Polynomials

by Pao_de_sal, Apr 1, 2025, 8:35 PM

find all natural numbers n such that the polynomial x²ⁿ + xⁿ + 1 is divisible by x² + x + 1

Polynomials

algebra

polynomial

inequalities

by Cobedangiu, Apr 1, 2025, 6:10 PM

$a,b>0$ and $a+b=1$ . Find min P:
$P=\sqrt{\frac{1-a}{1+7a}}+\sqrt{\frac{1-b}{1+7b}}$

This post has been edited 1 time. Last edited by Cobedangiu, 5 hours ago

inequalities

April Fools Geometry

by awesomeming327., Apr 1, 2025, 2:52 PM

Let $ABC$ be an acute triangle with $AB<AC$ , and let $D$ be the projection from $A$ onto $BC$ . Let $E$ be a point on the extension of $AD$ past $D$ such that $\angle BAC+\angle BEC=90^\circ$ . Let $L$ be on the perpendicular bisector of $AE$ such that $L$ and $C$ are on the same side of $AE$ and
$\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ$ Let the reflection of $D$ across $AB$ and $AC$ be $W$ and $Y$ , respectively. Let $X\in AW$ and $Z\in AY$ such that $\angle XBE=\angle ZCE=90^\circ$ . Let $EX$ and $EZ$ intersect the circumcircles of $EBD$ and $ECD$ at $J$ and $K$ , respectively. Let $LB$ and $LC$ intersect $WJ$ and $YK$ at $P$ and $Q$ . Let $PQ$ intersect $BC$ at $F$ . Prove that $FB/FC=DB/DC$ .

geometry

kind of well known?

by dotscom26, Apr 1, 2025, 4:11 AM

Let $y_1, y_2, ..., y_{2025}$ be real numbers satisfying
$y_1^2 + y_2^2 + \cdots + y_{2025}^2 = 1.$
Find the maximum value of
$|y_1 - y_2| + |y_2 - y_3| + \cdots + |y_{2025} - y_1|.$

I have seen many problems with the same structure, Id really appreciate if someone could explain which approach is suitable here

This post has been edited 1 time. Last edited by dotscom26, Today at 4:20 AM

algebra

algebra-geometry

sum(ab/4a^2+b^2) <= 3/5

by truongphatt2668, Mar 31, 2025, 1:23 PM

Let $a,b,c>0$ . Prove that:
$\dfrac{ab}{a^2+4b^2} + \dfrac{bc}{b^2+4c^2} + \dfrac{ca}{c^2+4a^2} \le \dfrac{3}{5}$

This post has been edited 1 time. Last edited by truongphatt2668, Yesterday at 1:47 PM

inequalities

Inequality

hard problem

by pennypc123456789, Mar 26, 2025, 11:39 AM

Let $\triangle ABC$ be an acute triangle inscribed in a circle $(O)$ with orthocenter $H$ and altitude $AD$ . The line passing through $D$ perpendicular to $OD$ intersects $AB$ at $E$ . The perpendicular bisector of $AC$ intersects $DE$ at $F$ . Let $OB$ intersect $DE$ at $K$ . Let $L$ be the reflection of $O$ across $EF$ . The circumcircle of triangle $BDE$ intersects $(O)$ at $G$ different from $B$ . Prove that $GF$ and $KL$ intersect on the circumcircle of triangle $DEH$ .

geometry

Reflections of AB, AC with respect to BC and angle bisector of A

by falantrng, Apr 29, 2024, 12:40 PM

Let $ABC$ be an acute-angled triangle with $AC > AB$ and let $D$ be the foot of the
$A$ -angle bisector on $BC$ . The reflections of lines $AB$ and $AC$ in line $BC$ meet $AC$ and $AB$ at points
$E$ and $F$ respectively. A line through $D$ meets $AC$ and $AB$ at $G$ and $H$ respectively such that $G$
lies strictly between $A$ and $C$ while $H$ lies strictly between $B$ and $F$ . Prove that the circumcircles of
$\triangle EDG$ and $\triangle FDH$ are tangent to each other.

This post has been edited 1 time. Last edited by falantrng, Apr 29, 2024, 12:40 PM

geometry

configurational geometry as usual

by GorgonMathDota, Nov 9, 2021, 5:10 AM

Given $\triangle ABC$ with circumcircle $\ell$ . Point $M$ in $\triangle ABC$ such that $AM$ is the angle bisector of $\angle BAC$ . Circle with center $M$ and radius $MB$ intersects $\ell$ and $BC$ at $D$ and $E$ respectively, $(B \not= D, B \not= E)$ . Let $P$ be the midpoint of arc $BC$ in $\ell$ that didn't have $A$ . Prove that $AP$ angle bisector of $\angle DPE$ if and only if $\angle B = 90^{\circ}$ .

very cute geo

by rafaello, Oct 26, 2021, 7:28 PM

Consider a triangle $ABC$ with incircle $\omega$ . Let $S$ be the point on $\omega$ such that the circumcircle of $BSC$ is tangent to $\omega$ and let the $A$ -excircle be tangent to $BC$ at $A_1$ . Prove that the tangent from $S$ to $\omega$ and the tangent from $A_1$ to $\omega$ (distinct from $BC$ ) meet on the line parallel to $BC$ and passing through $A$ .

geometry

incircle

Geometry

by Emirhan, Jan 30, 2016, 2:54 PM

Let $ABC$ be an equilateral triangle with side lenght is $1$ $cm$ .Let $D \in [AB]$ is a point. Perpendiculars from $D$ to $[AC]$ and $[BC]$ intersects with $[AC]$ and $[BC]$ at points $E$ and $F$ respectively. Perpendiculars from $E$ and $F$ to $[AB]$ intersects with $[AB]$ at points $E_1$ and $F_1$ . Prove that
$[E_1F_1]=\frac{3}{4}$