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Old High School Olympiads Regional, National, and International Math Olympiads before 2001 mostly, recent years have been posted in order to make contest collections richer
Regional, National, and International Math Olympiads before 2001 mostly, recent years have been posted in order to make contest collections richer
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Old High School Olympiads Regional, National, and International Math Olympiads before 2001 mostly, recent years have been posted in order to make contest collections richer
Regional, National, and International Math Olympiads before 2001 mostly, recent years have been posted in order to make contest collections richer
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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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0 replies
jlacosta
Apr 2, 2025
0 replies
J
H
k i testing latex post , please ignore
parmenides51   3
N Feb 23, 2025 by parmenides51
Source: used with https://artofproblemsolving.com/community/c3170239_latex_testing_for_post_collections
issue 1 ..,
Click to reveal hidden text
3 replies
parmenides51
Oct 10, 2022
parmenides51
Feb 23, 2025
J
H
i Geo Collections + Forums + Greek Aops Index
parmenides51   57
N Oct 12, 2024 by parmenides51
In order to keep an eye to any contest collection that I have created and is not in aops contest collections, I created a collection with all that in a link: Geo Collections + Forums + Greek Aops Index

It contains so far (in these categories):

$\bullet$ Forums
$\bullet \bullet$ Geometry Mathley
$\bullet \bullet$ Old High School Olympiads
$\bullet \bullet$ Old Russian Math Olympiads

All geometry mentioned below in one forum collection:
$\bullet \bullet$ Geometry Collections and Contests

$\bullet$ Aops Geo Mocks
$\bullet$ Aops Mock Contests - Geometry


$\bullet \bullet$ geo national (geometry from:)
$\bullet$ Argentina NMO (Round 2-4, Finals, Provincial, Regional Round 2)
$\bullet$ Armenia NMO (outside aops)
$\bullet$ Belarus NMO VIII (outside aops)
$\bullet$ Belarus NMO IX (>>)
$\bullet$ Belarus NMO X (>>)
$\bullet$ Belarus NMO XI (>>)
$\bullet$ Belgium Flanders Juniors JWO (outside aops)
$\bullet$ Belgium Flanders Seniors VWO (>>)
$\bullet$ Belgium OMB Juniors (outside aops)
$\bullet$ Belgium OMB Seniors (>>)
$\bullet$ Brazil NMO (Round 2 + Finals)
$\bullet$ Bulgaria NMO Plane Geometry (outside aops)
$\bullet$ Bulgaria NMO 3D Geometry (>>)
$\bullet$ Costa Rica NMO (OLCOMA)
$\bullet$ Croatia MO (outside aops)
$\bullet$ Chile NMO (outside aops)
$\bullet$ Denmark MO (Mohr Contest) (outside aops)
$\bullet$ Estonia Open Junior MO (outside aops)
$\bullet$ Estonia Open Senior MO (>>)
$\bullet$ Greece JMO (outside aops)
$\bullet$ Greece NMO (outside aops)
$\bullet$ Hungary OKTV I (outside aops)
$\bullet$ Hungary OKTV II (>>)
$\bullet$ Hungary OKTV III (>>)
$\bullet$ Iceland NMO (outside aops)
$\bullet$ Indonesia MO Province OSP SMA (outside aops)
$\bullet$ Kazakhstan NMO IX (outside aops)
$\bullet$ Kazakhstan NMO X (>>)
$\bullet$ Kazakhstan NMO XI (>>)
$\bullet$ Lithuania juniors (outside aops)
$\bullet$ Mexico NMO (States Semifinals + Finals, Finals)
$\bullet$ Moldova NMO VII-VIII (outside aops)
$\bullet$ Moldova NMO IX-XII Plane Geometry (>>)
$\bullet$ Moldova NMO 3D Geometry (>>)
$\bullet$ New Zealand MO (outside aops)
$\bullet$ Paraguay NMO (OMAPA) (outside aops)
$\bullet$ Peru NMO (ONEM) (Rounds 2-3, Finals)
$\bullet$ Poland Juniors , Round 1+2 (outside aops)
$\bullet$ Poland Juniors, Finals (>>)
$\bullet$ Portugal NMO (outside aops)
$\bullet$ Romania District 3D Geometry (outside aops)
$\bullet$ Romania District Plane Geometry (>>)
$\bullet$ Romania NMO 3D Geometry (outside aops)
$\bullet $ Romania NMO Plane Geometry (>>)
$\bullet$ Swiss NMO (outside aops)
$\bullet$ Ukraine NMO VIII (outside aops)
$\bullet$ Ukraine NMO IX (>>)
$\bullet $ Ukraine NMO X-XI Plane Geometry (>>)
$\bullet$ Ukraine NMO 3D Geometry (>>)


