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Quadrangle, nine-point conic, Steiner line
kosmonauten3114   0
15 minutes ago
Source: My own
Let $P_1P_2P_3P_4$ be a general quadrangle which does not form an orthocentric system. Let $P$, $I$, $M$, $T$ be the Euler-Poncelet point ($\text{QA-P2}$), isogonal center ($\text{QA-P4}$), midray homothetic center ($\text{QA-P8}$), inscribed square axes crosspoint ($\text{QA-P23}$) of $P_1P_2P_3P_4$, respectively.
Let $H_1$ be the orthocenter of $\triangle{P_2P_3P_4}$, and define $H_2$, $H_3$, $H_4$ cyclically.
Let $A_{ij}=P_iP_j \cap H_iH_j$ ($\{i, j\} \in \{1, 2, 3, 4\}, i<j$).
Let $B_{ij}=P_iP_j \cap H_kH_l$ ($\{i, j, k, l\} \in \{1, 2, 3, 4\}, i<j$).
Then, the 12 points $A_{12}$, $A_{13}$, $A_{14}$, $A_{23}$, $A_{24}$, $A_{34}$, $B_{12}$, $B_{13}$, $B_{14}$, $B_{23}$, $B_{24}$, $B_{34}$ lie on the same conic, here denoted by $\mathcal{C}_1$.
Let $\mathcal{C}_2$ be the nine-point conic of $P_1P_2P_3P_4$.
Suppose that $\mathcal{C}_1$ and $\mathcal{C}_2$ have 4 distinct intersection points, and let $U$, $V$, $W$ be the intersections, other than $P$, of $\mathcal{C}_1$ and $\mathcal{C}_2$.

Prove that the Steiner line of $P$ with respect to $\triangle{UVW}$ passes through $I$ and $M$, and show that the center of $\mathcal{C}_1$ and the orthocenter of $\triangle{UVW}$ coincide with $T$.
0 replies
kosmonauten3114
15 minutes ago
0 replies
Metric space
wiseman   3
N May 21, 2025 by alinazarboland
Source: IMS 2014 - Day1 - Problem4
Let $(X,d)$ be a metric space and $f:X \to X$ be a function such that $\forall x,y\in X : d(f(x),f(y))=d(x,y)$.
$\text{a})$ Prove that for all $x \in X$, $\lim_{n \rightarrow +\infty} \frac{d(x,f^n(x))}{n}$ exists, where $f^n(x)$ is $\underbrace{f(f(\cdots f(x)}_{n \text{times}} \cdots ))$.
$\text{b})$ Prove that the amount of the limit does not depend on choosing $x$.
3 replies
wiseman
Oct 2, 2014
alinazarboland
May 21, 2025
Double integration
Tricky123   2
N May 21, 2025 by Mathzeus1024
Q)
\[\iint_{R} \sin(xy) \,dx\,dy, \quad R = \left[0, \frac{\pi}{2}\right] \times \left[0, \frac{\pi}{2}\right]\]
How to solve the problem like this I am using the substitution method but its seems like very complicated in the last
Please help me
2 replies
Tricky123
May 18, 2025
Mathzeus1024
May 21, 2025
Find solution of IVP
neerajbhauryal   2
N May 15, 2025 by Mathzeus1024
Show that the initial value problem \[y''+by'+cy=g(t)\] with $y(t_o)=0=y'(t_o)$, where $b,c$ are constants has the form \[y(t)=\int^{t}_{t_0}K(t-s)g(s)ds\,\]

What I did
2 replies
neerajbhauryal
Sep 23, 2014
Mathzeus1024
May 15, 2025
fourier series?
keroro902   2
N May 15, 2025 by Mathzeus1024
f(x)=$\sum _{n=0}^{\infty } \text{cos}(nx)/2^{n}$
f(x) = ?
thanks
2 replies
keroro902
May 14, 2010
Mathzeus1024
May 15, 2025
uniformly continuous of multivariable function
keroro902   1
N May 14, 2025 by Mathzeus1024
How can I determine which of the following functions are uniformly continuous on the given domain A?

