RMM

Romanian Mathematical Magazine

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

collected inside aops here

Geometry Problems till year 2023 

(selected from problem column - not 2D 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 150 Let be $z_1,z_2, z_3 \in C^*$ different in pairs such that $|z_1| = |z_2| = |z_3|$. If $$(z_1 + z_2)(z_2 + z_3)(z_3 + z_1) + z_1z_2z_3 = 0$$, then $z_1,z_2,z_3$ are the affixes of an equilateral triangle.

by Marian Ursarescu - Romania

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

JP 167 Let $OABC$ be a tetrahedron with $\angle AOB = \angle BOC = \angle COA = 90^o$ and let $P$ be any point inside the triangle $ABC$. Denote respectively by $d_a, d_b, d_c$ the distances from $P$ to faces $(OBC)$, $(OCA)$, $(OAB)$. Prove that:

(a) $d^2_a + d^2_b+ d^2_c= OP^2$.

(b) $d_ad_bd_c \le \frac{OA \cdot OB\cdot OC}{27}$ .

(c) $OA \cdot  d^3_a + OB \cdot d^3_b+ OC \cdot d^3_c \ge OP^4$.

by Nguyen Viet Hung - Hanoi - Vietnama

JP 194 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

JP 199 Let $SABCD$ be a pyramid with the base $ABCD$ parallelogram and $E$ any point which belongs to the side $SC$ such that $\frac{SE}{SC} = k$. Through the vertex $A$ and the point $E$ we consider a variable plane which intersects the segment $SB$ in $M$ and the segment $SD$ in $N$. Prove that $$\frac{V_{SAEMN}}{V_{SABCD}}\ge \frac{2k^2}{k + 1}$$

by Marian Ursarescu - Romania

JP 260 In $\vartriangle ABC$, $N$ - Nagel’s point, $BQ, CP$ - symedians. Prove that 

$P, N, Q$ - collinear $\Leftrightarrow \frac{1}{b^2r_b}+\frac{1}{c^2r_c}=\frac{1}{a^2r_a}$

by Marian Ursarescu - Romania

JP 309 If $m \in N$, $h_A, h_B, h_C, h_D$ are the lengths of heights of a tetrahedron $[ABCD]$ having the radius of the inscribed sphere $ r$, then

$$m+ \frac14 \left( \left( \frac{h_A - 3r}{h_A + 3r}\right)^{m+1}+\left(\frac{h_B - 3r}{h_B + 3r}\right)^{m+1} +\left( \frac{h_C - 3r}{h_C + 3r}\right)^{m+1}+\left( \frac{h_D - 3r}{h_D + 3r}\right)^{m+1} \right) \ge \frac{m+1}{7}$$

by D.M. Batinetu - Giurgiu, Daniel Sitaru - Romania

JP 338 In $\vartriangle ABC$, $P,Q \in Int \,\, (\vartriangle ABC)$, $\alpha, \beta, \gamma \in R,\alpha, \gamma \ne 1$ such that $$\beta \overline{AB}+\gamma \overline{BP}+ \overline{PC} = \overline{0}$$ and

$$\overline{AQ}+\alpha \overline{QB}+\overline{BC} = \overline{0}.$$ Prove that $A,P,Q$ are collinear if and only if $\alpha+ \gamma = \beta +1$

by Florica Anastase - Romania

JP 342 Let $ABCDA'B'C'D'$ be a cube with length side $ 1$ and $M \in BC$,$N  \in DD'$, $P  \in A'B'$. Find minimum perimeter of $\vartriangle MNP$.

by Florentin Visescu - Romania

JP 352 If $a, b, c \in C$, $|a| = |b| = |c| = 1$ then $3|a + b + c| + 2(|a - b| + |b - c| + |c - a|) \ge 9$ 

by Daniel Sitaru - Romania

JP 353 In $\vartriangle ABC$, $P \in  Int (\vartriangle ABC)$, $\angle ABP  = 20^o$, $\angle PBC =\angle  PCB  = 10^o$, $\angle PCA  = 40^o$. Prove that $|AP|+ |BC| =\sqrt3 |AB|$.