$\bullet \bullet$ geo TST (geometry from:)
$\bullet$ Argentina Cono Sur TST (outside aops)
$\bullet$ Argentina Ibero TST (>>)
$\bullet$ Argentina IMO TST (>>)
$\bullet$ Argentina Training Lists Cono Sur (outside aops)
$\bullet$ Argentina Training Lists IberoAmerican (>>)
$\bullet$ Argentina Training Lists Rioplatense (>>)
$\bullet$ Bangladesh TST and Camp
$\bullet$ Belarus TST (outside aops)
$\bullet$ Brazil Training List Cono Sur 1999 - 2010 (outside aops)
$\bullet$ Brazil Training List Cono Sur 2011-20 (>>)
$\bullet$ Brazil Cono Sur TST (outside aops)
$\bullet$ Brazil IMO IberoAmerican TST (>>)
$\bullet$ Chile TST (outside aops)
$\bullet$ Croatia TST (outside aops)
$\bullet$ Cyprus TST (outside aops)
$\bullet$ Ecuador TST (outside aops)
$\bullet$ France TST (outside aops)
$\bullet$ Hong Kong TST (outside aops)
$\bullet$ Indonesia TST (outside aops)
$\bullet$ Italy TST (outside aops)
$\bullet$ Lithuania TST (outside aops)
$\bullet$ Peru Cono Sur TST (outside aops)
$\bullet$ Peru IberoAmerican TST (>>)
$\bullet$ Peru IMO TST (>>)
$\bullet$ Puerto Rico TST (outside aops)
$\bullet$ Serbia IMO TST (outside aops)
$\bullet$ Singapore TST (outside aops)
$\bullet$ Switzerland TST (outside aops)
$\bullet$ Taiwan TST (outside aops)
$\bullet$ Thailand TST (outside aops)
$\bullet$ Ukraine TST (outside aops)


