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Old Russian Math Olympiads All Russian, Moscow, Tournament of Towns,...
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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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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
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0 replies
jlacosta
Mar 2, 2025
0 replies
i Forum's purpose - Problem Index
parmenides51   18
N Jul 10, 2024 by awesomelion7
I created this forum in order to post all the old *Russian Math Olympiads and complete the contest collections in the future, without spamming the existing forums with so many posts in so little time.

Here shall be posted all the old problems from 6 Math Olympiads:
- ASU (All Soviet Union) 1961-1992 (aops contest collection) (completed)
- All-Russian Olympiad Regional Round 1993- 2006: (aops contest collection)
- Chisinau City Math Olympiad (Moldova): (aops contest collection)
- Moscow Math Olympiads 1935-1997 (aops contest collection)
- Leningrad Math Olympiad (renamed to Saint Petersburg) (aops contest collection)
- Tournament of Towns 1980-1997,1999, 2000, 2011-14 (aops contest collection)

contest collections created so far: (inside this forum)
All Russian: 1961, 1962, 1963, 1964, 1965, 1966,
All Soviet Union (ASU) : 1967, 1968, 1969, 1970, 1971, 1972, 1973, 1974 , 1975, 1976, 1977 , 1978, 1979, 1980, 1981, 1982, 1983, 1984, 1985, 1986, 1987, 1988, 1989, 1990, 1991
All Russian Regional: 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006
Chisinau City MO (Moldova): 1949-56, (1957-72 got lost) , 1973, 1974, 1975 , 1976, 1977, 1978, 1979
Commonwealth of Independent States: ASU 1992
Moscow MO (ΜΜΟ): 1935, 1936, 1937, 1938, 1939, 1940, 1941, (1942-44 did not take place) 1945, 1946, 1947 ,1948, 1949, 1950, 1951, 1952, 1953, 1954, 1955, 1956 , 1957
Tournament of Towns (ΤοΤ): 1980 , 1981, 1982, 1983, 1984, 1985, 1986, 1987, 1988,1989, 1990, 1991, 1992, 1993, 1994, 1995 ,1996 ,1997 , 1998, 1999, 2000, 2004 ,2006 ,2008 , 2011, 2012, 2013,2019

extra
$\bullet$ Leningrad 1965-92, 98 selected problems (besides Leningrad MO)
$\bullet$ Moscow Mathematical Circles - 99 Selected Problems (MMCircles)
The following are problems we find most interesting among those offered to the participants of mathematical clubs, to the winners of the Moscow Olympiads when they were coached to International Olympiads and also some problems from the archives of the Moscow Olympiad jury which were not used in any of the tournaments, and, therefore, are not well known.

under construction:
MMO: 1958, 1959-97 (1942-44 did not take place)
ToT: 2014 , 2015, 2016, 2017, 2018, 2020

In every problem source, the problem's numbering is continuous through years

sources:
1.All Soviet Union MO
$\bullet$ All Soviet Union Math Competitions 1961 - 1987 EN translated by S/W engineer Vladimir Pertsel (link)
$\bullet$ All Soviet Union Mathematical Olympiad 1961-1992 EN with solutions, by John Scholes (Kalva) (link)
2. All-Russian Olympiad Regional MO
$\bullet$ Всероссийских математических олимпиад школьников 1993–2006 (pdf)
3. Moscow City MO
$\bullet$ 60-odd YEARS of MOSCOW MATHEMATICAL OLYMPIADS, Edited by D. Leites (pdf)
$\bullet$ Московские математические олимпиады. 1935 - 1957 г. (pdf here)
$\bullet$ Московские математические олимпиады. 1958 - 1967 г. г (pdf here)
$\bullet$ Московские математические олимпиады. 1981 - 1992 г. (pdf here)
4. Tournamenent of Towns
$\bullet$ International Mathematical Tournamenent of Towns Book 1, 1980-1984 (AMT)
$\bullet$ International Mathematical Tournamenent of Towns Book 2, 1984-1989 (AMT)
$\bullet$ International Mathematical Tournamenent of Towns Book 3, 1989-1993 (AMT)
$\bullet$ International Mathematical Tournamenent of Towns Book 4, 1993-1997 (AMT)
$\bullet$ International Mathematical Tournamenent of Towns Book 5, 1997-2002 (AMT)
5. Chisinau City MO
$\bullet$ Кишиневские математические олимпиады Ю.М.Рябухин 1983
6. Leningrad proposed problems
$\bullet$ Санкт-Петербургские математические олимпиады by Фомин Д. В. (1961-93)

Related forums:
Old High School 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

PS. A Forum Collection for Aops Geo Mocks,

* Here Russian stands for ex - USSR countries.
18 replies
parmenides51
Jun 17, 2019
awesomelion7
Jul 10, 2024
TOT 2000 Spring AS4 convex polygon on lattice points
parmenides51   4
N Mar 3, 2025 by N3bula
Each vertex of a convex polygon has integer coordinates, and no side of this polygon is horizontal or vertical. Prove that the sum of the lengths of the segments of lines of the form $x = m$, $m$ an integer, that lie within the polygon is equal to the sum of the lengths of the segments of lines of the form $y = n$, $n$ an integer, that lie within the polygon.