by Mehmet Sahin - Turkey

JP 354 In acute $\vartriangle ABC,O$ - circumcenter, $F,K \in (AB)$, $M, L \in (BC)$, $E,N \in  (CA)$,$FOE$, $MON$, $LOK$ - are the antiparallels. Let $\rho_a$ , $\rho_b$ , $\rho_c$ - inradii of $\vartriangle AFE$, $\vartriangle BLK$, $\vartriangle CMN$. Prove that $\rho_a+\rho_b+\rho_c = R$.

by Mehmet Sahin - Turkey

JP 357 In $\vartriangle ABC$, prove that inscribed circle of $\vartriangle ABC$ passes through Nagel's point $N_a$ if and only if $s^2 +4r^2 = 16Rr$.

by Marian Ursarescu - Romania

JP 367 Let $a, b, c \in C^*$ be different in pairs, $A(a),B(b),C(c)$, $|a| = |b| = |c| = 1$. If $$(ab)^3 + (bc)^3 + (ca)^3 = 3(abc)^2$$ then $\vartriangle ABC$ is equilateral.

by Marian Ursarescu - Romania

JP 379 If $ABCD$ tetrahedron $AB = a$, $AD = b$ , $AC = c$ ,$BD = d$ , $DC = e$ , $CB = f$ , $F$ - total area, then $$a^4 + b^4 + c^4 + d^4 + e^4 + f^4 \ge 2F^2$$

by D.M. Batinetu-Giurgiu, Daniel Sitaru - Romania

JP 389 A right parallelepiped $ABCDA'B'C'D'$ has the basis $ABCD$ rhombus, and areas of the two diagonals sections of the parallelepiped are $F_1$ and $F_2$ respectively. Let $R_1$ be the circumradius of $\vartriangle ABC$, $R_2$ circumradius of $\vartriangle ABD$ and $V$ volume of the right parallelepiped . Prove that $R_1R_2F_1F_2 \ge V^2$.

by Radu Diaconu - Romania

JP 391 In $\vartriangle ABC, P$ - inner point, $M, L \in [AB]$, $D,E \in [BC]$, $F,K \in [CA]$, $AM = AF$, $BL = BE$ , $CK = CD$, $|DE| = a_1$, $|FK| = b_1$, $|LM| = c_1$, $(M,P,F)$, $(C, P, L)$, $(D, P, K)$ - are collinear. Prove that $$F =\frac12 (a_1r_a + b_1r_b + c_1r_c)$$

by Mehmet Sahin - Turkey

JP 392 Let $z_1, z_2, z_3 \in C^*$ be different in pairs such that $|z_1| = |z_2| = |z_3|=1$, $A(z_1),B(z_2),C(z_3)$. Prove that, if $$\sum_{cyc}\frac{z_2z_3}{14z_2z_3 - z_2^2 - z_3^2}=\frac15$$ then $AB = BC = CA$.

by Marian Ursarescu - Romania

Seniors

SP 122 If $z_1, z_2, z_3 \in C$ are different in pairs and $|z_1| = |z_2| = |z_3| = 1$ then $$|z_1 - z_3| + |z_2 - z_3| \le  3 + |z_1 + z_2|$$

by Marian Ursarescu - Romania

SP 235 Let be $A(z_1)$, $B(z_1)$, $C(z_3)$, $z_1, z_2, z_3 \in C-\{0\}$, $|z_1| = |z_2| = |z_3|$, $AB = c$, $BC = a$, $CA = b$. If $$(b+c)z_Bz_C+(c+a)z_Cz_A+(a+b)z_Az_B = 0$$ then $AB = BC = CA$.

