AoPS Academy
Forums
Geometry Mathley Old High School Olympiads Old Russian Math Olympiads Geometry Collections and Contests Aops Geo Mocks Aops Mock Contests - Geometry geo national (geometry from:) Argentina NMO (Round 2-4, Finals, Provincial, Regional Round 2) Armenia NMO (outside aops) Belarus NMO VIII (outside aops) Belarus NMO IX (>>) Belarus NMO X (>>) Belarus NMO XI (>>) Belgium Flanders Juniors JWO (outside aops) Belgium Flanders Seniors VWO (>>) Belgium OMB Juniors (outside aops) Belgium OMB Seniors (>>) Brazil NMO (Round 2 + Finals) Bulgaria NMO Plane Geometry (outside aops) Bulgaria NMO 3D Geometry (>>) Costa Rica NMO (OLCOMA) Croatia MO (outside aops) Chile NMO (outside aops) Denmark MO (Mohr Contest) (outside aops) Estonia Open Junior MO (outside aops) Estonia Open Senior MO (>>) Greece JMO (outside aops) Greece NMO (outside aops) Hungary OKTV I (outside aops) Hungary OKTV II (>>) Hungary OKTV III (>>) Iceland NMO (outside aops) Indonesia MO Province OSP SMA (outside aops) Kazakhstan NMO IX (outside aops) Kazakhstan NMO X (>>) Kazakhstan NMO XI (>>) Lithuania juniors (outside aops) Mexico NMO (States Semifinals + Finals, Finals) Moldova NMO VII-VIII (outside aops) Moldova NMO IX-XII Plane Geometry (>>) Moldova NMO 3D Geometry (>>) New Zealand MO (outside aops) Paraguay NMO (OMAPA) (outside aops) Peru NMO (ONEM) (Rounds 2-3, Finals) Poland Juniors , Round 1+2 (outside aops) Poland Juniors, Finals (>>) Portugal NMO (outside aops) Romania District 3D Geometry (outside aops) Romania District Plane Geometry (>>) Romania NMO 3D Geometry (outside aops)
Romania NMO Plane Geometry (>>) Swiss NMO (outside aops) Ukraine NMO VIII (outside aops) Ukraine NMO IX (>>) Ukraine NMO X-XI Plane Geometry (>>) Ukraine NMO 3D Geometry (>>) geo TST (geometry from:) Argentina Cono Sur TST (outside aops) Argentina Ibero TST (>>) Argentina IMO TST (>>) Argentina Training Lists Cono Sur (outside aops) Argentina Training Lists IberoAmerican (>>) Argentina Training Lists Rioplatense (>>) Bangladesh TST and Camp Belarus TST (outside aops) Brazil Training List Cono Sur 1999 - 2010 (outside aops) Brazil Training List Cono Sur 2011-20 (>>) Brazil Cono Sur TST (outside aops) Brazil IMO IberoAmerican TST (>>) Chile TST (outside aops) Croatia TST (outside aops) Cyprus TST (outside aops) Ecuador TST (outside aops) France TST (outside aops) Hong Kong TST (outside aops) Indonesia TST (outside aops) Italy TST (outside aops) Lithuania TST (outside aops) Peru Cono Sur TST (outside aops) Peru IberoAmerican TST (>>) Peru IMO TST (>>) Puerto Rico TST (outside aops) Serbia IMO TST (outside aops) Singapore TST (outside aops) Switzerland TST (outside aops) Taiwan TST (outside aops) Thailand TST (outside aops) Ukraine TST (outside aops) geo regional (geometry from:) 239 MO (St.Petersburg's Lyceum) (outside aops) Alberta High School Mathematics Competition (Canada) (outside aops) AESC MSU Internet Olympiad (Russia) (outside aops) All-Siberian Olympiad Finals (Russia) (outside aops) All-Siberian Olympiad Correspondence (Russia) (>>) All-Siberian Olympiad Qualifying (Russia) (>>) Almaty City MO (Kazakhstan) (outside aops) Association Mathematical Quebec Competition (Canada) (outside aops) Auckland MO (New Zealand) (outside aops) Champions Tournament (Ukraine) (Турнір чемпіонів) (outside aops) Clock-Tower School Junior (Romania) (outside aops) COMATEQ (Puerto Rico) (outside