Angle Relationships in Triangles

by steven_zhang123, May 14, 2025, 11:09 PM

In $\triangle ABC$ , $AB > AC$ . The internal angle bisector of $\angle BAC$ and the external angle bisector of $\angle BAC$ intersect the ray $BC$ at points $D$ and $E$ , respectively. Given that $CE - CD = 2AC$ , prove that $\angle ACB = 2\angle ABC$ .

angle

geometry

angle bisector

Long and wacky inequality

by Royal_mhyasd, May 12, 2025, 7:01 PM

Let $x, y, z$ be positive real numbers such that $x^2 + y^2 + z^2 = 12$ . Find the minimum value of the following sum :
$\sum_{cyc}\frac{(x^3+2y)^3}{3x^2yz - 16z - 8yz + 6x^2z}$ knowing that the denominators are positive real numbers.

Inequality

own

inequalities

Perpendicular passes from the intersection of diagonals, \angle AEB = \angle CED

by NO_SQUARES, May 5, 2025, 5:34 PM

Inside of convex quadrilateral $ABCD$ point $E$ was chosen such that $\angle DAE = \angle CAB$ and $\angle ADE = \angle CDB$ . Prove that if perpendicular from $E$ to $AD$ passes from the intersection of diagonals of $ABCD$ , then $\angle AEB = \angle CED$ .

geometry

A game with balls and boxes

by egxa, Apr 30, 2023, 11:24 AM

Initially, Aslı distributes $1000$ balls to $30$ boxes as she wishes. After that, Aslı and Zehra make alternated moves which consists of taking a ball in any wanted box starting with Aslı. One who takes the last ball from any box takes that box to herself. What is the maximum number of boxes can Aslı guarantee to take herself regardless of Zehra's moves?

combinatorics

Integer FE Again

by popcorn1, Jul 20, 2021, 9:18 PM

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:

$(i)$ $f(n) \neq 0$ for at least one $n$ ;
$(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$ ;
$(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$ .

concyclic wanted, PQ = BP, cyclic quadrilateral and 2 parallelograms related

by parmenides51, Sep 25, 2020, 4:27 AM

Let $ABCD$ be a cyclic quadrilateral in which the lines $BC$ and $AD$ meet at a point $P$ . Let $Q$ be the point of the line $BP$ , different from $B$ , such that $PQ = BP$ . We construct the parallelograms $CAQR$ and $DBCS$ . Prove that the points $C, Q, R, S$ lie on the same circle.

This post has been edited 1 time. Last edited by parmenides51, Sep 25, 2020, 4:30 AM

geometry

cyclic quadrilateral

parallelogram

help me please

by thuanz123, Jan 17, 2016, 2:08 PM

find all $a,b \in \mathbb{Z}$ such that:
a) $3a^2-2b^2=1$
b) $a^2-6b^2=1$

This post has been edited 1 time. Last edited by thuanz123, Jan 17, 2016, 2:13 PM

number theory

Easy functional equation

by fattypiggy123, Jul 5, 2014, 8:41 AM

Find all functions from the reals to the reals satisfying
$f(xf(y) + x) = xy + f(x)$

function

algebra proposed

algebra

Two circles, a tangent line and a parallel

by Valentin Vornicu, Oct 24, 2005, 10:15 AM

Two circles $G_1$ and $G_2$ intersect at two points $M$ and $N$ . Let $AB$ be the line tangent to these circles at $A$ and $B$ , respectively, so that $M$ lies closer to $AB$ than $N$ . Let $CD$ be the line parallel to $AB$ and passing through the point $M$ , with $C$ on $G_1$ and $D$ on $G_2$ . Lines $AC$ and $BD$ meet at $E$ ; lines $AN$ and $CD$ meet at $P$ ; lines $BN$ and $CD$ meet at $Q$ . Show that $EP = EQ$ .

Problem 5 (Second Day)

by darij grinberg, Jul 13, 2004, 2:49 PM

In a convex quadrilateral $ABCD$ , the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$ . The point $P$ lies inside $ABCD$ and satisfies $\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.$ Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$ .

Attachments:

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Check out the Pascal's Law post. I included a cartoon from the xkcd serial webcomic.
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