by Marian Ursarescu - Romania

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

SP 250/ 323 (250) Let $z_1, z_2, z_3 \in C-\{0\}$ be different in pairs, $|z_1| = |z_2| = |z_3|=1$, $A(z_1)$, $B(z_1)$, $C(z_3)$. If $$|z_1 - z_2 - z_3| +|z_2 - z_1 - z_3| +|z_3 - z_2 - z_1| = 6$$ then $AB = BC = CA$.

reposted and rephrased as

(323) Let $z_A, z_B, z_V \in C^*$ be different in pairs, $|z_A| = |z_B| = |z_B|=1$. If $$|z_A - z_B - z_C| +|z_B - z_C - z_A| +|z_C - z_A - z_B| = 6$$ then $\vartriangle ABC$ is an equilateral triangle.

by Marian Ursarescu - Romania

SP 259 In $\vartriangle ABC$, $\Gamma$ - Gergonne’s point and $BN, CM$ symedians, $M \in (AB), N \in (AC)$. Prove that $B, \Gamma, N$ - collinear $\Leftrightarrow \frac{r_b}{b^2}+\frac{r_c}{c^2} =\frac{r_a}{a^2}$

by Marian Ursarescu - Romania

SP 310 In $\vartriangle ABC$, $B' \in (AC)$ the contact point of the external circumscription circle of side $AC$ and $C'$ the contact point of the external circumscription circle of side $AB$. Prove that $B'C'$ is tangent of the inscribed circle in $ABC$ if and only if $(s - b)^2 + (s - c)^2 = (s - a)^2$

by Marian Ursarescu - Romania

My note: $B'$ is touchpoint of $B$-excircle with $AC$, $C'$ is touchpoint of $C$-excircle with $AB$, $s$ is the semiperimeter

SP 348 In $\vartriangle ABC$ prove that inscribed circle of $\vartriangle ABC$ passes through the centroid $G$ if and only if $s^2 = 16Rr + 4r^2$.

by Marian Ursarescu - Romania

SP 360 Let $z_1, z_2, z_3 \in C^*$ be different in pairs such that $|z_1| = |z_2| = |z_3|$. If

$$\sum_{cyc}\frac{2z_1 - z_2 - z_3}{(z_1 - z_2)|z_1 - z_3| + (z_1 - z_3)|z_1 - z_2|}=\frac{1}{|z_1 - z_2|}+\frac{1}{|z_2 - z_3|}+\frac{1}{|z_3 - z_1|}$$, then $z_1, z_2,z_3$ are affixes on equilateral triangle.

by Marian Ursarescu - Romania

SP 371 Let $ABCD$ be a tetrahedron, and let $M$ be a point in space, $M \not\in  \{A,B,C\}$. Prove that $$\frac{MA}{MB +MC+MD}+\frac{MB}{MC +MD +MA}+\frac{MC}{MD +MA +MB}+$$

$$+\frac{MD}{MA +MB +MC} \ge \frac{R + r}{R} \ge \frac{4r}{R}$$

by D.M. Batinetu - Giurgiu, Neculai Stanciu - Romania

SP 382 Let $z_1, z_2, z_3 \in C^*$ be different in pairs such that $|z_1| = |z_2| = |z_3|=1$, $A(z_1),B(z_2),C(z_3)$. Prove that, if $$\sum_{cyc}\frac{z_2z_3}{(z_2 - z_3)^2[z_2(z_1 - z_3)^2 - z_3(z_1 + z_2)^2]}=\frac{1}{4z_1z_2z_3}$$ then $AB = BC = CA$.

by Marian Ursarescu - Romania

SP 410 Let $z_1, z_2, z_3 \in C^*$ be different in pairs such that $|z_1| = |z_2| = |z_3|=1$, $A(z_1),B(z_2),C(z_3)$. Prove that:

$$\sum_{cyc}|2z_1-z_2-z_3|^4=243 \Rightarrow AB = BC = CA$$.