aops) Cordoba pre TSTs (Argentina) (outside aops) Dürer Math Competition (Hungary) (outside aops) El Número de Oro Certamen (Argentina) (outside aops) Euler Olympiad Remote (Russia) (outside aops) Euler Olympiad Regional & Final (Russia) (>>) Euler Teachers' MO (Russia) (outside aops) European Math Tournament (Belarus) (outside aops) Formula of Unity / Third Millennium Olympiad (Russia) (outside aops) HSGS Class 9 Contest (Vietnam) (outside aops) High Standards Olympiad (Russia) (Высшая проба) (outside aops) Innopolis University Open MO (Russia) (outside aops) Izumrud Olympiad Emerald by UrFU (Russia) (outside aops) Kharkiv City MO (Ukraine) (outside aops) Kharkiv Lyceum No 27 Olympiad (Ukraine) (outside aops) Kharkiv Masters Tournament (Ukraine) (outside aops) Kolomogorov Cup / Tournament (Russia) Kukin MO by OmSU (Russia) (outside aops) Kurchatov Olympiad (Russia) (outside aops) Kyiv City MO 1984-93 (Ukraine) (outside aops) Kyiv City MO 2003+ Round 1 (Ukraine) (>>) Kyiv City MO 2010+ Round 2 (Ukraine) (>>) Kyiv TST 2005-16 (for Ukraine MO) (outside aops) Kyiv TST 2017+ (for Ukraine MO) (>>) League of Winners Tournament (Russia) (outside aops) Lomonosov Tournament (Russia) (outside aops) Mathematical Multiathlon Tournament Juniors (Russia) (outside aops) Mathematical Multiathlon Tournament Seniors (Russia) (>>) Maths Beyond Limits Camp (Poland) (outside aops) May Olympiad, level 1 (Latin America) (outside aops) May Olympiad, level 2 (Latin America) (>>) Minsk City Internet Olympiad (Junior) (Belarus) (outside aops) MIPT Metropolitan Olympiad (Russia) (outside aops) Moscow Corespondence Competition 6-8 MO (Russia) (outside aops) Moscow Teachers' MO (Russia) (outside aops) OMAFOROS - OFO,FOFO, COFFEE (Argentina) (outside aops) Oral Moscow City MO VI-VII (Russia) (outside aops) Oral Moscow City Team MO 1999 - 2004 (Russia) (outside aops) Oral Moscow City Team MO 2005-19 (Russia) (>>) Oral Moscow City Team MO 2020+ (Russia) (>>) Rio de Janeiro MO (OMERJ) (Brazil) (outside aops) Rioplatense Olympiad, level 1 (Latin America) (outside aops) Rioplatense Olympiad, level 2 (Latin America) (>>) Rioplatense Olympiad, level 3 (Latin America) (>>) Rusanovsky Lyceum, Kyiv Olympiad(Ukraine) (outside aops) Savin Competition 1990 - 2015 (Russia) (outside aops) Savin Competition 2016+ (Russia) (>>) Savin Tournament (Russia) (>>) SPBU School Olympiad besides Finals X-XI (Russia) (outside aops) SPBU School Olympiad, Finals X-XI - plane geometry (Russia) (>>) SPBU School Olympiad, Finals X-XI - 3D geometry (Russia) (>>) Ukraine Correspodence MO (outside aops) Ukrainian From Tasks to Tasks (outside aops) Ukrainian Tournament of Young Mathematicians Qualifying (outside aops) Ukrainian Tournament of Young Mathematicians Finals (>>) UNSW School Mathematics Competition Juniors (Australia) (outside aops) UNSW School Mathematics Competition Seniors (Australia) (>>) UQ/QAMT Problem Solving Competition (Australia) (outside aops) VU MIF Olympiad - geometry (Lithuania) (outside aops) Youth Mathematical School Olympiad ordinary (Russia) (outside aops) Youth Mathematical School Olympiad research (Russia) (>>) Zhautykov City Olympiad - geometry (Kazakstan) (outside aops) geo USA (geometry from:) BMT / Geometry Rounds BMT / Geometry problems from non Geometry Rounds CMIMC / Geometry Rounds CMIMC / Geometry problems from non Geometry Rounds HMMT / Geometry Rounds 2014-21 PUMaC / Geometry