multiple of 15-15 positive factors

by britishprobe17, Apr 18, 2025, 6:23 AM

Find the sum of all natural numbers $n$ such that $n$ is a multiple of $15$ and has exactly $15$ positive factors.
number theory
L

general form

by pennypc123456789, Apr 18, 2025, 6:18 AM

If $a,b,c$ are positive real numbers, $k \ge 3$ then
$$ \frac{a + b}{a + kb + c} + \dfrac{b + c}{b + kc + a}+\dfrac{c + a}{c + ka + b} \geq \dfrac{6}{k+2}$$
inequalities
L

Beautiful geometry

by m4thbl3nd3r, Apr 17, 2025, 4:41 PM

Let $\omega$ be the circumcircle of triangle $ABC$, $M$ is the midpoint of $BC$ and $E$ be the second intersection of $AM$ and $\omega$. Tangent line of $\omega$ at $E$ intersects $BC$ at $P$, let $PKL$ be a transversal of $\omega$ and $X,Y$ be intersections of $AK,AL$ with $BC$. Let $PF$ be a tangent line of $\omega$. Prove that $LYFP$ is cyclic
geometry
circumcircle
L

Function equation

by luci1337, Apr 17, 2025, 3:01 PM

find all function $f:R \rightarrow R$ such that:
$2f(x)f(x+y)-f(x^2)=\frac{x}{2}(f(2x)+f(f(y)))$ with all $x,y$ is real number
functional equation
algebra
L

Multi-equation

by giangtruong13, Apr 17, 2025, 12:30 PM

Solve equations: $$\begin{cases} x^4+x^3y+x^2y^2=7x+9 \\ x(y-x+1)=3 \end{cases} $$
equation
number theory
L

a+b+c=abc

by KhuongTrang, Apr 16, 2025, 11:51 AM

Problem. Let $a,b,c$ be three positive real numbers satisfying $a+b+c=abc.$ Prove that$$\sqrt{a^2+b^2+3}+\sqrt{b^2+c^2+3}+\sqrt{c^2+a^2+3}\ge4\cdot \frac{a^2b^2c^2-3}{ab+bc+ca-3}-7.$$There is a very elegant proof :-D Could anyone think of it?
inequalities
L

Circumcenter of reflection of collinear points over sides

by a1267ab, Jan 11, 2025, 11:04 PM

Let $ABC$ be a triangle, and let $X$, $Y$, and $Z$ be collinear points such that $AY=AZ$, $BZ=BX$, and $CX=CY$. Points $X'$, $Y'$, and $Z'$ are the reflections of $X$, $Y$, and $Z$ over $BC$, $CA$, and $AB$, respectively. Prove that if $X'Y'Z'$ is a nondegenerate triangle, then its circumcenter lies on the circumcircle of $ABC$.

Michael Ren
geometry
circumcircle
geometric transformation
reflection
L

BMO Shortlist 2021 A5

by Lukaluce, May 8, 2022, 5:01 PM

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$f(xf(x + y)) = yf(x) + 1$$holds for all $x, y \in \mathbb{R}^{+}$.

Proposed by Nikola Velov, North Macedonia
This post has been edited 2 times. Last edited by Lukaluce, May 10, 2022, 2:31 PM
Balkan
shortlist
2021
algebra
functional equation
L

Maximum with positive integers

by SMOJ, Mar 31, 2020, 7:04 AM

Let $a,b,c,d$ be positive integers such that $a+c=20$ and $\frac{a}{b}+\frac{c}{d}<1$. Find the maximum possible value of $\frac{a}{b}+\frac{c}{d}$.
This post has been edited 1 time. Last edited by SMOJ, Mar 31, 2020, 7:10 AM
algebra
inequalities
High school olympiad
singapore
L

Right-angled triangle if circumcentre is on circle

by liberator, Jan 4, 2016, 9:41 PM

Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$, respectively. Suppose that the circumcentre of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$. Prove that triangle $ABC$ is right-angled.

Proposed by Alexander A. Polyansky, Russia
geometry
IMO
conditional geometry
L

Fun with math!

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  • WOAH I JUST CAME HERE, CSS IS CRAZY

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  • Thanks! I'm happy to hear that! How is the new CSS? If you don't like it, I can go back.

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  • This is such a cool blog! Just a suggestion, but I feel like it would look a bit better if the entries were wider. They're really skinny right now, which makes the posts seem a lot longer.

    by Catcumber, Apr 4, 2025, 11:16 PM

  • The first few posts for April are out!

    by aoum, Apr 1, 2025, 11:51 PM

  • Sure! I understand that it would be quite a bit to take in.

    by aoum, Apr 1, 2025, 11:08 PM

  • No, but it is a lot to take in. Also, could you do the Gamma Function next?

    by HacheB2031, Apr 1, 2025, 3:04 AM

  • Am I going too fast? Would you like me to slow down?

    by aoum, Mar 31, 2025, 11:34 PM

  • Seriously, how do you make these so fast???

    by HacheB2031, Mar 31, 2025, 6:45 AM

  • I am now able to make clickable images in my posts! :)

    by aoum, Mar 29, 2025, 10:42 PM

  • Am I doing enough? Are you all expecting more from me?

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