and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
,
are all greater than 
.
,
.
and 
to be the remainder when 
is divided by 
.
and 
is odd for 
.
or an equivalent formulation. To prove this construction works, we need to show that
is always an integer between 
and 
inclusive.
is divided by 
.
,
since it telescopes.
,![\[\frac{r_{i+1} n - r_i}{2^i} > \frac{r_{i+1} n}{2^i} - 1 \geq \frac{n}{2^i} - 1 \geq \frac{n}{2^k} - 1.\]](http://latex.artofproblemsolving.com/4/2/0/420d26b6099ca264846885e7eef23b2c9c7b2dd0.png)
Thus, each digit is lower bounded by 
,
becomes arbitrarily large.
fails.
.
be a polynomial of degree 
such that the polynomial 
divides 
is zero. Prove that 
,
such that 
has no nonreal roots. We can also assume 
case is trivial as the constant term must be nonzero, so fix 
.
.
.
which has the 
for all integers 
.
be the coefficient of the 
term of 
for all 
,
for which 
.
and 
.
and 
.
is the coefficient of the 
term in 
and 
.
,
.
,
)
has 
distinct real roots as well, along with two consecutive nonleading coefficients equal to zero, but one degree lower, by the Power Rule. By doing this repeatedly, we eventually end up with a polynomial where the two consecutive coefficients have degree 
,
and 
in the plane and a subset 
of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair 
of cities, they are connected with a road along the line segment 
if and only if the following condition holds:
distinct from 
and 
,
such[/center]
is directly similar to either 
or 
.
is directly similar to 
if there exists a sequence of rotations, translations, and dilations sending 
to 
,
to 
,
to 
.
to be the set of points strictly outside the circle with diameter 
.
is always acute. Suppose two roads 
and 
cross. Then, 
is a convex quadrilateral. Since 
.
is not acute but there does not exist an 
such that 
are connected by some path. This is trivially true for 
,
are also connected.
is obtuse, we have 
.
,
,
is not mentioned. Do not reward points if the induction argument is incomplete or incorrect.
be the orthocenter of acute triangle 
,
be the foot of the altitude from 
.
intersects line 
.
be the center of 
.
be the intersection of 
and the line through 
,
.
be the foot of the altitude from 
and 
,
is a midline and thus 
.
is a midline of 
,
and thus 
.
(by definition), by either HL congruence or the Pythagorean theorem, we must have 
.
.
and 
to attempt to identify 
.
or 
,
.
.![\[\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k\]](http://latex.artofproblemsolving.com/3/c/5/3c5984d0003c1ca2df2e68841dd9e93ed6c2d27c.png)
is an integer for every positive integer 
is divisible by 
,
be a prime power dividing 
.
.
,
if 
and 
otherwise, so for each term, either both the numerator are divisible by 
,
,
,
.
has an inverse modulo 
.![\[\binom ni \equiv \pm 1 \binom{\lfloor n/p \rfloor}{\lfloor i/p \rfloor} \pmod{p^a}.\]](http://latex.artofproblemsolving.com/0/1/c/01c00beb6efc07ed028fd0273f262b8bfb965d33.png)
modulo 
.
,![\[\sum_{i=0}^n \binom ni^k \equiv \sum_{i=0}^n (\pm 1)^k \equiv p \equiv 0 \pmod p.\]](http://latex.artofproblemsolving.com/2/e/c/2ececb6afa8a10e116056cdc57a6c948298d0c29.png)
where 
,
satisfies the induction hypothesis for the prime 
.![\[\sum_{i=0}^n \binom ni^k \equiv \sum_{i=0}^n (\pm 1)^k \binom{\lfloor n/p \rfloor}{\lfloor i/p \rfloor}^k \equiv p\sum_{i=0}^d \binom di^k \pmod {p^a}\]](http://latex.artofproblemsolving.com/6/c/9/6c9a4b6aac62b1befef1511451c9fa30993d2cf7.png)
By the induction hypothesis, 
divides the inner binomial sum, so since we are multiplying it by 
.
.
is incorrect, reward up to 
points total.
and 
.
between the set of people except Evan 
if the total score of the cupcakes in that bubble with respect to 
.
denotes the set of neighbors of 
people.
.
from the graph, noting that no person in 
then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
is the incentre and point 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
and 
are perpendicular.
.
-
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
.
meets the midpoint of 
.
,
.
.
