inequality

by danilorj, May 14, 2025, 9:08 PM

Let $a, b, c$ be nonnegative real numbers such that $a + b + c = 3$ . Prove that
$\frac{a}{4 - b} + \frac{b}{4 - c} + \frac{c}{4 - a} + \frac{1}{16}(1 - a)^2(1 - b)^2(1 - c)^2 \leq 1,$ and determine all such triples $(a, b, c)$ where the equality holds.

inequalities

Imtersecting two regular pentagons

by Miquel-point, May 14, 2025, 6:27 PM

The intersection of two congruent regular pentagons is a decagon with sides of $a_1,a_2,\ldots ,a_{10}$ in this order. Prove that
$a_1a_3+a_3a_5+a_5a_7+a_7a_9+a_9a_1=a_2a_4+a_4a_6+a_6a_8+a_8a_{10}+a_{10}a_2.$

geometry

pentagon

D1031 : A general result on polynomial 1

by Dattier, May 14, 2025, 5:14 PM

Let $P(x,y) \in \mathbb Q(x,y)$ with $\forall (a,b) \in \mathbb Z^2, P(a,b) \in \mathbb Z$ .

Is it true that $P(x,y) \in \mathbb Q[x,y]$ ?

algebra

polynomial

Nice one

by imnotgoodatmathsorry, May 2, 2025, 2:10 PM

With $x,y,z >0$ .Prove that: $\frac{xy}{4y+4z+x} + \frac{yz}{4z+4x+y} +\frac{zx}{4x+4y+z} \le \frac{x+y+z}{9}$

inequalities

inequalities proposed

inequalities unsolved

Old hard problem

by ItzsleepyXD, Apr 25, 2025, 4:15 AM

Let $ABC$ be a triangle and let $O$ be its circumcenter and $I$ its incenter.
Let $P$ be the radical center of its three mixtilinears and let $Q$ be the isogonal conjugate of $P$ .
Let $G$ be the Gergonne point of the triangle $ABC$ .
Prove that line $QG$ is parallel with line $OI$ .

radical axis

geometry

Easy Geometry

by pokmui9909, Mar 30, 2025, 5:18 AM

Triangle $ABC$ satisfies $\overline{CA} > \overline{AB}$ . Let the incenter of triangle $ABC$ be $\omega$ , which touches $BC, CA, AB$ at $D, E, F$ , respectively. Let $M$ be the midpoint of $BC$ . Let the circle centered at $M$ passing through $D$ intersect $DE, DF$ at $P(\neq D), Q(\neq D)$ , respecively. Let line $AP$ meet $BC$ at $N$ , line $BP$ meet $CA$ at $L$ . Prove that the three lines $EQ, FP, NL$ are concurrent.

This post has been edited 1 time. Last edited by pokmui9909, Mar 30, 2025, 5:29 AM

geometry

incenter

Asymmetric FE

by sman96, Feb 8, 2025, 5:11 PM

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(xf(y)-y) + f(xy-x) + f(x+y) = 2xy$ for all $x, y \in \mathbb{R}$ .

algebra

functional equation

monving balls in 2018 boxes

by parmenides51, Sep 6, 2022, 11:50 PM

There are $2018$ boxes $C_1$ , $C_2$ , $C_3$ ,.., $C_{2018}$ . The $n$ -th box $C_n$ contains $n$ balls.
A move consists of the following steps:
a) Choose an integer $k$ greater than $1$ and choose $m$ a multiple of $k$ .
b) Take a ball from each of the consecutive boxes $C_{m-1}$ , $C_m$ , $C_{m+1}$ and move the $3$ balls to the box $C_{m+k}$ .
With these movements, what is the largest number of balls we can get in the box $2018$ ?

This post has been edited 1 time. Last edited by parmenides51, Sep 7, 2022, 12:36 AM

combinatorics

P,Q,B are collinear

by MNJ2357, Nov 21, 2020, 6:43 PM

$H$ is the orthocenter of an acute triangle $ABC$ , and let $M$ be the midpoint of $BC$ . Suppose $(AH)$ meets $AB$ and $AC$ at $D,E$ respectively. $AH$ meets $DE$ at $P$ , and the line through $H$ perpendicular to $AH$ meets $DM$ at $Q$ . Prove that $P,Q,B$ are collinear.

geometry

Pascal s theorem

Chinese Girls Mathematical Olympiad 2017, Problem 7

by Hermitianism, Aug 16, 2017, 9:29 AM

This is a very classical problem.
Let the $ABCD$ be a cyclic quadrilateral with circumcircle $\omega_1$ .Lines $AC$ and $BD$ intersect at point $E$ ,and lines $AD$ , $BC$ intersect at point $F$ .Circle $\omega_2$ is tangent to segments $EB,EC$ at points $M,N$ respectively,and intersects with circle $\omega_1$ at points $Q,R$ .Lines $BC,AD$ intersect line $MN$ at $S,T$ respectively.Show that $Q,R,S,T$ are concyclic.

Attachments:

This post has been edited 1 time. Last edited by Hermitianism, Aug 17, 2017, 9:04 AM

geometry

Submit

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math_exploration

by danilorj, May 14, 2025, 9:08 PM

by Miquel-point, May 14, 2025, 6:27 PM

by Dattier, May 14, 2025, 5:14 PM

by imnotgoodatmathsorry, May 2, 2025, 2:10 PM

by ItzsleepyXD, Apr 25, 2025, 4:15 AM

by pokmui9909, Mar 30, 2025, 5:18 AM

by sman96, Feb 8, 2025, 5:11 PM

by parmenides51, Sep 6, 2022, 11:50 PM

by MNJ2357, Nov 21, 2020, 6:43 PM

by Hermitianism, Aug 16, 2017, 9:29 AM