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Let 
a point in the space and 
the centroid of a tetrahedron 
. Prove that:
a point in the space and 
the centroid of a tetrahedron 
. Prove that:
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Iranian tough nut: AA', BN, CM concur in Gergonne picture
grobber 69
N
by zuat.e
Source: Iranian olympiad/round 3/2002
Let 
be a triangle. The incircle of triangle touches the side 
at 
, and the line 
meets the incircle again at a point 
. Let the lines 
and 
meet the incircle of triangle again at 
and , respectively. Prove that the lines , 
and 
are concurrent.
be a triangle. The incircle of triangle touches the side 
at 
, and the line 
meets the incircle again at a point 
. Let the lines 
and 
meet the incircle of triangle again at 
and , respectively. Prove that the lines , 
and 
are concurrent.
69 replies
"all of the stupid geo gets sent to tst 2/5" -allen wang
pikapika007 27
N
by HamstPan38825
Source: USA TST 2024/2
Let be a triangle with incenter 
. Let segment 
intersect the incircle of triangle at point 
. Suppose that line 
is perpendicular to line 
. Let be a point such that 
. Point 
lies on segment such that the circumcircle of triangle 
is tangent to line 
. Point 
lies on line 
such that 
. Prove that 
.
Luke Robitaille
be a triangle with incenter 
. Let segment 
intersect the incircle of triangle at point 
. Suppose that line 
is perpendicular to line 
. Let be a point such that 
. Point 
lies on segment such that the circumcircle of triangle 
is tangent to line 
. Point 
lies on line 
such that 
. Prove that 
.Luke Robitaille
27 replies
+1 wAnother integral limit
RobertRogo 3
N
by Bayram_Turayew
Source: "Traian Lalescu" student contest 2025, Section A, Problem 3
Let 
be a function differentiable at 0 with 
. Find
be a function differentiable at 0 with 
. Find
3 replies
Derivatives on a Functional Equation
Kingofmath101 1
N
by Mathzeus1024
Let 
be a smooth function on 
where 
for all 
and satisfies
for all
. Prove that 
for all 
.
be a smooth function on 
where 
for all 
and satisfies
for all
. Prove that 
for all 
.1 reply
Infinite series with Pell numbers
Entrepreneur 12
N
by Entrepreneur
Source: Own
Evaluate the sum 
Where 
denotes the n-th Pell Number given by 
and 
Where 
denotes the n-th Pell Number given by 
and 
12 replies
3xn matrice with combinatorical property
Sebaj71Tobias 2
N
by c00lb0y
Let"s have a 3xn matrice with the following properties:
The firs row of the matrice is 1,2,3,... ,n in this order.
The second and the third rows are permutations of the first.
Very important, that in each column thera are different entries.
How many matrices with thees properties are there?
The answer for 2xn matrices is well-known, but what is the answer for 3xn, or for kxn ( k<=n) ?
The firs row of the matrice is 1,2,3,... ,n in this order.
The second and the third rows are permutations of the first.
Very important, that in each column thera are different entries.
How many matrices with thees properties are there?
The answer for 2xn matrices is well-known, but what is the answer for 3xn, or for kxn ( k<=n) ?
2 replies
The matrix in some degree is a scalar
FFA21 5
N
by c00lb0y
Source: MSU algebra olympiad 2025 P2
invertible, for an infinite number of 
:
Is it true that
such that 
is a scalar
5 replies
Approximate the integral
ILOVEMYFAMILY 2
N
by ILOVEMYFAMILY
Approximate the integral
![\[
I = \int_0^1 \frac{(2^x + 2)\, dx}{1 + x^4}
\]](//latex.artofproblemsolving.com/e/3/9/e391d4208a008146dc3922ff31ac1f544172b5b4.png)
using the trapezoidal rule with accuracy 
.
![\[
I = \int_0^1 \frac{(2^x + 2)\, dx}{1 + x^4}
\]](http://latex.artofproblemsolving.com/e/3/9/e391d4208a008146dc3922ff31ac1f544172b5b4.png)
using the trapezoidal rule with accuracy 
.