.
as 
.
Y by greenturtle3141, AKS_9_54_61, fuzimiao2013, suvamkonar, Mango247, Mango247, Mango247
I just created and posted here a list of math olympiads abbreviations
that list is also here split into two categories National & International Contests
any suggestions are welcome
that list is also here split into two categories National & International Contests
• 239 MO = Open Math Olympiad of 239 St. Petersburg Lycuem (Russia)
• AIME = American Invitational Mathematics Examination
• AMC = American or Australian Mathematics Competitions
• AMO = Australian Mathematical Olympiad
• APMC = Austrian-Polish Mathematical Competition
• APMO = Asian Pacific Mathematics Olympiad
• ARMO = All - Russian Mathematics Olympiad (also ARO)
• ARO = All - Russian Mathematics Olympiad (also ARMO)
• ASU = All - Soviet Union Mathematical Olympiad
• BMO = Balkan or British Mathematical Olympiad (also BrMO)
• BrMO = British Mathematical Olympiad (also BMO)
• BW = Baltic Way Mathematical Team Contest
• BWM = Bundeswettbewerb Mathematik (Germany Federal Mathematical Competition )
• BxMO = Benelux Mathematical Olympiad
• CGMO = China Girls' Mathematical Olympiad
• CHKMO = China Hong Kong Mathematical Olympiad
• CMO = Canadian or Chinese Mathematical Olympiad
• CMOQR = Canadian Mathematical Olympiad Qualifying Repêchage
• CNMO= China Northern Mathematical Olympiad
• COMC = Canadian Open Math Challenge
• CPS = Czech-Polish-Slovak Match
• CPSJ = Czech-Polish-Slovak Junior Match
• CWMI = China Western Mathematical Invitation (also CWMO)
• CWMO = China Western Mathematical Invitation (also CWMI)
• DMO= Dutch Mathematical Olympiad
• EGMO = European Girls' Mathematical Olympiad
• ELMO = annual math Olympiad at MOP (USA)
• EMC = European Mathematical Cup
• EMIC = Elementary Mathematics International Contest
• FHSMC = Finnish National High School Mathematics Competition
• FKMO = Final Korean South Mathematical Olympiad
• FMC = Final Mathematical Cup
• GMO = Gulf Mathematical Olympiad
• HJMO = Croatia Junior Math Olympiad
• HMO = Croatia Mathematical Olympiad
• HOMC = Hanoi Open Mathematics Competitions (Vietnam)
• IFYM = International Festival of Young Mathematicians (Bulgaria)
• IGO = Iranian Geometry Olympiad
• ILL = International Longlist, used for IMO
• IMAC = International Mathematical Arhimede Contest (Romania)
• IMAR = IMAR Mathematical Competition (Romania)
• IMC = International Mathematics Competition for University Students or International Mathematics Competitions or Intermediate Mathematical Challenge
• IMEO = International Mathematical Excellence Olympiad
• IMO = International Mathematical Olympiad
• IMOR = International Olympic Revenge
• IMOTC = India TST (IMO Training Camp) (also INMOTC)
• IMTS = International Mathematical Talent Search
• INAMO = Indonesian National Science Olympiad
• INMO = Indian National Mathematical Olympiad
• INMOTC = India TST (IMO Training Camp) (also IMOTC)
• IOM = Metropolises International Olympiad
• IrMO = Irish Mathematical Olympiad
• ISL = International Shortlist, used for IMO
• ITAMO = Italian Mathematical Olympiad
• IWYMIC = International World Youth Mathematics Intercity Competition
• IZhO = Zhautykov Olympiad International
• JBMO= Junior Balkan Mathematical Olympiad
• JJMO = Japan Junior Math Olympiad
• JMC = Junior Mathematical Challenge
• JMO = Junior or Japan Mathematical Olympiad
• JTST = Junior Team Selection Test
• JWO = Flanders Juniors Mathematical Olympiad (Belgium)
• KJMO = Korean Junior Math Olympiads
• KMO = Korean South Mathematical Olympiad 2nd Round
• LMO = Leningrad Mathematical Olympiad (Russia)
• MEMO= Middle European Mathematical Olympiad
• MMC = Mediterranean Mathematical Competition
• MOCP = Mathematical Olympiads Correspondence Program (Canada)
• MODS = Mathematics Olympiads Discord Server
