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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 March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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0 replies
jlacosta
Mar 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
KöMaL (Hungarian Magazine)

For the friends of geometry:
Olympiad Geometry Collections + Forums
A Beautiful Journey Through Olympiad Geometry - solutions forum
Evan Chen's EGMO study group
Lemmas in Olympiad Geometry - active forum
Tran Quang Hung's geometry group

uc = Under Construction

PS. It is not permitted to create contest collections of each year inside aops from National Contests of UK and Australia.

PS.2 Test
29 replies
parmenides51
Feb 11, 2020
huashiliao2020
Mar 30, 2023
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7^p - p - 16 is a perfect square.
parmenides51   2
N Today at 5:39 AM by MuradSafarli
Source: 2018 Romania JBMO TST 4.1
Determine the prime numbers $p$ for which the number $a = 7^p - p - 16$ is a perfect square.

Lucian Petrescu
2 replies
parmenides51
Jun 19, 2020
MuradSafarli
Today at 5:39 AM
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5 (a^2 + b^2 + c^2) = 6 (ab + bc + ca) if centroid lies on incircle
parmenides51   3
N Mar 8, 2025 by Marcus_Zhang
Source: 2001 Argentina IberoAmerican TST p5
Given a triangle with sides $a, b, c$ such that the centroid of the triangle lies on the circle inscribed in the triangle. Prove that $$5 (a^2 + b^2 + c^2) = 6 (ab + bc + ca)$$
3 replies
parmenides51
Jul 20, 2021
Marcus_Zhang
Mar 8, 2025
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1/x_1 + 1/x_2 +... + 1/x_n = 1997/1998
parmenides51   1
N Mar 8, 2025 by grupyorum
Source: 1998 Swedish Mathematical Competition p5
Show that for any $n > 5$ we can find positive integers $x_1, x_2, ... , x_n$ such that $\frac{1}{x_1} + \frac{1}{x_2} +... + \frac{1}{x_n} = \frac{1997}{1998}$. Show that in any such equation there must be two of the $n$ numbers with a common divisor ($> 1$).
1 reply
parmenides51
Apr 2, 2021
grupyorum
Mar 8, 2025
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c >= (a+b) sin(C/2)
parmenides51   2
N Mar 8, 2025 by grupyorum
Source: 1998 Swedish Mathematical Competition p2
$ABC$ is a triangle. Show that $c \ge (a+b) \sin \frac{C}{2}$
2 replies
parmenides51
Apr 2, 2021
grupyorum
Mar 8, 2025
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concyclic wanted, OD _|_ BC, intersecting circles related
parmenides51   2
N Mar 8, 2025 by parmenides51
Source: Singapore Open Math Olympiad 2004 2nd Round p3 SMO
Let $AD$ be the common chord of two circles $\Gamma_1$ and $\Gamma_2$. A line through $D$ intersects $\Gamma_1$ at $B$ and $\Gamma_2$ at $C$. Let $E$ be a point on the segment $AD$, different from $A$ and $D$. The line $CE$ intersect $\Gamma_1$ at $P$ and $Q$. The line $BE$ intersects $\Gamma_2$ at $M$ and $N$.
(i) Prove that $P,Q,M,N$ lie on the circumference of a circle $\Gamma_3$.
(ii) If the centre of $\Gamma_3$ is $O$, prove that $OD$ is perpendicular to $BC$.
2 replies
parmenides51
Apr 2, 2020
parmenides51
Mar 8, 2025
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necklace of marbles of n inhabitants of an island, friends of enemies
parmenides51   1
N Mar 8, 2025 by Bata325
Source: Norwegian Mathematical Olympiad 2004 - Abel Competition p4
Among the $n$ inhabitants of an island, where $n$ is even, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that two necklaces have a marble of the same type if and only if their owners are friends.
(a) Show that the chief’s order can be achieved by using $n^2/4$ different types of stones.
(b) Prove that this is not necessarily true with less than $n^2/4$ types.
1 reply
parmenides51
Feb 11, 2020
Bata325
Mar 8, 2025
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diophantine (x^2 + y^2)^n = (xy)^{2016} with no solutions
parmenides51   2
N Mar 6, 2025 by John_Mgr
Source: New Zealand NZMOC Camp Selection Problems 2016 p7
Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$has positive integer solutions.
2 replies
parmenides51
Sep 19, 2021
John_Mgr
Mar 6, 2025
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$a_1=n^2-10n+23, a_1=n^2-9n+31, a_1=n^2-12n+46.$
augustin_p   4
N Mar 5, 2025 by magnusarg
Source: Albania JTST 2015
For every positive integer $n{}$, consider the numbers $a_1=n^2-10n+23, a_2=n^2-9n+31, a_3=n^2-12n+46.$
a) Prove that $a_1+a_2+a_3$ is even.
b) Find all positive integers $n$ for which $a_1, a_2$ and $a_3$ are primes.
4 replies
augustin_p
Oct 4, 2023
magnusarg
Mar 5, 2025
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p divides F_{2p} - F_p
parmenides51   1
N Mar 3, 2025 by shanelin-sigma
Source: 2011 Saudi Arabia BMO TST 3.4 - Balkan MO
Let $(F_n )_{n\ge o}$ be the sequence of Fibonacci numbers: $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ , for every $n \ge 0$.
Prove that for any prime $p \ge 3$, $p$ divides $F_{2p} - F_p$ .
1 reply
parmenides51
Dec 29, 2021
shanelin-sigma
Mar 3, 2025
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x^[x]=9/2
parmenides51   2
N Mar 2, 2025 by parmenides51
Source: 2002 Austria Beginners' Competition p2
Prove that there are no $x\in\mathbb{R}^+$ such that $$x^{\lfloor x \rfloor }=\frac92.$$
2 replies
parmenides51
Oct 4, 2022
parmenides51
Mar 2, 2025
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1= d_1 <d_2 < ... <d_k = n , and d_2d_3 + d_3d_5 + d_5d_2 = n.
parmenides51   2
N Dec 19, 2022 by charme7269
Source: Estonia IMO TST 2018 p8
Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors
$1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$.
2 replies
parmenides51
Jul 23, 2020
charme7269
Dec 19, 2022
J
1= d_1 <d_2 < ... <d_k = n , and d_2d_3 + d_3d_5 + d_5d_2 = n.
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number theory
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Source: Estonia IMO TST 2018 p8
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parmenides51
30627 posts
parmenides51
#1 h  Jul 23, 2020, 11:22 AM • 1 Y
Y by Mango247
Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors
$1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$.
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ILOVEMYFAMILY
641 posts
ILOVEMYFAMILY
#2 h  Feb 8, 2022, 7:45 AM
Y by
bummp bump
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charme7269
7 posts
charme7269
#3 h  Dec 19, 2022, 9:21 PM
Y by
Let $p$ be the smallest prime divisor of $n$. From the given equation, since $d_2 \mid n$, $d_3 \mid n$ and $d_5 \mid n$, it follows that $d_2 \mid d_3d_5$, $d_3 \mid d_2d_5$ and $d_5 \mid d_2d_3$. In particular, $d_2 = p \mid d_3d_5$.

