G
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Last Poster
and 
Prove that
four points lie on a circle
pohoatza 78
N
by ezpotd
Source: IMO Shortlist 2006, Geometry 2, AIMO 2007, TST 1, P2
Let 
be a trapezoid with parallel sides 
. Points 
and 
lie on the line segments 
and 
, respectively, so that 
. Suppose that there are points 
and 
on the line segment 
satisfying ![\[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]](//latex.artofproblemsolving.com/b/1/a/b1ae208955104081bafe221832509d9dd20eeb83.png)
Prove that the points , , 
and 
are concyclic.
Proposed by Vyacheslev Yasinskiy, Ukraine
be a trapezoid with parallel sides 
. Points 
and 
lie on the line segments 
and 
, respectively, so that 
. Suppose that there are points 
and 
on the line segment 
satisfying ![\[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]](http://latex.artofproblemsolving.com/b/1/a/b1ae208955104081bafe221832509d9dd20eeb83.png)
Prove that the points , , 
and 
are concyclic.Proposed by Vyacheslev Yasinskiy, Ukraine
78 replies
JBMO TST Bosnia and Herzegovina 2023 P4
FishkoBiH 2
N
by Stear14
Source: JBMO TST Bosnia and Herzegovina 2023 P4
Let 
be a positive integer. A board with a format 
is divided in equal squares.Determine all integers ≥3 such that the board can be covered in 
(or 
) pieces so that there is exactly one empty square in each row and each column.
be a positive integer. A board with a format 
is divided in equal squares.Determine all integers ≥3 such that the board can be covered in 
(or 
) pieces so that there is exactly one empty square in each row and each column.
2 replies
Does there exist 2011 numbers?
cyshine 8
N
by TheBaiano
Source: Brazil MO, Problem 4
Do there exist 
positive integers 
such that 
for any 
, 
such that 
?
positive integers 
such that 
for any 
, 
such that 
?
8 replies
1 viewing
D1036 : Composition of polynomials
Dattier 1
N
by Dattier
Source: les dattes à Dattier
Find all ![$A \in \mathbb Q[x]$](//latex.artofproblemsolving.com/d/1/5/d158f344cf30f8beb5f2199da53272bd8e304704.png)
with ![$\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$](//latex.artofproblemsolving.com/1/1/9/119a7b617e09ba16bb266b44ab3c85acc529f88f.png)
and 
.
![$A \in \mathbb Q[x]$](http://latex.artofproblemsolving.com/d/1/5/d158f344cf30f8beb5f2199da53272bd8e304704.png)
with ![$\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$](http://latex.artofproblemsolving.com/1/1/9/119a7b617e09ba16bb266b44ab3c85acc529f88f.png)
and 
.
1 reply
number sequence contains every large number
mathematics2003 3
N
by sttsmet
Source: 2021ChinaTST test3 day1 P2
Given distinct positive integer 
. For 
, 
is the smallest number different from 
which doesn't divide 
. Proof that every number large enough appears in the sequence.
. For 
, 
is the smallest number different from 
which doesn't divide 
. Proof that every number large enough appears in the sequence.
3 replies
IMO ShortList 2002, geometry problem 2
orl 28
N
by ezpotd
Source: IMO ShortList 2002, geometry problem 2
Let 
be a triangle for which there exists an interior point 
such that 
. Let the lines 
and 
meet the sides 
and 
at 
and 
respectively. Prove that ![\[ AB+AC\geq4DE. \]](//latex.artofproblemsolving.com/3/a/4/3a49c5f8bd2930adca7bde2601dd045d4bf05ff6.png)
be a triangle for which there exists an interior point 
such that 
. Let the lines 
and 
meet the sides 
and 
at 
and 
respectively. Prove that ![\[ AB+AC\geq4DE. \]](http://latex.artofproblemsolving.com/3/a/4/3a49c5f8bd2930adca7bde2601dd045d4bf05ff6.png)
28 replies
Arc Midpoints Form Cyclic Quadrilateral
ike.chen 56
N
by ezpotd
Source: ISL 2022/G2
In the acute-angled triangle , the point is the foot of the altitude from 
, and 
is a point on the segment 
. The lines through parallel to and meet 
at and , respectively. Points 
and 
lie on the circles 
and 
, respectively, such that 
and 
.
Prove that
and 
are concyclic.
, the point is the foot of the altitude from 
, and 
is a point on the segment 
. The lines through parallel to and meet 
at and , respectively. Points 
and 
lie on the circles 
and 
, respectively, such that 
and 
.Prove that
and 
are concyclic.
56 replies
Non-linear Recursive Sequence
amogususususus 4
N
by GreekIdiot
Given 
and the recursive relation
for all natural number . Find the general form of .
Is there any way to solve this problem and similar ones?
and the recursive relation
for all natural number . Find the general form of .Is there any way to solve this problem and similar ones?
4 replies
Russian Diophantine Equation
LeYohan 2
N
by RagvaloD
Source: Moscow, 1963
Find all integer solutions to
.
.
2 replies