2 replies
Approximate the integral
ILOVEMYFAMILY 0
Approximate the integral
![\[
I = \int_0^1 \frac{(2x^2 + 1)\, dx}{\sqrt{(1 + x^2)(2 - x^2)}}
\]](//latex.artofproblemsolving.com/3/7/4/3744a151b0383f98b63359cec6947395881e7496.png)
using the parabola method (Simpson's rule) with accuracy 
.
![\[
I = \int_0^1 \frac{(2x^2 + 1)\, dx}{\sqrt{(1 + x^2)(2 - x^2)}}
\]](http://latex.artofproblemsolving.com/3/7/4/3744a151b0383f98b63359cec6947395881e7496.png)
using the parabola method (Simpson's rule) with accuracy 
.0 replies
ISI UGB 2025
Entrepreneur 5
N
by Gauler
Source: ISI UGB 2025
1.)
Suppose
is differentiable and 
Show that for some 
3.)
Suppose![$f:[0,1]\to\mathbb R$](//latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)
is differentiable with 
If ![$|f'(x)|\le f(x)\;\forall\;x\in[0,1],$](//latex.artofproblemsolving.com/b/9/d/b9d6088657cd85b37190b98abac1d94a18bb5916.png)
then show that 
4.)
Let
be the unit circle in the complex plane. Let 
be the map given by 
We define 
and 
for 
The smallest positive integer 
such that 
is called period of 
Determine the total number of points 
of period 
6.)
Let
denote the set of natural numbers, and let 
be nine distinct tuples in 
Show that there are 
distinct elements in the set 
whose product is a perfect cube.
8.)
Let
and let 
be positive integers such that 
Prove that 
and determine when equality holds.
Suppose
is differentiable and 
Show that for some 
3.)
Suppose
![$f:[0,1]\to\mathbb R$](http://latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)
is differentiable with 
If ![$|f'(x)|\le f(x)\;\forall\;x\in[0,1],$](http://latex.artofproblemsolving.com/b/9/d/b9d6088657cd85b37190b98abac1d94a18bb5916.png)
then show that 
4.)
Let
be the unit circle in the complex plane. Let 
be the map given by 
We define 
and 
for 
The smallest positive integer 
such that 
is called period of 
Determine the total number of points 
of period 
6.)
Let
denote the set of natural numbers, and let 
be nine distinct tuples in 
Show that there are 
distinct elements in the set 
whose product is a perfect cube.8.)
Let
and let 
be positive integers such that 
Prove that 
and determine when equality holds.
5 replies
sequence of highly correlated rv pairs
Konigsberg 1
N
by yofro
For a given 
, construct a sequence of random variables 
such that
Let
Find positive constants 
and 
such that
, construct a sequence of random variables 
such that
Let
Find positive constants 
and 
such that
1 reply
functional analysis
ILOVEMYFAMILY 1
N
by alexheinis
Let 
, 
be normed spaces with a Banach space. Suppose 
is a family of continuous linear maps. Prove that the set
![\[
X = \left\{ x \in E \mid \sup_{n\geq 1}|| A_n(x)||< +\infty \right\}
\]](//latex.artofproblemsolving.com/e/c/d/ecd2cd83e90604ce128e8efb5e5b0748ad354ee3.png)
is either of first category in or is equal to the whole space .
, 
be normed spaces with a Banach space. Suppose 
is a family of continuous linear maps. Prove that the set![\[
X = \left\{ x \in E \mid \sup_{n\geq 1}|| A_n(x)||< +\infty \right\}
\]](http://latex.artofproblemsolving.com/e/c/d/ecd2cd83e90604ce128e8efb5e5b0748ad354ee3.png)
is either of first category in or is equal to the whole space .
1 reply
3D geometry theorem
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Y by KAME06
Just use the vectors (like in Leibniz's formula for a triangle).
Or, see this: https://vixra.org/pdf/1003.0187v1.pdf
Or, see this: https://vixra.org/pdf/1003.0187v1.pdf
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