• MOP = Mathematical Olympiad Summer Program (also MOSP)
• MOSP = Mathematical Olympiad Summer Program (also MOP)
• MYMC = Mediterranean Youth Mathematical Championship
• NIMO = National Internet Mathematical Olympiad
• NMC = Nordic Mathematical Contest
• OBM = Brazil Mathematical Olympiad
• OFM = Francophone Mathematical Olympiad
• OIM = Iberoamerican Mathematical Olympiad
• OMA = Argentina Mathematical Olympiad
• OMB = Belgium National Mathematical Olympiad
• OMCC = Central American & Caribbean Mathematical Olympiad
• OMCPLP = Lusophon Mathematical Olympiad
• OMCS = Cono Sur Math Olympiad / Southern Cone
• OMM = Mexican Mathematical Olympiad
• OMO = Online Math Open
• OMR = Rioplatense Mathematical Olympiad
• ONEM = Peruvian Mathematical Olympiad
• OPM = Portugal Mathematical Olympiad
• PAMO = Pan African Mathematics Olympiad
• PMO = Philippines Mathematical Olympiad
• PRMO = Indian Pre-Regional Mathematical Olympiad
• RMM = Romanian Master of Mathematics
• RMO = Indian Regional Mathematical Olympiad or Romanian Mathematical Olympiad
• SAMO = South African Mathematics Olympiad
• SHL = Shortlist (also SL)
• SL = Shortlist (also SHL)
• SMC = Senior Mathematical Challenge
• SMO = Senior or Serbian or Singapore or Swiss Mathematical Olympiad
• SRMC = Silk Road Mathematics Competition
• TMO = Thailand Mathematical Olympiad
• ToT = Tournament of Towns International Mathematics (also TT)
• TST = Team Selection Test
• TSTST = Team Selection Test for the Selection Team
• TT= Tournament of Towns International Mathematics (also ToT)
• UKMOG = UK Mathematical Olympiad for Girls
• USAJMO = United States of America Junior Mathematical Olympiad
• USAMO = United States of America Mathematical Olympiad
• USAMTS = USA Mathematical Talent Search
• USEMO = US Ersatz Math Olympiad
• USMCA = US Math Competition Association
• VMO = Vietnamese Mathematical Olympiad
• VWO = Flanders Mathematical Olympiad (Belgium)
• WMTC = World Math Team Competition
and USA Contests• AIME = American Invitational Mathematics Examination
• AMC = American or Australian Mathematics Competitions
• AMO = Australian Mathematical Olympiad
• APMC = Austrian-Polish Mathematical Competition
• APMO = Asian Pacific Mathematics Olympiad
• ARMO = All - Russian Mathematics Olympiad (also ARO)
• ARO = All - Russian Mathematics Olympiad (also ARMO)
• ASU = All - Soviet Union Mathematical Olympiad
• BMO = Balkan or British Mathematical Olympiad (also BrMO)
• BrMO = British Mathematical Olympiad (also BMO)
• BW = Baltic Way Mathematical Team Contest
• BWM = Bundeswettbewerb Mathematik (Germany Federal Mathematical Competition )
• BxMO = Benelux Mathematical Olympiad
• CGMO = China Girls' Mathematical Olympiad
• CHKMO = China Hong Kong Mathematical Olympiad
• CMO = Canadian or Chinese Mathematical Olympiad
• CMOQR = Canadian Mathematical Olympiad Qualifying Repêchage
• CNMO= China Northern Mathematical Olympiad
• COMC = Canadian Open Math Challenge
• CPS = Czech-Polish-Slovak Match
• CPSJ = Czech-Polish-Slovak Junior Match
• CWMI = China Western Mathematical Invitation (also CWMO)
• CWMO = China Western Mathematical Invitation (also CWMI)
• DMO= Dutch Mathematical Olympiad
• EGMO = European Girls' Mathematical Olympiad
• ELMO = annual math Olympiad at MOP (USA)
• EMC = European Mathematical Cup
• EMIC = Elementary Mathematics International Contest
• FHSMC = Finnish National High School Mathematics Competition
• FKMO = Final Korean South Mathematical Olympiad
• FMC = Final Mathematical Cup
• GMO = Gulf Mathematical Olympiad
• HJMO = Croatia Junior Math Olympiad
• HMO = Croatia Mathematical Olympiad
• HOMC = Hanoi Open Mathematics Competitions (Vietnam)
• IFYM = International Festival of Young Mathematicians (Bulgaria)
• IGO = Iranian Geometry Olympiad
