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RMM

Romanian Mathematical Magazine


Here are gonna be collected all the Problem Collections and the Marathons from the online magazine ''Romanian Mathematical Magazine''.



Geometry Problems 
(selected from problem column - not geometric inequalities)

Juniors

JP 057 Let $ABC$ be an arbitrary triangle and $I_a, I_b, I_c$ are excenters of $ABC$. $I_aBC$, $I_bCA$, $I_cAB$ are the extriangles of ABC. Let $h_i$ ($i = 1, 2,3,...,9$) the altitudes of extriangles. Prove that $$\prod _{i=1}^9 h_i =\left(\prod _{a,b,c}r_a \right)^3$$
by Mehmet Sahin - Ankara - Turkey

JP 156 Let $ABC$ be a triangle having the area $S$. Let be $A' \in  (BC)$ such that the incircles of $\vartriangle AA'B$, $\vartriangle AA'C$ have the same radius. Analogous, we obtain the points $B' \in (AC)$, $C' \in (AB)$. Prove that: $$S =\frac{AA' \cdot BB' \cdot CC' }{s}$$ where $s$ is the semiperimeter of $\vartriangle ABC$.
by Marian Ursarescu - Romania

JP194 In $\vartriangle ABC, BE, CF$ are internal bisectors, $E \in (AC), F \in (AB),O$ is circumcentre.  Prove that:
$E,O, F$ collinear  $\Leftrightarrow  \cos A =\ cos B +\ cos C$
by Marian Ursarescu - Romania
Seniors

SP 246 If $ABCD$ bicentric quadrilateral, $ I$ incenter then:
$$(IA^2 + IC^2)(IB^2 + ID^2) \ge  AB \cdot BC \cdot CD \cdot  DA$$
by Daniel Sitaru - Romania
Undergraduate 

UP 063 Let $SABC$ be a tetrahedron and let $M$ be any point  inside the triangle $ABC$. The lines through $M$ parallel with the planes $SBC, SCA,SAB$ intersect $SA,SB,SC$ at $X,Y,Z$, respectively.  Prove that:  Vol $(MXY Z) \le \frac{2}{27}$ Vol $(SABC)$.
Determine position of the point $M$ such that the equality holds.

by Nguyen Viet Hung - Hanoi - Vietnam
UP 203 Given a triangle $ABC$ with incenter $I$. The lines $AI,BI,CI $ meet the sides $BC,CA,AB$ at $A',B',C'$ and meet the circumcircle at the second points $A_1,B_1,C_1$ respectively. Prove that:
(a) $\frac{AI}{AA'} + \frac{BI}{BB'} + \frac{CI}{CC'} = 2$,

(b) $\frac{A_1I}{AI} + \frac{B_1I}{BI }+\frac{C_1I}{CI} = \frac{2R}{r} - 1$.

by Nguyen Viet Hung - Hanoi - Vietnam


Geometry articles




Marathons
Problem Column

2016: Problems & Solutions  problems 001-045
2017: Problems & Solutions  problems 046-105                                         
2018: Problems & Solutions  problems 106-165
2019: Problems & Solutions  problems 166-225
2020: Problems & Solutions  problems 226-285


latest issues - year 2021:
Spring:    Problems & Solutions problems 286-300
Summer: Problems & Solutions problems 301-315
Autumn:  Problems & Solutions  problems
Winter:    Problems & Solutions problems  



sources:
www.cut-the-knot.org  (Alexander Bogomolny)
www.ssmrmh.ro   (RMM = )



https://www.ssmrmh.ro/

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