latex.artofproblemsolving.com/4/0/7/4072235afbaa1f2d8d4c9ca044615a113a4306ce.png)
for 
, where 
is the binomial coefficient 
choose 
.
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
April Fools Geometry
by awesomeming327., Apr 1, 2025, 2:52 PM
Let 
be an acute triangle with 
, and let 
be the projection from 
onto 
. Let 
be a point on the extension of 
past such that 
. Let 
be on the perpendicular bisector of 
such that and 
are on the same side of and
![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](//latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of across and 
be 
and , respectively. Let 
and 
such that 
. Let 
and 
intersect the circumcircles of 
and 
at 
and 
, respectively. Let 
and 
intersect 
and 
at and . Let 
intersect at 
. Prove that 
.
be an acute triangle with 
, and let 
be the projection from 
onto 
. Let 
be a point on the extension of 
past such that 
. Let 
be on the perpendicular bisector of 
such that and 
are on the same side of and![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of across and 
be 
and , respectively. Let 
and 
such that 
. Let 
and 
intersect the circumcircles of 
and 
at 
and 
, respectively. Let 
and 
intersect 
and 
at and . Let 
intersect at 
. Prove that 
.
such that 
A board with crosses that we color
by nAalniaOMliO, Mar 28, 2025, 8:37 PM
In some cells of the table 
crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least 
crosses.
Find the minimal for which it is always possible to color crosses in two colors such that any balanced set has crosses of both colors.
crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least 
crosses.Find the minimal
for which it is always possible to color crosses in two colors such that any balanced set has crosses of both colors.This post has been edited 1 time. Last edited by nAalniaOMliO, Mar 29, 2025, 10:41 AM
Conditional maximum
by giangtruong13, Mar 22, 2025, 4:26 PM
Let 
satisfy that: 
and 
. Find the maximum: 
satisfy that: 
and 
. Find the maximum: 
This post has been edited 1 time. Last edited by giangtruong13, Mar 22, 2025, 4:26 PM
H not needed
by dchenmathcounts, May 23, 2020, 11:00 PM
Let be a cyclic quadrilateral. A circle centered at 
passes through 
and and meets lines 
and again at points and (distinct from 
). Let 
denote the orthocenter of triangle 
Prove that if lines 
are concurrent, then triangle and 
are similar.
Robin Son
be a cyclic quadrilateral. A circle centered at 
passes through 
and and meets lines 
and again at points and (distinct from 
). Let 
denote the orthocenter of triangle 
Prove that if lines 
are concurrent, then triangle and 
are similar.Robin Son
This post has been edited 2 times. Last edited by v_Enhance, Oct 25, 2020, 6:01 AM
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