$\bullet \bullet$ geo regional (geometry from:)
$\bullet$ 239 MO (St.Petersburg's Lyceum) (outside aops)
$\bullet$ Alberta High School Mathematics Competition (Canada) (outside aops)
$\bullet$ AESC MSU Internet Olympiad (Russia) (outside aops)
$\bullet$ All-Siberian Olympiad Finals (Russia) (outside aops)
$\bullet$ All-Siberian Olympiad Correspondence (Russia) (>>)
$\bullet$ All-Siberian Olympiad Qualifying (Russia) (>>)
$\bullet$ Almaty City MO (Kazakhstan) (outside aops)
$\bullet$ Association Mathematical Quebec Competition (Canada) (outside aops)
$\bullet$ Auckland MO (New Zealand) (outside aops)
$\bullet$ Champions Tournament (Ukraine) (Турнір чемпіонів) (outside aops)
$\bullet$ Clock-Tower School Junior (Romania) (outside aops)
$\bullet$ COMATEQ (Puerto Rico) (outside aops)
$\bullet$ Cordoba pre TSTs (Argentina) (outside aops)
$\bullet$ Dürer Math Competition (Hungary) (outside aops)
$\bullet$ El Número de Oro Certamen (Argentina) (outside aops)
$\bullet$ Euler Olympiad Remote (Russia) (outside aops)
$\bullet$ Euler Olympiad Regional & Final (Russia) (>>)
$\bullet$ Euler Teachers' MO (Russia) (outside aops)
$\bullet$ European Math Tournament (Belarus) (outside aops)
$\bullet$ Formula of Unity / Third Millennium Olympiad (Russia) (outside aops)
$\bullet$ HSGS Class 9 Contest (Vietnam) (outside aops)
$\bullet$ High Standards Olympiad (Russia) (Высшая проба) (outside aops)
$\bullet$ Innopolis University Open MO (Russia) (outside aops)
$\bullet$ Izumrud Olympiad Emerald by UrFU (Russia) (outside aops)
$\bullet$ Kharkiv City MO (Ukraine) (outside aops)
$\bullet$ Kharkiv Lyceum No 27 Olympiad (Ukraine) (outside aops)
$\bullet$ Kharkiv Masters Tournament (Ukraine) (outside aops)
$\bullet$ Kolomogorov Cup / Tournament (Russia)
$\bullet$ Kukin MO by OmSU (Russia) (outside aops)
$\bullet$ Kurchatov Olympiad (Russia) (outside aops)
$\bullet$ Kyiv City MO 1984-93 (Ukraine) (outside aops)
$\bullet$ Kyiv City MO 2003+ Round 1 (Ukraine) (>>)
$\bullet$ Kyiv City MO 2010+ Round 2 (Ukraine) (>>)
$\bullet$ Kyiv TST 2005-16 (for Ukraine MO) (outside aops)
$\bullet$ Kyiv TST 2017+ (for Ukraine MO) (>>)
$\bullet$ League of Winners Tournament (Russia) (outside aops)
$\bullet$ Lomonosov Tournament (Russia) (outside aops)
$\bullet$ Mathematical Multiathlon Tournament Juniors (Russia) (outside aops)
$\bullet$ Mathematical Multiathlon Tournament Seniors (Russia) (>>)
$\bullet$ Maths Beyond Limits Camp (Poland) (outside aops)
$\bullet$ May Olympiad, level 1 (Latin America) (outside aops)
$\bullet $ May Olympiad, level 2 (Latin America) (>>)
$\bullet$ Minsk City Internet Olympiad (Junior) (Belarus) (outside aops)
$\bullet$ MIPT Metropolitan Olympiad (Russia) (outside aops)
$\bullet$ Moscow Corespondence Competition 6-8 MO (Russia) (outside aops)
$\bullet$ Moscow Teachers' MO (Russia) (outside aops)
$\bullet$ OMAFOROS - OFO,FOFO, COFFEE (Argentina) (outside aops)
$\bullet$ Oral Moscow City MO VI-VII (Russia) (outside aops)
$\bullet$ Oral Moscow City Team MO 1999 - 2004 (Russia) (outside aops)
$\bullet$ Oral Moscow City Team MO 2005-19 (Russia) (>>)
$\bullet$ Oral Moscow City Team MO 2020+ (Russia) (>>)
$\bullet$ Rio de Janeiro MO (OMERJ) (Brazil) (outside aops)
$\bullet$ Rioplatense Olympiad, level 1 (Latin America) (outside aops)
$\bullet$ Rioplatense Olympiad, level 2 (Latin America) (>>)
$\bullet$ Rioplatense Olympiad, level 3 (Latin America) (>>)
$\bullet$ Rusanovsky Lyceum, Kyiv Olympiad(Ukraine) (outside aops)
$\bullet$ Savin Competition 1990 - 2015 (Russia) (outside aops)
$\bullet$ Savin Competition 2016+ (Russia) (>>)
$\bullet$ Savin Tournament (Russia) (>>)$\bullet$ SPBU School Olympiad besides Finals X-XI (Russia) (outside aops)
$\bullet$ SPBU School Olympiad, Finals X-XI - plane geometry (Russia) (>>)
$\bullet$ SPBU School Olympiad, Finals X-XI - 3D geometry (Russia) (>>)
$\bullet$ Ukraine Correspodence MO (outside aops)
$\bullet$ Ukrainian From Tasks to Tasks (outside aops)
$\bullet$ Ukrainian Tournament of Young Mathematicians Qualifying (outside aops)
$\bullet$ Ukrainian Tournament of Young Mathematicians Finals (>>)
$\bullet$ UNSW School Mathematics Competition Juniors (Australia) (outside aops)
$\bullet$ UNSW School Mathematics Competition Seniors (Australia) (>>)
$\bullet$ UQ/QAMT Problem Solving Competition (Australia) (outside aops)
$\bullet$ VU MIF Olympiad - geometry (Lithuania) (outside aops)
$\bullet$ Youth Mathematical School Olympiad ordinary (Russia) (outside aops)
$\bullet$ Youth Mathematical School Olympiad research (Russia) (>>)
$\bullet$ Zhautykov City Olympiad - geometry (Kazakstan) (outside aops)