Rounds 2006-14 PUMaC / Geometry Rounds 2005-20 PUMaC / Geometry problems from non Geometry Round geo olympiads Adygea Teachers' Geometry Olympiad (outside aops) Cabri Clubs (outside aops) Iranian Geometry Olympiad (outside aops) Mexican Geometry Olympiad (outside aops) Novosibirsk Oral Olympiad in Geometry (outside aops) Oral Moscow Geometry Olympiad (outside aops) Sharygin Geometry Olympiad (outside aops) Yaskinsky Geometry Olympiad (outside aops) Ukrainian Geometry Olympiad (outside aops) geometry olympiads not yet in contest collections geo shortlists (not IMO, BMO, JBMO, ELMO) Baltic Way Shortlist 2009-16 (outside aops) Baltic Way Shortlist 2017+ (outside aops) CentroAmerican Shortlist (outside aops) Cono Sur Olympiad Shortlist (outside aops) IberoAmerican Shortlist 1992 - 2010 (outside aops) IberoAmerican Shortlist 2011+ (>>) Indonesia Shortlist (INAMO OSN) (outside aops) IMOC Shortlist (Taiwan) (outside aops) Real IMO Shortlist (International Monster's Olympiad) (outside aops) Romanian Master in Mathematics Shortlist (outside aops) Thailand MO Shortlist (outside aops) geo IMO Longlists (outside aops) IMO Longlist I 1966-70 Geometry IMO Longlist II 1971-76 Geometry IMO Longlist III 1977-82 Geometry IMO Longlist IV 1983-85 Geometry IMO Longlist V 1986-88 Geometry IMO Longlist VI 1989-82 Geometry geo magazines Mathematical Excalibur - geometry (outside aops) Mathematical Reflections Junior - geometry (outside aops) Mathematical Reflections Senior - geometry (>>) Mathematical Reflections Olympiad - geometry (>>) Mathematical Reflections Undergraduate - geometry (>>) Quantum English - geometry (outside aops) Romanian Mathematical Magazine (outside aops) U sviti matematyky , part 1- geometry (In the World of Mathematics) (outside aops) U sviti matematyky , part 2 - geometry (In the World of Mathematics) (>>) Greek Aops Index Greek Aops Index - Άλγεβρα Greek Aops Index - Ανισότητες Greek Aops Index - Γεωμετρία Greek Aops Index - Θεωρία Αριθμών Greek Aops Index - Συνδυαστική
satisfying the equality 
for every 
.
such that 
is divisible by 
. Justify your answer.
, where 
.
be a positive integer and 
be the remainder when 
is divided by . for which the sequence 
is eventually periodic.
such that 
to 
are to be put into two identical boxes so that each box contains at least one ball and the greatest common divisor of the product of the labels of all the balls in one box and the product of the labels of all the balls in the other box is . Determine the number of ways that this can be done.
with real coefficients such that for all real 
, 
.
and 
such that the equation 
at least one solution 
where and 
are positive integers.
is inscribed in a circle. Show that the line connecting the midpoints of the arcs 
and 
and the line connecting the midpoints of the arcs 
and 
are perpendicular.
positive integers into 
pairs is called square-free if the product of numbers in each pair is not a perfect square.Prove that if for distinct positive integers, there exists one square-free partition, then there exists at least 
square-free partitions.
teams (every team plays with every other team for a win or loss), there exist 
teams such that the first seven teams have each defeated the remaining teams.
teams (every team plays with every other team for a win or loss), there exist teams such that the first seven teams have each defeated the remaining teams.
