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PQ bisects AC if <BCD=90^o, A, B,C,D concyclic
parmenides51 2
N
by venhancefan777
Source: Mathematics Regional Olympiad of Mexico Northeast 2020 P2
Let 
, 
, 
and 
be points on the same circumference with 
. Let 
and 
be the projections of onto 
and 
, respectively. Prove that 
cuts the segment 
into equal parts.
, 
, 
and 
be points on the same circumference with 
. Let 
and 
be the projections of onto 
and 
, respectively. Prove that 
cuts the segment 
into equal parts.
2 replies
Inequality with three conditions
oVlad 3
N
by sqing
Source: Romania EGMO TST 2019 Day 1 P3
Let 
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](//latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
3 replies
standard Q FE
jasperE3 2
N
by jasperE3
Source: gghx, p19004309
Find all functions 
such that for any 
:
such that for any 
:
2 replies
abc(a+b+c)=3, show that prod(a+b)>=8 [Indian RMO 2012(b) Q4]
Potla 29
N
by sqing
Let be positive real numbers such that 
Prove that we have
![\[(a+b)(b+c)(c+a)\geq 8.\]](//latex.artofproblemsolving.com/2/a/6/2a63f19de634fce70ff4ebc02ad13d442a20c378.png)
Also determine the case of equality.
be positive real numbers such that 
Prove that we have![\[(a+b)(b+c)(c+a)\geq 8.\]](http://latex.artofproblemsolving.com/2/a/6/2a63f19de634fce70ff4ebc02ad13d442a20c378.png)
Also determine the case of equality.
29 replies
Functional Equation Problem
dimi07 2
N
by dimi07
Source: Pang Chung Wu FE Book
Could someone please solve this problem?
Find all functions
that satisfy 
and
![\[
f(f(n)) = f(f(n+2)+2) = n
\]](//latex.artofproblemsolving.com/4/d/8/4d8d7420e51a7529996473badd8a4e9976c2f01a.png)
for all integers 
.
Find all functions
that satisfy 
and![\[
f(f(n)) = f(f(n+2)+2) = n
\]](http://latex.artofproblemsolving.com/4/d/8/4d8d7420e51a7529996473badd8a4e9976c2f01a.png)
for all integers 
.
2 replies
Nondecreasing FE
pieater314159 16
N
by jasperE3
Source: 2019 ELMO Shortlist A4
Find all nondecreasing functions 
such that, for all 
, 
Proposed by Carl Schildkraut
such that, for all 
, 
Proposed by Carl Schildkraut
16 replies
Ez induction to start it off
alexanderhamilton124 21
N
by NerdyNashville
Source: Inmo 2025 p1
Consider the sequence defined by 
, 
, and
![\[
a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1},
\]](//latex.artofproblemsolving.com/4/e/c/4ec892df19b4ec97af9bb1e5d1954e6c26be3172.png)
for all integers 
. Determine all positive integers 
such that
![\[
\frac{a_n}{n}
\]](//latex.artofproblemsolving.com/3/d/1/3d1d012e7c41e1e09abd14626a79b1fb1cbab34d.png)
is an integer.
Proposed by Niranjan Balachandran, SS Krishnan, and Prithwijit De.
, 
, and![\[
a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1},
\]](http://latex.artofproblemsolving.com/4/e/c/4ec892df19b4ec97af9bb1e5d1954e6c26be3172.png)
for all integers 
. Determine all positive integers 
such that![\[
\frac{a_n}{n}
\]](http://latex.artofproblemsolving.com/3/d/1/3d1d012e7c41e1e09abd14626a79b1fb1cbab34d.png)
is an integer.Proposed by Niranjan Balachandran, SS Krishnan, and Prithwijit De.
21 replies
fun set problem
iStud 1
N
by GreenTea2593
Source: Monthly Contest KTOM April P2 Essay
Given a set 
with exactly 9 elements that is subset of 
. Prove that there exist two subsets and that satisfy the following:
- and are non-empty subsets from ,
- the sum of all elements in each of and are equal, and
-
is an empty subset.
with exactly 9 elements that is subset of 
. Prove that there exist two subsets and that satisfy the following:-
and are non-empty subsets from ,- the sum of all elements in each of
and are equal, and-
is an empty subset.
1 reply
Why is the old one deleted?
EeEeRUT 12
N
by John_Mgr
Source: EGMO 2025 P1
For a positive integer 
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
Here
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
Proposed by Paulius Aleknavičius, Lithuania
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
Here
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.Proposed by Paulius Aleknavičius, Lithuania
12 replies
trolling geometry problem
iStud 4
N
by GreenTea2593
Source: Monthly Contest KTOM April P3 Essay
Given a cyclic quadrilateral 
with 
and 
. Lines 
and 
intersect at 
, and lines and 
intersect at 
. Let 
be the midpoints of sides 
, respectively. Let and 
be points on segment 
and 
, respectively, so that 
is the angle bisector of 
and 
is the angle bisector of 
. Prove that 
is parallel to if and only if divides into two triangles with equal area.
with 
and 
. Lines 
and 
intersect at 
, and lines and 
intersect at 
. Let 
be the midpoints of sides 
, respectively. Let and 
be points on segment 
and 
, respectively, so that 
is the angle bisector of 
and 
is the angle bisector of 
. Prove that 
is parallel to if and only if divides into two triangles with equal area.
4 replies
