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Topic
First Poster
Last Poster
Number Theory Chain!
JetFire008 60
N
by whwlqkd
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!
?60 replies
3 right-angled triangle area
NicoN9 1
N
by Mathzeus1024
Source: Japan Junior MO Preliminary 2020 P1
Right angled triangle 
, and a square are drawn as shown below. Three numbers written below implies each of the area of shaded small right angled triangle. Find the value of 
.
IMAGE
, and a square are drawn as shown below. Three numbers written below implies each of the area of shaded small right angled triangle. Find the value of 
.IMAGE
1 reply
two sequences of positive integers and inequalities
rmtf1111 50
N
by math-olympiad-clown
Source: EGMO 2019 P5
Let 
be an integer, and let 
be positive integers. Show that there exist positive integers 
satisfying the following three conditions:
for 
the remainders of on division by 
are pairwise different; and
(Here,
denotes the integer part of real number 
, that is, the largest integer that does not exceed .)
be an integer, and let 
be positive integers. Show that there exist positive integers 
satisfying the following three conditions:
for 
the remainders of on division by 
are pairwise different; and
(Here,
denotes the integer part of real number 
, that is, the largest integer that does not exceed .)
50 replies
Problem 6
SlovEcience 2
N
by mashumaro
Given two points A and B on the unit circle. The tangents to the circle at A and B intersect at point P. Then:
![\[ p = \frac{2ab}{a + b} \]](//latex.artofproblemsolving.com/0/f/4/0f4297ec59fe97c92fda75af4cc835be6b9a2507.png)
, ![\[ p, a, b \in \mathbb{C} \]](//latex.artofproblemsolving.com/b/4/4/b4499970ac49d770a810d7345fedef7773cd03f5.png)
![\[ p = \frac{2ab}{a + b} \]](http://latex.artofproblemsolving.com/0/f/4/0f4297ec59fe97c92fda75af4cc835be6b9a2507.png)
, ![\[ p, a, b \in \mathbb{C} \]](http://latex.artofproblemsolving.com/b/4/4/b4499970ac49d770a810d7345fedef7773cd03f5.png)
2 replies
A coincidence about triangles with common incenter
flower417477 3
N
by mashumaro
have the same incenter 
.Prove that 
is concyclic iff 
is concurrent
3 replies
this geo is scarier than the omega variant
AwesomeYRY 11
N
by LuminousWolverine
Source: TSTST 2021/6
Triangles and 
share circumcircle 
and incircle 
so that points 
and 
occur in this order along . Let 
be the triangle formed by lines 
and 
and define triangles 
similarly. Furthermore, let 
and 
be the circumcircle and incircle of triangle , respectively, and define circles 
similarly.
(a) Prove that the two common external tangents to circles and 
and the two common external tangents to and 
are either concurrent or pairwise parallel.
(b) Suppose that these four lines meet at point
, and define points 
and 
similarly. Prove that points 
, and are collinear.
Nikolai Beluhov
and 
share circumcircle 
and incircle 
so that points 
and 
occur in this order along . Let 
be the triangle formed by lines 
and 
and define triangles 
similarly. Furthermore, let 
and 
be the circumcircle and incircle of triangle , respectively, and define circles 
similarly.(a) Prove that the two common external tangents to circles
and 
and the two common external tangents to and 
are either concurrent or pairwise parallel.(b) Suppose that these four lines meet at point
, and define points 
and 
similarly. Prove that points 
, and are collinear.Nikolai Beluhov
11 replies
No function f on reals such that f(f(x))=x^2-2
N.T.TUAN 17
N
by Assassino9931
Source: VietNam TST 1990
Prove that there does not exist a function 
such that 
for all 
.
such that 
for all 
.
17 replies
Hard diophant equation
MuradSafarli 4
N
by iniffur
Find all positive integers 
such that the equation
is satisfied.
such that the equation
is satisfied.
4 replies
Geometry that "looks" hard
Pmshw 3
N
by Lemmas
Source: Iran 2nd round 2022 P6
we have an isogonal triangle such that 
. take a random 
on the altitude from 
to 
.
The circle
intersects second time in 
. Take 
such that it's on the segment and 
and 
.The second intersection of 
and circle 
is 
, (
) and the second intersection of 
and circle 
is 
,(
).The tangent from 
to the circle intersects the altitude from at 
.
Prove that
is tangent to circle 
.
such that 
. take a random 
on the altitude from 
to 
.The circle
intersects second time in 
. Take 
such that it's on the segment and 
and 
.The second intersection of 
and circle 
is 
, (
) and the second intersection of 
and circle 
is 
,(
).The tangent from 
to the circle intersects the altitude from at 
.Prove that
is tangent to circle 
.
3 replies