by Marian Ursarescu - Romania

SP 411 Let $z_1, z_2, z_3 \in C^*$ be different in pairs such that $|z_1| = |z_2| = |z_3|=1$, $A(z_1),B(z_2),C(z_3)$. Prove that:

$$\sum_{cyc}\frac{1}{8z_1z_2z_3 - (z_1^2+z_2z_3)(z_2+z_3)}=\frac{3}{10z_1z_2z_3} \Rightarrow AB = BC = CA$$.

by Marian Ursarescu - Romania

SP 423 Let $z_1, z_2, z_3 \in C^*$ be different in pairs such that $|z_1| = |z_2| = |z_3|=1$, $A(z_1),B(z_2),C(z_3)$. Prove that:

$$\sum_{cyc}\frac{z_2z_3}{3z_2z_3 - z_2^2 - z_3^2}=\frac34 \Leftrightarrow AB = BC = CA$$.

by Marian Ursarescu - Romania

SP 451 If $ABCD$ is a convex quadrilateral such that $AC \cap BD =\{O\}$, $AE = EC$, $BF = FD$ with order $A - O - E - C$ respectively , $B -F -O - D$, $EF \cap AB = \{J\}$, $EF \cap CD = \{K\}$, $CJ \cap BK = \{L\}$ and $M$ the midpoint of $KJ$, then prove that $O, M$ and $L$ are collinear.

by Marius Dragan, Neculai Stanciu - Romania

SP 462 Let $ABC$ be an triangle, $D$ be a point on side $BC$ and $M$ be the symmetrical of $A$ with respect to $D$. If $\frac{BM^2}{AB} +\frac{CM^2}{AC} = AB+AC$, then prove that $AD$ is the bisector of the angle $\angle A$, or is the altitude from the vertex $A$.

by Neculai Stanciu - Romania

Undergraduate 

UP 056 Let $ABC$ be a triangle and $\Omega$ the first Brocard point of $ABC$. Let $D,E, F$ are on the sides $BC$, $CA$, $AB$ of $ABC$ respectively. If $\angle B \Omega D  =\angle  C \Omega E=\angle A \Omega F  = 90^o$ then prove that $$\frac{BD}{BC} + \frac{CE}{CA} + \frac{AF}{AB} = 2$$

by Mehmet Sahin - Ankara - Turkey

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 162 If $ABCD$ is tetrahedron $AB = a_1$, $AC = a_2$, $AD = a_3$, $BC = a_4$, $BD = a_5$ ,$CD = a_6$ then $$\sum_{1\le i <j \le 6}(a_i + a_j)^2 \ge 4\sqrt3 S[ABCD]$$ where $S[ABCD]$ is total area of tetrahedron $ABCD$.

by Daniel Sitaru - Romania

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

UP 294 In $\vartriangle ABC$, $AD,BE,CF$ -medians, $G$ centroid, $AM = MG$ , $M \in  (AG)$, and $2 \cot A = \cot B + \cot C$. Prove that $DEMF$ is a cyclic quadrilateral.

by Marian Ursarescu - Romania

UP 321 Let $A_0A_1...A_n$ be an Euclidean $n$-simplex. We will use the following notations:

- $O, V,R, r$ the centre if its circumscribed hypersphere, its volume, its circumradius and its inradius, respectively.

- $O_i,R_i$ the centre and the radius of the hypersphere tangent to the circumscribed sphere of $A_0,A_1,...,A_n$ in the vertex $A_i$ and to the hyperplane $A_0A_1...A_{i-1}A_{i+1}...A_n$ simultaneously.

With the above notations, the following identity holds: $$\sum_{i=0}^{n} \frac{1}{R_i}= \frac{n}{R} +\frac{1}{r}$$

by Vasile Jiglau - Romania

UP415 Let $ABC$ denote a triangle and $H$ its orthocenter. Let point $M$ be the midpoint of the segment $AH$. Prove that:

(a) angle $\angle BMC$ is acute.