Case 1: $p \mid d_3$. Notice that $d_3 = pq$ for $q \neq p$ and $q$ prime immediately yields a contradiction. So, $d_3 = p^2$. Moreover, we have that $d_5 \mid d_2d_3 = p^3$, so $d_5 = p^3$. Thus, $n = d_2d_3+d_3d_5+d_5d_2 = p^3(p^2+p+1)$.
Now since for any prime $p$ the inequality $p^2 < p^2+p+1 < p^3$ clearly holds and $p^2+p+1 \mid n$, $d_4$ must be $p^2+p+1$.
We have found two divisors $d_4$ and $d_5$ such that $d_4d_5 = n$. From here, the list of divisors of $n$ (in ascending order) is
$$ 1, p, p^2, p^2+p+1, p^3, p(p^2+p+1), p^2(p^2+p+1), p^3(p^2+p+1) $$It can be readily seen that $p^2+p+1$ must be prime and $n = p^3(p^2+p+1)$ with $p$ and $p^2+p+1$ prime satisfies the conditions.

Case 2: $p \mid d_5$. Observe that we get a contradiction if $d_5$ and $d_2$ differ by at least three prime factors. If they differ by two prime factors, we again get a contradiction (just check manually the possible subcases). Since $p \mid d_5$, $d_5 = p^2$ or $d_5 = pq$ for some prime $q \neq p$. If $d_5 = p^2$, $d_3 \mid d_5d_2 = p^3$, so $d_3 = p$ or $d_3 = p^2$, contradiction. Therefore, $d_5 = pq$ for a prime $q \neq p$. Now, $d_3 \mid d_5d_2 = p^2q$ and $d_5 = pq \mid pd_3 = d_2d_3$, $q \mid d_3$ and $d_3 = q$. Then, $n = d_2d_3+d_2d_5+d_3d_5 = pq(p+q+1)$, so $d_4 = p+q+1$ unless $(p, q)$ is $(2, 3)$. Note, however, that for $(p, q) = (2, 3)$, $n = 36$ satisfies the condition. For all other prime pairs $(p, q)$, $p+q+1 < pq$. Therefore, as we did in Case 1, there are exactly eight divisors of n so $p+q+1$ must be prime. Hence, similar to the above case, $n = pq(p+q+1)$ with $p$, $q$ and $p+q+1$ prime satisfies the conditions.

In conclusion, all numbers satisfying our condition are $n = 36$, $n = p^3(p^2+p+1)$ ($p$ and $p^2+p+1$ are prime) and $n = pq(p+q+1)$ ($p$, $q$ and $p+q+1$ are prime). Therefore, all possible values of $k$ are $8$, $9$.
This post has been edited 2 times. Last edited by charme7269, Dec 19, 2022, 10:00 PM
Reason: Proofreading solution
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