Romanian Mathematical Competitions 2015 Pdf

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

• Daniel Sitaru - 6 areas of 6 famous pedal triangles (Romania)
• Daniel Sitaru - Napoleon outer triangle revisited (Romania)
• Daniel Sitaru,Claudia Nănuți - The heptagonal triangle revisited (Romania)
• Jose Ferreira de Queiroz Filho - Gergonne point of a triangle and it's distance from any point in the plane (Brazil)
• Nguyen Ngoc Giang - The creation of the Steiner-Lehmus' theorem (Ho Chi Minh City - Vietnam)
• Thanasis Gakopoulos - Gakopoulos' Lemmas (Greece)
• Thanasis Gakopoulos - Gakopoulos' Lemma II (Greece)
• Thanasis Gakopoulos,Dimitris Blatsis  - Plagiogonal Plane Coordinate System, / area of polygon (n-sided) (Greece)
• Thanasis Gakopoulos - Metric Relations for mixtlinear incircles and excircles (Greece)
• Thanasis Gakopoulos -Gakopoulos' special transversals(Greece)

Marathons

• Abstract Algebra Marathon 001-100
• Abstract Algebra Marathon 101-200
• Abstract Algebra Marathon 201-300
• Abstract Algebra Marathon 301-400
• Calculus Marathon 001-100
• Calculus Marathon 101-200
• Calculus Marathon 201-300
• Calculus Marathon 301-400
• Calculus Marathon 401-500
• Calculus Marathon 501-600
• Calculus Marathon 601-700
• Calculus Marathon 701-800
• Calculus Marathon 801-900
• Calculus Marathon 901-1000
• Calculus Marathon 1001-1100
• Calculus Marathon 1101-1200
• Calculus Marathon 1201-1300
• Calculus Marathon 1301-1400
• Calculus Marathon 1401-1500
• Calculus Marathon 1501-1600
• Calculus Marathon 1601-1700
• Cyclic Inequalities Marathon 001-100
• Cyclic Inequalities Marathon 101-200
• Cyclic Inequalities Marathon 201-300
• Cyclic Inequalities Marathon 301-400
• Cyclic Inequalities Marathon 401-500
• Cyclic Inequalities Marathon 501-600
• Cyclic Inequalities Marathon 601-700
• Cyclic Inequalities Marathon 701-800
• Cyclic Inequalities Marathon 801-900
• Famous Inequalities Marathon 01-100
• Geometry Marathon 001-100
• Geometry Marathon 101-200
• Geometry Marathon 201-300
• Inequalities Marathon 001-100
• Inequalities Marathon 101-200
• Inequalities Marathon 201-300
• Inequalities Marathon 301-400
• Inequalities Marathon 401-500
• Inequalities Marathon 501-600
• Inequalities Marathon 601-700
• Math Adventures On CutTheKnot 01-50 (with CutTheKnot)
• Math Adventures On CutTheKnot 50-100 (with CutTheKnot)
• Math Adventures On CutTheKnot 101-150 (with CutTheKnot)
• Math Adventures On CutTheKnot 151- 200 (with CutTheKnot)
• Triangle Marathon 001-100 (Geometric Inequalities)
• Triangle Marathon 101-200 (Geometric Inequalities)
• Triangle Marathon 201-300 (Geometric Inequalities)
• Triangle Marathon 301-400 (Geometric Inequalities)
• Triangle Marathon 401-500 (Geometric Inequalities)
• Triangle Marathon 501-600 (Geometric Inequalities)
• Triangle Marathon 601-700 (Geometric Inequalities)
• Triangle Marathon 701-800 (Geometric Inequalities)
• Triangle Marathon 801-900 (Geometric Inequalities mostly)
• Triangle Marathon 901-1000 (Geometric Inequalities mostly)
• Triangle Marathon 1001-1100 (Geometric Inequalities mostly)
• Triangle Marathon 1101-1200 (Geometric Inequalities mostly)
• Triangle Marathon 1201-1200 (Geometric Inequalities mostly)
• Triangle Marathon 1301-1400 (Geometric Inequalities mostly)
• Triangle Marathon 1401-1500 (Geometric Inequalities mostly)
• Triangle Marathon 1501-1600 (Geometric Inequalities mostly)
• Triangle Marathon 1601-1700 (Geometric Inequalities mostly)
• Triangle Marathon 1701-1800 (Geometric Inequalities mostly)
• Triangle Marathon 1801-1900 (Geometric Inequalities mostly)
• Triangle Marathon 1901-2000 (Geometric Inequalities mostly)
• Triangle Marathon 2001-2100 (Geometric Inequalities mostly)
• Triangle Marathon 2101-2200 (Geometric Inequalities mostly)
• Triangle Marathon 2201-2300 (Geometric Inequalities mostly)
• Triangle Marathon 2301-2400 (Geometric Inequalities mostly)
• Triangle Marathon 2401-2500 (Geometric Inequalities mostly)
• Triangle Marathon 2501-2600 (Geometric Inequalities mostly)
• Triangle Marathon 2601-2700 (Geometric Inequalities mostly)
• Triangle Marathon 2701-2800 (Geometric Inequalities mostly)
• Triangle Marathon 2801-2900 (Geometric Inequalities mostly)

Problem Column

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 = )

Romanian Mathematical Competitions 2015 Pdf

Source: https://imogeometry.blogspot.com/p/romanian-mathematical-magazine-problem.html

Posted by: gonzaleztheast.blogspot.com

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