$f \left( x, y \right) = \frac{x^3 + y^2}{x^2 + y}$ , $A = \left\{ \left( x, y
\right) \in \mathbb m{R}^2 \left|  \right.  \left| y \right| \leq \frac{x^2}{2}
%Error. "nocomma" is a bad command.
, x^2 + y^2 < 1 \right\}$

$g \left( x, y \right) = \frac{y^2 + 4 x^2}{y^2 - 4 x^2 - 1}$, $A = \left\{
\left( x, y \right) \in \mathbb m{R}^2 \left| 0 \leq x^2 - y^2 \leqslant 1
\right\} \right.$
1 reply
keroro902
Nov 2, 2012
Mathzeus1024
May 14, 2025
Miklos Schweitzer 1971_5
ehsan2004   2
N May 9, 2025 by pi_quadrat_sechstel
Let $ \lambda_1 \leq \lambda_2 \leq...$ be a positive sequence and let $ K$ be a constant such that \[  \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\] Prove that there exists a constant $ K'$ such that \[  \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\]

L. Leindler
2 replies
ehsan2004
Oct 29, 2008
pi_quadrat_sechstel
May 9, 2025
Miklos Schweitzer 1968_9
ehsan2004   1
N May 8, 2025 by pi_quadrat_sechstel
Let $ f(x)$ be a real function such that
\[ \lim_{x \rightarrow +\infty} \frac{f(x)}{e^x}=1\]
and $ |f''(x)|\leq c|f'(x)|$ for all sufficiently large $ x$. Prove that \[ \lim_{x \rightarrow +\infty} \frac{f'(x)}{e^x}=1.\]

P. Erdos
1 reply
ehsan2004
Oct 8, 2008
pi_quadrat_sechstel
May 8, 2025
Range of 2 parameters and Convergency of Improper Integral
Kunihiko_Chikaya   3
N May 1, 2025 by Mathzeus1024
Source: 2012 Kyoto University Master Course in Mathematics
Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.
3 replies
Kunihiko_Chikaya
Aug 21, 2012
Mathzeus1024
May 1, 2025
f'(1)>1 implies f has a fixed point in (0,1)
Sayan   13
N Apr 27, 2025 by Apple_maths60
Source: ISI(BS) 2010 #4
A real valued function $f$ is defined on the interval $(-1,2)$. A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$. Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$. Show that if $f'(1)>1$, then $f$ has a fixed point in the interval $(0,1)$.
13 replies
Sayan
May 17, 2012
Apple_maths60
Apr 27, 2025
Maximum value of ∫_0^1 e^x logf(x) dx when ∫_0^1 f(x) dx =1
tom-nowy   2
N Apr 26, 2025 by Dattier
Source: 1982 Niigata University entrance exam
Let $\mathcal{F}$ be the set of continuous functions $f: [0,1] \to (0, \infty )$ such that $ \int_0^1 f(x) \, \mathrm dx =1 $. For $f \in \mathcal{F}$, let $$I(f)=\int_0^1 e^x \log f(x) \, \mathrm dx .$$Determine $\max_{f \in \mathcal{F}}I(f)$.
2 replies
tom-nowy
Jul 7, 2013
Dattier
Apr 26, 2025
Parallel condition and isogonal
ItzsleepyXD   1
N Apr 30, 2025 by moony_
Source: Own , Mock Thailand Mathematic Olympiad P5
Let $ABC$ be triangle and point $D$ be $A-$ altitude of $\triangle ABC$ .
Let $E,F$ be a point on $AC$ and $AB$ such that $DE\parallel AB$ and $DF\parallel AC$ . Point $G$ is the intersection of $(AEF)$ and $(ABC)$ . Point $P$ be intersection of $(ADG)$ and $BC$ . Line $GD$ intersect circumcircle of $\triangle ABC$ again at $Q$ .
Prove that
(a) $\angle BAP = \angle QAC$ .
(b) $AQ$ bisect $BC$ .
1 reply
ItzsleepyXD
Apr 30, 2025
moony_
Apr 30, 2025
Parallel condition and isogonal
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Source: Own , Mock Thailand Mathematic Olympiad P5
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ItzsleepyXD
147 posts
#1
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Let $ABC$ be triangle and point $D$ be $A-$ altitude of $\triangle ABC$ .
Let $E,F$ be a point on $AC$ and $AB$ such that $DE\parallel AB$ and $DF\parallel AC$ . Point $G$ is the intersection of $(AEF)$ and $(ABC)$ . Point $P$ be intersection of $(ADG)$ and $BC$ . Line $GD$ intersect circumcircle of $\triangle ABC$ again at $Q$ .
Prove that
(a) $\angle BAP = \angle QAC$ .
(b) $AQ$ bisect $BC$ .
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moony_
22 posts
#2 • 1 Y
Y by Funcshun840
cool ratio lemma... >w<
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