$\bullet \bullet$ geo USA (geometry from:)
$\bullet$ BMT / Geometry Rounds
$\bullet$ BMT / Geometry problems from non Geometry Rounds
$\bullet$ CMIMC / Geometry Rounds
$\bullet$ CMIMC / Geometry problems from non Geometry Rounds
$\bullet$ HMMT / Geometry Rounds 2014-21
$\bullet$ PUMaC / Geometry Rounds 2006-14
$\bullet$ PUMaC / Geometry Rounds 2005-20
$\bullet$ PUMaC / Geometry problems from non Geometry Round


$\bullet \bullet$ geo olympiads
$\bullet$ Adygea Teachers' Geometry Olympiad (outside aops)
$\bullet$ Cabri Clubs (outside aops)
$\bullet$ Iranian Geometry Olympiad (outside aops)
$\bullet$ Mexican Geometry Olympiad (outside aops)
$\bullet$ Novosibirsk Oral Olympiad in Geometry (outside aops)
$\bullet$ Oral Moscow Geometry Olympiad (outside aops)
$\bullet$ Sharygin Geometry Olympiad (outside aops)
$\bullet$ Yaskinsky Geometry Olympiad (outside aops)
$\bullet$ Ukrainian Geometry Olympiad (outside aops)
$\bullet \bullet$ geometry olympiads not yet in contest collections


$\bullet \bullet$ geo shortlists (not IMO, BMO, JBMO, ELMO)
$\bullet$ Baltic Way Shortlist 2009-16 (outside aops)
$\bullet$ Baltic Way Shortlist 2017+ (outside aops)
$\bullet$ CentroAmerican Shortlist (outside aops)
$\bullet$ Cono Sur Olympiad Shortlist (outside aops)
$\bullet$ IberoAmerican Shortlist 1992 - 2010 (outside aops)
$\bullet$ IberoAmerican Shortlist 2011+ (>>)
$\bullet$ Indonesia Shortlist (INAMO OSN) (outside aops)
$\bullet$ IMOC Shortlist (Taiwan) (outside aops)
$\bullet$ Real IMO Shortlist (International Monster's Olympiad) (outside aops)
$\bullet$ Romanian Master in Mathematics Shortlist (outside aops)
$\bullet$ Thailand MO Shortlist (outside aops)


$\bullet \bullet$ geo IMO Longlists (outside aops)
$\bullet$ IMO Longlist I 1966-70 Geometry
$\bullet$ IMO Longlist II 1971-76 Geometry
$\bullet$ IMO Longlist III 1977-82 Geometry
$\bullet$ IMO Longlist IV 1983-85 Geometry
$\bullet$ IMO Longlist V 1986-88 Geometry
$\bullet$ IMO Longlist VI 1989-82 Geometry


$\bullet \bullet$ geo magazines
$\bullet$ Mathematical Excalibur - geometry (outside aops)
$\bullet$ Mathematical Reflections Junior - geometry (outside aops)
$\bullet$ Mathematical Reflections Senior - geometry (>>)
$\bullet$ Mathematical Reflections Olympiad - geometry (>>)
$\bullet$ Mathematical Reflections Undergraduate - geometry (>>)
$\bullet$ Quantum English - geometry (outside aops)
$\bullet$ Romanian Mathematical Magazine (outside aops)
$\bullet$ U sviti matematyky , part 1- geometry (In the World of Mathematics) (outside aops)
$\bullet$ U sviti matematyky , part 2 - geometry (In the World of Mathematics) (>>)


Akopyan's Geometry in Figures (random problems posted in aops)

$\bullet \bullet$ Greek Aops Index
$\bullet$ Greek Aops Index - Άλγεβρα
$\bullet$ Greek Aops Index - Ανισότητες
$\bullet$ Greek Aops Index - Γεωμετρία
$\bullet$ Greek Aops Index - Θεωρία Αριθμών
$\bullet$ Greek Aops Index - Συνδυαστική

PS.1. The links outside aops link to my blogspot page, in each competition's tab, imogeometry.blogspot.com.

