Drawing equilateral triangle
by xeroxia, May 11, 2025, 7:14 AM
Equilateral triangle 
is given. Let 
and 
be the midpoints of 
and 
, respectively.
A point
on segment 
is given. Draw equilateral 
such that 
is on and 
is on 
.
is given. Let 
and 
be the midpoints of 
and 
, respectively.A point
on segment 
is given. Draw equilateral 
such that 
is on and 
is on 
.This post has been edited 1 time. Last edited by xeroxia, an hour ago
The familiar right angle from the orthocenter
by buratinogigle, May 11, 2025, 5:33 AM
Let be a triangle inscribed in a circle 
with orthocenter 
and altitude 
. Let 
be the midpoint of 
. Line 
meets again at 
. Line 
meets again at 
. Let 
be the orthogonal projection of on the line . Line 
meets again at 
. Prove that 
.
be a triangle inscribed in a circle 
with orthocenter 
and altitude 
. Let 
be the midpoint of 
. Line 
meets again at 
. Line 
meets again at 
. Let 
be the orthogonal projection of on the line . Line 
meets again at 
. Prove that 
.Attachments:
Divisibility NT
by reni_wee, May 11, 2025, 5:11 AM
Suppose that 
and 
are natural numbers such that
is a prime number. Find all possible values of ,,
.
and 
are natural numbers such that
is a prime number. Find all possible values of ,,
.Divisibilty...
by Sadigly, May 10, 2025, 9:07 PM
Find all 
consecutive even numbers, such that the sum of their squares divides the square of their product.
consecutive even numbers, such that the sum of their squares divides the square of their product.This post has been edited 1 time. Last edited by Sadigly, an hour ago
Inequality
by Sadigly, May 9, 2025, 7:59 AM
For positive real numbers 
satisfying 
, find the biggest value the following equation could acquire:
satisfying 
, find the biggest value the following equation could acquire:
such that
for all 
.D'B, E'C and l are congruence.
by cronus119, May 22, 2022, 7:03 PM
Let and on 
and respectively in 
such that 
then draw line 
through 
such that 
let 
and 
reflection of and to respectively prove that 
and are congruence.
and on 
and respectively in 
such that 
then draw line 
through 
such that 
let 
and 
reflection of and to respectively prove that 
and are congruence.a set of $9$ distinct integers
by N.T.TUAN, Mar 31, 2007, 5:13 AM
Let 
be a set of 
distinct integers all of whose prime factors are at most 
Prove that contains 
distinct integers such that their product is a perfect cube.
be a set of 
distinct integers all of whose prime factors are at most 
Prove that contains 
distinct integers such that their product is a perfect cube."Where wisdom and valor fail, all that remains is faith. . . And it can overcome all." -Toa Mata Tahu
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