(G Galperin)
4 replies
parmenides51
May 11, 2020
N3bula
Mar 3, 2025
f(a(x + y)) = f(x) + f(y) - All-Russian MO 1997 Regional (R4) 11.8
parmenides51   5
N Feb 28, 2025 by navier3072
For which $a$, there is a function $f: R \to R$, different from a constant, such that
$$f(a(x + y)) = f(x) + f(y) ?$$
5 replies
parmenides51
Sep 24, 2024
navier3072
Feb 28, 2025
TOT 2004 Spring - Senior A-Level p5 common tangent circle and y = x^2
parmenides51   3
N Feb 25, 2025 by Mathandski
The parabola $y = x^2$ intersects a circle at exactly two points $A$ and $B$. If their tangents at $A$ coincide, must their tangents at $B$ also coincide?
3 replies
parmenides51
Feb 25, 2020
Mathandski
Feb 25, 2025
TOT 2006 Fall - Junior O-Level p4 cyclic wanted
parmenides51   1
N Feb 9, 2025 by ehuseyinyigit
Given triangle $ABC, BC$ is extended beyond $B$ to the point $D$ such that $BD = BA$. The bisectors of the exterior angles at vertices $B$ and $C$ intersect at the point $M$. Prove that quadrilateral $ADMC$ is cyclic. (4)
1 reply
parmenides51
Feb 25, 2020
ehuseyinyigit
Feb 9, 2025
MMO 121 Moscow MO 1946 Fibonacci sequence a term that ends in 4 zeros
parmenides51   1
N Feb 5, 2025 by xytan0585
Given the Fibonacci sequence $0, 1, 1, 2, 3, 5, 8, ... ,$ ascertain whether among its first $(10^8+1)$ terms there is a number that ends with four zeros.
1 reply
parmenides51
Jul 19, 2019
xytan0585
Feb 5, 2025
Leningrad proposed 24.1972 >= N -2 triangles by N lines on plane
parmenides51   1
N Jan 24, 2025 by ppvmiki2002
$N$ straight lines in general position are drawn on the plane. Prove that among the parts into which they divide the plane, there are at least $N -2$ triangles
1 reply
parmenides51
Jun 25, 2021
ppvmiki2002
Jan 24, 2025
p + q = (p -q)^3 - All-Russian MO 2001 Regional (R4) 11.1
parmenides51   3
N Jan 23, 2025 by aether_1729
Find all prime numbers $p$ and $q$ such that $p + q = (p -q)^3.$
3 replies
parmenides51
Sep 26, 2024
aether_1729
Jan 23, 2025
TOT 087 1985 Spring J3 class of 32 pupils is organised into 33 clubs
parmenides51   2
N Jan 21, 2025 by de-Kirschbaum
A certain class of $32$ pupils is organised into $33$ clubs , so that each club contains $3$ pupils and no two clubs have the same composition. Prove that there are two clubs which have exactly one common member.
2 replies
parmenides51
Aug 24, 2019
de-Kirschbaum
Jan 21, 2025
TOT 1999 Spring AS1 convex polyhedron is floating in a sea
parmenides51   4
N Jan 21, 2025 by sopaconk
A convex polyhedron is floating in a sea. Can it happen that $90\%$ of its volume is below the water level, while more than half of its surface area is above the water level?

(A Shapovalov)
4 replies
parmenides51
May 11, 2020
sopaconk
Jan 21, 2025
2011 ToT Fall Senior A p7 100 red points divide a blue circle into 100 arcs
parmenides51   5
N Dec 27, 2024 by HamstPan38825
$100$ red points divide a blue circle into $100$ arcs such that their lengths are all positive integers from $1$ to $100$ in an arbitrary order. Prove that there exist two perpendicular chords with red endpoints.
5 replies
parmenides51
Mar 22, 2020
HamstPan38825
Dec 27, 2024
1 rectangle intersects all rectangles - All-Russian MO 2002 Regional (R4) 9.4
parmenides51   0
Sep 17, 2024
Located on the plane $\left[ \frac43 n \right]$ rectangles with sides parallel to the coordinate axes. It is known that any rectangle intersects at least n rectangles. Prove that exists a rectangle that intersects all rectangles.
0 replies
parmenides51
Sep 17, 2024
0 replies
1 rectangle intersects all rectangles - All-Russian MO 2002 Regional (R4) 9.4
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parmenides51
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Located on the plane $\left[ \frac43 n \right]$ rectangles with sides parallel to the coordinate axes. It is known that any rectangle intersects at least n rectangles. Prove that exists a rectangle that intersects all rectangles.
This post has been edited 1 time. Last edited by parmenides51, Sep 26, 2024, 11:19 PM
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