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F=(F^3+F^3)/9-2F^3
Yiyj1 1
N
by sp0rtman00000
Source: 101 Algebra Problems for the AMSP
Define the Fibonacci sequence 
as 
fir 
. Prove that 
for all 
.
as 
fir 
. Prove that 
for all 
.
1 reply
,s.t :
4 lines concurrent
Zavyk09 3
N
by ItzsleepyXD
Source: Homework
Let 
be triangle with circumcenter 
and orthocenter 
. 
intersect again at 
respectively. Lines through parallel to 
intersects 
at 
respectively. Point 
such that 
is a parallelogram. Prove that lines 
and 
are concurrent at a point on 
.
be triangle with circumcenter 
and orthocenter 
. 
intersect again at 
respectively. Lines through parallel to 
intersects 
at 
respectively. Point 
such that 
is a parallelogram. Prove that lines 
and 
are concurrent at a point on 
.
3 replies
be real numbers . Prove that
A Interesting Puzzle Again
Iwato 0
Source: One of my classmates, Zhu MingYu
Given n∈ N+, for all m≤n, proof that: At least one of the mininum points of φ(m) is staying in [n,2n]. For more, to check the bound value and if it's strict. By the way, to consider the situation under the range that m is bigger than n, and the total condition or the particular ones(about Euler's function's Value Distribution). At least, the Betrand-Chebyshv and Legendre's Conjecture(the latter idea is a improvement of the beyond one, which wassuspected by myself before I know this :D) are deserved to consider.
0 replies
PD is the angle bisector of <BPC
the_universe6626 3
N
by ja.
Source: Janson MO 6 P1
Let be an acute triangle with 
. The angle bisector of 
intersects 
at , and the perpendicular to at intersects 
and 
at respectively. Suppose 
and 
intersect again at 
, prove that 
is the angle bisector of 
.
(Proposed by quacksaysduck)
be an acute triangle with 
. The angle bisector of 
intersects 
at , and the perpendicular to at intersects 
and 
at respectively. Suppose 
and 
intersect again at 
, prove that 
is the angle bisector of 
.(Proposed by quacksaysduck)
3 replies
help me guyss
Bet667 1
N
by Bet667
i want to learn functionl equation.Can you guys give me some advise to learn functional equations :starwars:
1 reply
Inspired by old results
sqing 2
N
by sqing
Source: Own
Let 
and 
Prove that
Let and 
Prove that
and 
Prove that
Let and 
Prove that
2 replies
China Mathematical Olympiad 1986 problem3
jred 3
N
by L13832
Source: China Mathematical Olympiad 1986 problem3
Let 
be complex numbers satisfying 
. Show that there exist some among the 
complex numbers such that the modulus of the sum of these complex numbers is not less than 
.
be complex numbers satisfying 
. Show that there exist some among the 
complex numbers such that the modulus of the sum of these complex numbers is not less than 
.
3 replies
3-digit palindrome and binary expansion \overline {xyx}
parmenides51 6
N
by imzzzzzz
Source: RMM Shortlist 2017 N2
Let 
and 
be three positive integers. Prove that there exist a positive integer 
and a set of 
positive integers 
, such that, for every 
, the 
-ary expansion of is a 
-digit palindrome, and the 
-ary expansion is exactly 
.
proposed by Bojan Basic, Serbia
and 
be three positive integers. Prove that there exist a positive integer 
and a set of 
positive integers 
, such that, for every 
, the 
-ary expansion of is a 
-digit palindrome, and the 
-ary expansion is exactly 
.proposed by Bojan Basic, Serbia
6 replies