• ILL = International Longlist, used for IMO
• IMAC = International Mathematical Arhimede Contest (Romania)
• IMAR = IMAR Mathematical Competition (Romania)
• IMC = International Mathematics Competition for University Students or International Mathematics Competitions or Intermediate Mathematical Challenge
• IMEO = International Mathematical Excellence Olympiad
• IMO = International Mathematical Olympiad
• IMOR = International Olympic Revenge
• IMOTC = India TST (IMO Training Camp) (also INMOTC)
• IMTS = International Mathematical Talent Search
• INAMO = Indonesian National Science Olympiad
• INMO = Indian National Mathematical Olympiad
• INMOTC = India TST (IMO Training Camp) (also IMOTC)
• IOM = Metropolises International Olympiad
• IrMO = Irish Mathematical Olympiad
• ISL = International Shortlist, used for IMO
• ITAMO = Italian Mathematical Olympiad
• IWYMIC = International World Youth Mathematics Intercity Competition
• IZhO = Zhautykov Olympiad International
• JBMO= Junior Balkan Mathematical Olympiad
• JJMO = Japan Junior Math Olympiad
• JMC = Junior Mathematical Challenge
• JMO = Junior or Japan Mathematical Olympiad
• JTST = Junior Team Selection Test
• JWO = Flanders Juniors Mathematical Olympiad (Belgium)
• KJMO = Korean Junior Math Olympiads
• KMO = Korean South Mathematical Olympiad 2nd Round
• LMO = Leningrad Mathematical Olympiad (Russia)
• MEMO= Middle European Mathematical Olympiad
• MMC = Mediterranean Mathematical Competition
• MOCP = Mathematical Olympiads Correspondence Program (Canada)
• MODS = Mathematics Olympiads Discord Server
• MOP = Mathematical Olympiad Summer Program (also MOSP)
• MOSP = Mathematical Olympiad Summer Program (also MOP)
• MYMC = Mediterranean Youth Mathematical Championship
• NIMO = National Internet Mathematical Olympiad
• NMC = Nordic Mathematical Contest
• OBM = Brazil Mathematical Olympiad
• OFM = Francophone Mathematical Olympiad
• OIM = Iberoamerican Mathematical Olympiad
• OMA = Argentina Mathematical Olympiad
• OMB = Belgium National Mathematical Olympiad
• OMCC = Central American & Caribbean Mathematical Olympiad
• OMCPLP = Lusophon Mathematical Olympiad
• OMCS = Cono Sur Math Olympiad / Southern Cone
• OMM = Mexican Mathematical Olympiad
• OMO = Online Math Open
• OMR = Rioplatense Mathematical Olympiad
• ONEM = Peruvian Mathematical Olympiad
• OPM = Portugal Mathematical Olympiad
• PAMO = Pan African Mathematics Olympiad
• PMO = Philippines Mathematical Olympiad
• PRMO = Indian Pre-Regional Mathematical Olympiad
• RMM = Romanian Master of Mathematics
• RMO = Indian Regional Mathematical Olympiad or Romanian Mathematical Olympiad
• SAMO = South African Mathematics Olympiad
• SHL = Shortlist (also SL)
• SL = Shortlist (also SHL)
• SMC = Senior Mathematical Challenge
• SMO = Senior or Serbian or Singapore or Swiss Mathematical Olympiad
• SRMC = Silk Road Mathematics Competition
• TMO = Thailand Mathematical Olympiad
• ToT = Tournament of Towns International Mathematics (also TT)
• TST = Team Selection Test
• TSTST = Team Selection Test for the Selection Team
• TT= Tournament of Towns International Mathematics (also ToT)
• UKMOG = UK Mathematical Olympiad for Girls
• USAJMO = United States of America Junior Mathematical Olympiad
• USAMO = United States of America Mathematical Olympiad
• USAMTS = USA Mathematical Talent Search
• USEMO = US Ersatz Math Olympiad
• USMCA = US Math Competition Association
• VMO = Vietnamese Mathematical Olympiad
• VWO = Flanders Mathematical Olympiad (Belgium)
• WMTC = World Math Team Competition
• ABMC = Acton-Boxborough Math Competition
• AIME = American Invitational Mathematics Examination
• AMC = Australian Mathematics Competitions
• ARML = American Regions Mathematics League
• BAMO = Bay Area Mathematical Olympiad
