Simple but hard
by Lukariman, May 16, 2025, 2:47 AM
Given triangle ABC. Outside the triangle, construct rectangles ACDE and BCFG with equal areas. Let M be the midpoint of DF. Prove that CM passes through the center of the circle circumscribing triangle ABC.
This post has been edited 2 times. Last edited by Lukariman, 6 hours ago
Good Permutations in Modulo n
by swynca, Apr 27, 2025, 2:03 PM
An integer
is called
if there exists a permutation
of the numbers
, such that:
and
have different parities for every
;
the sum
is a quadratic residue modulo
for every
.
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.












Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
This post has been edited 2 times. Last edited by swynca, Apr 27, 2025, 4:15 PM
sqrt(2)<=|1+z|+|1+z^2|<=4
by SuiePaprude, Jan 23, 2025, 10:53 PM
let z be a complex number with |z|=1 show that sqrt2 <=|1+z|+|1+z^2|<=4
concyclic wanted, diameter related
by parmenides51, May 5, 2024, 1:26 AM
As shown in the figure,
is the diameter of circle
, and chords
and
intersect at point
,
intersects at point
, and
intersects
at point
. Point
lies on
such that
. Prove that points
,
,
,
lies on a circle.



















This post has been edited 1 time. Last edited by parmenides51, May 5, 2024, 1:27 AM
Grid combo with tilings
by a_507_bc, Apr 23, 2023, 3:40 PM
A square grid
is tiled in two ways - only with dominoes and only with squares
. What is the least number of dominoes that are entirely inside some square
?



<ACB=90^o if AD = BD , <ACD = 3 <BAC, AM=//MD, CM//AB,
by parmenides51, Oct 7, 2022, 8:15 PM
In the convex quadrilateral
,
and
. Let
be the midpoint of side
. If the lines
and
are parallel, prove that the angle
is right.








Concurrency in Parallelogram
by amuthup, Jul 12, 2022, 12:25 PM
Let
be a parallelogram with
A point
is chosen on the extension of ray
past
The circumcircle of
meets the segment
again at
The circumcircle of triangle
meets the segment
at
Prove that lines
are concurrent.












This post has been edited 1 time. Last edited by amuthup, Jul 15, 2022, 4:57 PM
bulgarian concurrency, parallelograms and midpoints related
by parmenides51, May 28, 2019, 2:10 PM
In a triangle
points
and
lie on the segments
and
, respectively, and are such that
is a parallelogram. The circle with center the midpoint
of the segment
and radius
and the circle of diameter
intersect for the second time at the point
. Prove that the lines
and
intersect in a point.













Reel Math
by rrusczyk, Nov 12, 2011, 11:43 PM
Quick reminder for those of you participating in the Reel Math Challenge from MATHCOUNTS: the public voting starts Tuesday. Public voting is part of the evaluation of the videos, so obviously students who get their videos in early will have an advantage! That said, if you don't have your video finished by Tuesday, you can still enter. The voting runs until February 1st of next year, so there's still plenty of time to finish your video and rally your voting forces.
MATHCOUNTS Movies
by rrusczyk, Sep 28, 2011, 11:10 PM
MATHCOUNTS has launched a new contest in which students are challenged to make movies out of MATHCOUNTS problems. (Hey, that sounds familiar!) The program is called Reel Math, and you can check it out here. Winners get a free trip to Nationals in Orlando!
Concurrency
by Omid Hatami, May 20, 2008, 9:29 AM
Suppose that
is incenter of triangle
and
is a line tangent to the incircle. Let
be another line such that intersects
respectively at
. We draw a tangent from
to the incircle other than
, and this line intersects with
at
.
are similarly defined. Prove that
are concurrent.












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