(b) area $\vartriangle BMC = 1/8 \cdot  AH^2 \cdot \tan \angle BMC$.

by George Apostolopoulos - Greece

Geometry articles ,  a selection

(to be updated)

• 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)
• Jose Ferreira de Queiroz Filho - Bevan's point in a brazilian approach (Brazil)
• Nguyen Ngoc Giang - The creation of the Steiner-Lehmus’ theorem (Ho Chi Minh City - Vietnam)

Gakopoulos articles  (soon more)

• 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)
• Thanasis Gakopoulos, Kousik Sett - The GakSett Circle (Greec, India) 

Collected books

                             (to be added)

Marathons (google drive folder here)

Abstract Algebra Marathon Google Drive

• Abstract Algebra Marathon 001-100
• Abstract Algebra Marathon 101-200
• Abstract Algebra Marathon 201-300
• Abstract Algebra Marathon 301-400
• Abstract Algebra Marathon 401-500
• Abstract Algebra Marathon 501-600
• Abstract Algebra Marathon 601-700

Calculus Marathon Google Drive

• 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 
• Calculus Marathon 1701-1800
• Calculus Marathon 1801-1900
• Calculus Marathon 1901-2000
• Calculus Marathon 2001-2100
• Calculus Marathon 2101-2200
• Calculus Marathon 2201-2300
• Calculus Marathon 2301-2400
• Calculus Marathon 2401-2500

Cyclic Inequalities Marathon Google Drive

• 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
• Cyclic Inequalities Marathon 901-1000
• Cyclic Inequalities Marathon 1001-1100
• Cyclic Inequalities Marathon 1101-1200
• Cyclic Inequalities Marathon 1201-1300
• Cyclic Inequalities Marathon 1301-1400
• Cyclic Inequalities Marathon 1401-1500
• Cyclic Inequalities Marathon 1501-1600

• Famous Inequalities Marathon 01-100 

Geometry Marathon Google Drive

• Geometry Marathon 001-100
• Geometry Marathon 101-200
• Geometry Marathon 201-300
• Geometry Marathon 301-400
• Geometry Marathon 401-500
• Geometry Marathon 501-600
• Geometry Marathon 601-700
• Geometry Marathon 701-800
• Geometry Marathon 801-900
• Geometry Marathon 901-1000
• Geometry Marathon 1001-1100
• Geometry Marathon 1101-1200
• Geometry Marathon 1201-1300
• Geometry Marathon 1301-1400
• Geometry Marathon 1401-1500
• Geometry Marathon 1501-1600
  
• Geometry Marathon 1601-1700
  
• Geometry Marathon 1701-1800
  
• Geometry Marathon 1801-1900
  
• Geometry Marathon 1901-2000
  
• Geometry Marathon 2001-2100
  

Inequalities Marathon Google Drive

• 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
• Inequalities Marathon 701-800
• Inequalities Marathon 801-900
• Inequalities Marathon 901-1000
• Inequalities Marathon 1001-1100
• Inequalities Marathon 1101-1200
• Inequalities Marathon 1201-1300
• Inequalities Marathon 1301-1400
• Inequalities Marathon 1401-1500
  

Math Adventures On CutTheKnot Google Drive

• 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  Google Drive

• Triangle Marathon 001-100 (Geometric Inequalities mostly)
• Triangle Marathon 101-200 (Geometric Inequalities mostly)
• Triangle Marathon 201-300 (Geometric Inequalities mostly) 
• Triangle Marathon 301-400 (Geometric Inequalities mostly)
• Triangle Marathon 401-500 (Geometric Inequalities mostly)
• Triangle Marathon 501-600 (Geometric Inequalities mostly)
• Triangle Marathon 601-700 (Geometric Inequalities mostly)
• Triangle Marathon 701-800 (Geometric Inequalities mostly)
• 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) 
• Triangle Marathon 2901-3000 (Geometric Inequalities mostly) 

Problem Column 2016-2023

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

2021: Problems & Solutions  problems 286-345
2022: Problems & Solutions  problems 346-405
2023: Problems & Solutions  problems 406-465



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