PS.2. The geometry problems I have solved inside aops are collected here.
57 replies
parmenides51
Mar 7, 2020
parmenides51
Oct 12, 2024
J
H
i Forum's Purpose - Index of posted Contest Collections
parmenides51   29
N Mar 30, 2023 by huashiliao2020
I created this forum in order to post High School Math Olympiads before 2001 (not present in aops at the moment) in order to add more Contests to the existing Aops Contest Collections (without spamming the existing forums by posting too many posts in so little time). I also post recent problems not available at the moment at contest collections, in oder to create those post collections.
Anyone is welcome to post problems from a national competition (final round or TST) of his / her country not already posted in Aops Contest Collections, to get those links (just not post random problem but all from each year's competition). There is no limit to how many problems anyone may post in this forum, as it serves as a new problem collection source for more (older usually) posts collection (e.g. anyone may post in a day 40 different problems from 10 years of his \ her 4 problem's olympiad in 40 different posts - one problem at each post).

Forum Contests Collected:
- Mathcenter Contest (Thai)
- Mathlinks Contests (old name of aops)
- OIFMAT (FMAT - Chilean)
- Oliforum Contest (Italian)
- VMEO (VMF - Vietnamese)


Below are the contest collections I have created using this forum :
- Arab Math Olympiad
- Argentina NMO Finals: 2008
- Argentina NMO Round 4 / Regional Geo, L1, L2, L3, years 1995-2022
- Argentina Provincional Geo: L1, L2, L3 , 1995-2022
- Argentina TST / geometry : Ibero , IMO, Cono Sur
- Austrian - Polish Competition : 1979-2000
- Austria Beginners' Competition: 2001-08
- Belarus National MO: 1995, 1997 missing problems , 1998 , 2001, 2002, 2003,2006 , 2014
- Belarus TST : 1998 , 2000 , 2009 , 2010, 2011, 2012, 2014, 2015, 2016
- Bosnia and Herzegovina JBMO TST: 2003-07
- Brazil TSTs: EGMO, Geometry : Cono Sur, Cono Sur Training List , ΙΜΟ + Ibero ,
- Brazil IMO TST: 2007-16
- Bulgaria National MO: 1993-96
- Chile Contests / Chile NMO 1989 -2015
- Czech and Slovak Olympiad III A: so far 1986, 1991- 2001
- Czech-Polish-Slovak Junior Match: all
- Croatia National MO: 1996-2004
- Croatia TST: so far 2001, 2002, 2003
- Estonia Open Junior: 2001-2005, 2006, 2007, 2008-2019
- Estonia Open Senior: 2001-2005, 2006, 2007, 2008-2019
- Estonia National MO: so far 1996-2007, (soon 2008-2021)
- Estonia TST: 1992-95, 1998-2000, 2001-19
- French National MO: 1986, 1988-2003
- Final Mathematical Cup
- Germany / Bundeswettbewerb Mathematik : 1974-1982 , 1990
- Germany / National MO: so far 1965, 1996- 2001, 2009
- Germany / QUEDMO (11 /15 editions so far)
- Greece Junior: 1996 - 2020
- Hungary / Kürschák: 1947-84
- Hungary / Dürer / Finals: 2019-21
- India / LIMIT : 2020
- India / Postal Coaching: 2007-09 , 2012
- Israel / National (Gillis) : 1995-98
- Israel / Grosman: so far 1995, 1997, 1999, 2001
- Italy TST: so far 1993-94, 1998
- Kyrgystan TST: 2010, 2018
- Lithuania / Grand Duchy of Lithuania 2009-20
- Macedonia North National MO: so far 1994, 1995, 1996, 1997, 1998, 1999, 2002, 2003, 2004, 2005
- May Olympiad all
- Moldova Contests /JBMO TST: 1999- 2020
- Moldova National MO
- Mongolia National MO: 1999-2002, 2007
- Netherlands / Dutch ΙΜΟ TST: 2021
- New Zealand / NZMOC Camp Selection Problems, New Zealand MO
- Norway / Abels Math Contest (Final Round): 1993- 2006
- Polish MO Finals: 1966-84
- Rioplatense L3 1990-99
- Romania MOP: ROMOP 2011 Juniors
- Romania JBMO TST : 2002-19
- Romania TST: so far 1989, 1990 , 1991 (missing), 1992, 1993, 1995
- Saudi Arabia / BMO TST: 2010-11, 2013, 2015-19
/ GMO TST: 2013-16, 2018
/ IMO TST: 2011.2013, 2015-19
/ Pre-TST + Training Tests: 2010-11, 2021
- Saudi Arabia JBMO TST: 2017, 2019
- Serbia National MO: 1974 - 2006 (also known as Yugoslavia before 1990)
- Serbia TST:
- Singapore Junior : 2006-2019 collected
- Singapore Open: 1995 - 2004, 2014, 2016-17
- Singapore Senior : 1996 - 2008, 2011-13, 2015-19,
- Singapore TST: 1995 -2003
- Slovenia National MO: 1997-2001, 2005
- Slovenia TST: so far 1997, 1998, 2005
- Soros Olympiad
- Spanish MO: 1966-83
- Sweden National MO: 1961 - 2020
- Switzerland TST : so far 1997 - 2004, 2015 .
- Thailand MO: 2004- 15
- Ukraine National MO: 1997, 1998, 1999, 2005
- Ukraine TST: 1999, 2009-18
- USA / iΤest: 2005