• BMT = Berkley Math Tournament
• CCAMB = Canyon Crest Academy Math Bonanza
• CCMM = College of Charleston Math Meet
• CHMMC = Caltech Harvey Mudd Math Competition
• CMIMC = Carnegie Mellon Informatics and Mathematics Competition
• DMC = Deerfield Math Competition
• DMM = Duke Math Meet
• ELMO = annual math Olympiad at MOP
• EMCC = Exeter Math Club Competition
• HMIC = Harvard-MIT Mathematics Tournament Invitational Competition
• HMMT = Harvard-MIT Mathematics Tournament
• IMTS = International Mathematical Talent Search
• JHMMC = Joe Holbrook Memorial Math Competition
• JHMT = Johns Hopkins Math Tournament
• JMO = Junior Mathematical Olympiad
• LMT = Lexington Math Tournament
• MBMT = Montgomery Blair Math Tournament
• MCAMC = Middlesex County Academy Math Competition
• MMATHS = Math Majors of America Tournament for High Schools
• MOAA = Math Open at Andover by Phillips Academy
• MOEMS = Math Olympiads for Elementary and Middle Schools
• MOP = Mathematical Olympiad Summer Program (also MOSP)
• MOSP = Mathematical Olympiad Summer Program (also MOP)
• MPfG = Math Prize for Girls
• NIMO = National Internet Mathematical Olympiad
• OMO = Online Math Open
• PUMaC = Princeton University Mathematics Competition
• RCML = Rocket City Math League
• SBMT = South Bay Math Tournament
• SMT= Stanford Math Tournament
• TAMU = Texas A&M High School Mathematics Contest
• TST = Team Selection Test
• TSTST = Team Selection Test for the Selection Team
• USAJMO = United States of America Junior Mathematical Olympiad
• USAMO = United States of America Mathematical Olympiad
• USAMTS = USA Mathematical Talent Search
• USEMO = US Ersatz Math Olympiad
• USMCA = US Math Competition Association
• ZIML = Zoom International Math League
• AIME = American Invitational Mathematics Examination
• AMC = Australian Mathematics Competitions
• ARML = American Regions Mathematics League
• BAMO = Bay Area Mathematical Olympiad
• BMT = Berkley Math Tournament
• CCAMB = Canyon Crest Academy Math Bonanza
• CCMM = College of Charleston Math Meet
• CHMMC = Caltech Harvey Mudd Math Competition
• CMIMC = Carnegie Mellon Informatics and Mathematics Competition
• DMC = Deerfield Math Competition
• DMM = Duke Math Meet
• ELMO = annual math Olympiad at MOP
• EMCC = Exeter Math Club Competition
• HMIC = Harvard-MIT Mathematics Tournament Invitational Competition
• HMMT = Harvard-MIT Mathematics Tournament
• IMTS = International Mathematical Talent Search
• JHMMC = Joe Holbrook Memorial Math Competition
• JHMT = Johns Hopkins Math Tournament
• JMO = Junior Mathematical Olympiad
• LMT = Lexington Math Tournament
• MBMT = Montgomery Blair Math Tournament
• MCAMC = Middlesex County Academy Math Competition
• MMATHS = Math Majors of America Tournament for High Schools
• MOAA = Math Open at Andover by Phillips Academy
• MOEMS = Math Olympiads for Elementary and Middle Schools
• MOP = Mathematical Olympiad Summer Program (also MOSP)
• MOSP = Mathematical Olympiad Summer Program (also MOP)
• MPfG = Math Prize for Girls
• NIMO = National Internet Mathematical Olympiad
• OMO = Online Math Open
• PUMaC = Princeton University Mathematics Competition
• RCML = Rocket City Math League
• SBMT = South Bay Math Tournament
• SMT= Stanford Math Tournament
• TAMU = Texas A&M High School Mathematics Contest
• TST = Team Selection Test
• TSTST = Team Selection Test for the Selection Team
• USAJMO = United States of America Junior Mathematical Olympiad
• USAMO = United States of America Mathematical Olympiad
• USAMTS = USA Mathematical Talent Search
• USEMO = US Ersatz Math Olympiad
• USMCA = US Math Competition Association
• ZIML = Zoom International Math League
any suggestions are welcome
This post has been edited 15 times. Last edited by parmenides51, Dec 28, 2020, 8:17 AM