Basic Source: Imomath

PS. My posting ability in public forums is as everyone’s else limited. Therefore when I post in public forums, I post only the geometry problems. When my source is written not in English, only the geometry ones are posted as I spend time translating them by google translate. When my source is written in English, I post the whole problem set in aops, posting the non-geometry problems in this forum and adding all the problems in contest collections.
In this forum’s index, anyone may find which contests’ years have been posted inside this forum.
In the future, each year's post collections shall be available in the Contests Collections, till then anyone may find them above.

Related forums:
Old Russian Math Olympiads
China High School Contests
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PS.2 Test
29 replies
parmenides51
Feb 11, 2020
huashiliao2020
Mar 30, 2023
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n^2 + 24n + 35 is a perfect square
parmenides51   5
N Mar 31, 2025 by lightsynth123
Source: Singapore Junior Math Olympiad 2016 2nd Round p1 SMO
Find all integers$ n$ such that $n^2 + 24n + 35$ is a square.
5 replies
parmenides51
Mar 26, 2020
lightsynth123
Mar 31, 2025
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integer p(x), p (1) = p (2) = 0, p (n) is a prime ,
parmenides51   2
N Mar 28, 2025 by AshAuktober
Source: 2010 Swedish Mathematical Competition p3
Find all natural numbers $n \ge 1$ such that there is a polynomial $p(x)$ with integer coefficients for which $p (1) = p (2) = 0$ and where $p (n)$ is a prime number .
2 replies
parmenides51
Apr 30, 2021
AshAuktober
Mar 28, 2025
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no of words of the maximum length in a language with n letters
parmenides51   2
N Mar 28, 2025 by NicoN9
Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 2001 p4 / Croatia TST 2002 p1
In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.
2 replies
parmenides51
Feb 11, 2020
NicoN9
Mar 28, 2025
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f(xy) f( f(y)/x) = 1 , monotonic
parmenides51   8
N Mar 28, 2025 by navier3072
Source: 1990 Swedish Mathematical Competition p5
Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$.
8 replies
parmenides51
Apr 2, 2021
navier3072
Mar 28, 2025
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5^p + 4 p^4 is a perfect square
parmenides51   7
N Mar 27, 2025 by Chanome
Source: Singapore Math Olympiad 2008 2nd Round SMO, Senior p2, Junior p5
Determine all primes $p$ such that $5^p + 4 p^4$ is a perfect square, i.e., the square of an integer.
7 replies
parmenides51
Mar 25, 2020
Chanome
Mar 27, 2025
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largest multiple of 55 using once digits 0,1,2,3,4,5,6
parmenides51   1
N Mar 27, 2025 by lightsynth123
Source: Singapore Junior Math Olympiad 2014 2nd Round p1 SMO
Consider the integers formed using the digits $0,1,2,3,4,5,6$, without repetition. Find the largest multiple of $55$. Justify your answer.
1 reply
parmenides51
Mar 26, 2020
lightsynth123
Mar 27, 2025
J
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f(f(x)) +f(x) = 12x , for all x>=0
parmenides51   2
N Mar 27, 2025 by jasperE3
Source: 2002 Singapore TST 2.3
Find all functions $f : [0,\infty) \to [0,\infty)$ such that $f(f(x)) +f(x) = 12x$, for all $x \ge 0$.
2 replies
parmenides51
Dec 29, 2020
jasperE3
Mar 27, 2025
J
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a\sqrt{b}=a+c, b\sqrt{c}=b+a, c\sqrt{a}=c+b , system
parmenides51   2
N Mar 26, 2025 by lightsynth123
Source: Singapore Junior Math Olympiad 2014 2nd Round p4 SMO
Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations:
$$\begin{cases} a\sqrt{b}=a+c \\ b\sqrt{c}=b+a \\ c\sqrt{a}=c+b \end{cases}$$
2 replies
parmenides51
Mar 26, 2020
lightsynth123
Mar 26, 2025
J
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square is cut into rectangles with // sides, sum of ratio of sides >=1
parmenides51   1
N Mar 26, 2025 by lightsynth123
Source: Singapore Junior Math Olympiad 2017 2nd Round p1 SMO
A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.
1 reply
parmenides51
Mar 26, 2020
lightsynth123
Mar 26, 2025
J
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f(f(a)) = a but not f(a) = a if f(x) = x^2 - 2x
parmenides51   2
N Mar 25, 2025 by jasperE3
Source: 2011 Denmark MO - Mohr Contest p4
A function $f$ is given by $f(x) = x^2 - 2x$ .
Prove that there exists a number a which satisfies $f(f(a)) = a$ without satisfying $f(a) = a$ .
2 replies
parmenides51
Jan 12, 2023
jasperE3
Mar 25, 2025
J
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J
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6 triangles with equal areas
parmenides51   2
N Feb 25, 2022 by RavenKing34
Source: Singapore Senior Math Olympiad 2000 2nd Round p1 SMO
In $\vartriangle ABC$, the points $D, E$ and $F$ lie on $AB, BC$ and $CA$ respectively. The line segments $AE, BF$ and $CD$ meet at the point $G$. Suppose that the area of each of $\vartriangle BGD, \vartriangle ECG$ and $\vartriangle GFA$ is $1$ cm$^2$. Prove that the area of each of $\vartriangle BEG, \vartriangle GCF$ and $\vartriangle ADG$ is also $1$ cm$^2$.
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2 replies
parmenides51
Apr 4, 2020
RavenKing34
Feb 25, 2022
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6 triangles with equal areas
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Reveal topic tags
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equal areas
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Source: Singapore Senior Math Olympiad 2000 2nd Round p1 SMO
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parmenides51
30629 posts
parmenides51
#1 h  Apr 4, 2020, 5:35 PM
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In $\vartriangle ABC$, the points $D, E$ and $F$ lie on $AB, BC$ and $CA$ respectively. The line segments $AE, BF$ and $CD$ meet at the point $G$. Suppose that the area of each of $\vartriangle BGD, \vartriangle ECG$ and $\vartriangle GFA$ is $1$ cm$^2$. Prove that the area of each of $\vartriangle BEG, \vartriangle GCF$ and $\vartriangle ADG$ is also $1$ cm$^2$.
https://cdn.artofproblemsolving.com/attachments/e/7/ec090135bd2e47a9681d767bb984797d87218c.png
This post has been edited 1 time. Last edited by parmenides51, Apr 4, 2020, 5:36 PM
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parmenides51
30629 posts
parmenides51
#2 h  Apr 4, 2020, 5:36 PM
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posted for the image link
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RavenKing34
6 posts
RavenKing34
#3 h  Feb 25, 2022, 6:03 AM
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Let areas of ∆ADG be x, ∆BGE be y and ∆CGF be z. By Ceva's theorem, we get x*y*z=1---(1).

[..] Denotes area here
Now [ABG]/[AGF]=BG/GF. So x+1/1=AG/GF. Also [BGC]/[FCG]=y+1/z=AG/GF. Therefore we can say that x+1/1=y+1/z.

So y+1=z(x+1)
Similarly x+1=y(z+1) & z+1=x(y+1)
Adding these 3 equations we get
x + y + z + 3 = xz + z + yz + y + xy + x
=> xz + yz + xy = 3, recalling that we got x*y*z=1 above, substitute xy=1/z, yz=1/x and xz=1/y.

Hence we get 3 = 1/x + 1/y + 1/z--(2)
Take lcm and then we get 3=x+y+z/xyz, again from eq. (1), we get xyz as 1
So x + y + z =3 --(3)

Now based on what we get from equation (2) and (3), we can say that this is only possible if x=y=z=1
So area of remaining triangles is 1 respectively and that's what we needed prove.
This post has been edited 1 time. Last edited by RavenKing34, Feb 25, 2022, 6:03 AM
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