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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

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

x^2 + ax + b, x^2 + cx + d - All-Russian MO 1996 Regional (R4) 9.1

parmenides51 1

N Apr 17, 2025 by Mathzeus1024

Find all pairs of square trinomials $x^2 + ax + b$ , $x^2 + cx + d$ such that $a$ and $b$ are the roots of the second trinomial, $c$ and $d$ are the roots of the first.

1 reply

parmenides51

Sep 23, 2024

Mathzeus1024

Apr 17, 2025

sum 1/\sqrt{1+x^2} <=2/\sqrt{1+xy} - All-Russian MO 2000 Regional (R4) 11.5

parmenides51 1

N Apr 17, 2025 by Mathzeus1024

For non-negative numbers $x$ and $y$ not exceeding $1$ , prove that
$\frac{1}{\sqrt{1+x^2}}+\frac{1}{\sqrt{1+y^2}} \le \frac{2}{\sqrt{1 + xy}},$

1 reply

parmenides51

Sep 26, 2024

Mathzeus1024

Apr 17, 2025

| sin nx| >= \sqrt3 / 2 - All-Russian MO 2006 Regional (R4) 10.5

parmenides51 1

N Apr 17, 2025 by Mathzeus1024

Prove that for every $x$ such that $\sin x \ne 0$ , there is such natural $n$ , which $| \sin nx| \ge \frac{\sqrt3}{2}.$

1 reply

parmenides51

Sep 27, 2024

Mathzeus1024

Apr 17, 2025

Ice cream costs 2000 rubles - All-Russian MO 1996 Regional (R4) 8.1

parmenides51 1

N Apr 15, 2025 by Mathzeus1024

Ice cream costs $2000$ rubles. Petya has $400^5 - 399^2\cdot (400^3 + 2\cdot 400^2 + 3\cdot 400 + 4)$ rubles. Does Petya have enough money for ice cream?

1 reply

parmenides51

Sep 23, 2024

Mathzeus1024

Apr 15, 2025

ASU 550 All Soviet Union MO 1991 100x100 array, equal product each column

parmenides51 2

N Apr 15, 2025 by zhoujef000

a) $r_1, r_2, ... , r_{100}, c_1, c_2, ... , c_{100}$ are distinct reals. The number $r_i + c_j$ is written in position $i, j$ of a $100 \times 100$ array. The product of the numbers in each column is $1$ . Show that the product of the numbers in each row is $-1$ .

b) $r_1, r_2, ... , r_{2n}, c_1, c_2, ... , c_{2n}$ are distinct reals. The number $r_i + c_j$ is written in position $i, j$ of a $2n \times 2n$ array. The product of the numbers in each column is the same. Show that the product of the numbers in each row is also the same.

2 replies

parmenides51

Aug 14, 2019

zhoujef000

Apr 15, 2025

ax^2+bx+c, (c-b)x^2 + (c- a)x +(a + b) - All-Russian MO 2001 Regional (R4) 10.5

parmenides51 1

N Apr 15, 2025 by Mathzeus1024

Given integers $a$ , $b$ and $c$ , $c\ne b$ . It is known that the square trinomials $ax^2 + bx + c$ and $(c-b)x^2 + (c- a)x + (a + b)$ have a common root (not necessarily integer). Prove that $a+b+2c$ is divisible by $3$ .

1 reply

parmenides51

Sep 26, 2024

Mathzeus1024

Apr 15, 2025

cosa+cosb+cosc>sina+sinb+sinc- All-Russian MO 2004 Regional (R4) 10.1

parmenides51 1

N Apr 15, 2025 by Mathzeus1024

The sum of positive numbers $a, b, c$ is equal to $\pi/2$ . Prove that
$\cos a + \cos b + \cos c > \sin a + \sin b + \sin c.$

1 reply

parmenides51

Sep 27, 2024

Mathzeus1024

Apr 15, 2025

MMO 372 Moscow MO 1957 divide a_i <= a_{i+1} <= 2a_i, into 2 groups

parmenides51 10

N Apr 14, 2025 by InterLoop

Given $n$ integers $a_1 = 1, a_2,..., a_n$ such that $a_i \le a_{i+1} \le 2a_i$ ( $i = 1, 2, 3,..., n - 1$ ) and whose sum is even. Find whether it is possible to divide them into two groups so that the sum of numbers in one group is equal to the sum of numbers in the other group.

10 replies

parmenides51

Mar 20, 2021

InterLoop

Apr 14, 2025

Help needed

JetFire008 2

N Apr 5, 2025 by JetFire008

Does there exist a finite set $M$ of nonzero real numbers, such that for any natural number $n$ a polynomial of degree no less than $n$ with coefficients in $M$ , all of whose roots are real and belong to $M$ ?
Can someone explain the solution of this question?

2 replies

JetFire008

Mar 23, 2025

JetFire008

Apr 5, 2025

TOT 377 1993 Spring A S5 piecewise linear function f(f(x)) = -x

parmenides51 1

N Mar 26, 2025 by jasperE3

Does there exist a piecewise linear function $f$ defined on the segment [ $-1,1]$ (including the ends) such that $f(f(x)) = -x$ for all x? (A function is called piecewise linear if its graph is the union of a finite set of points and intervals; it may be discontinuous).

1 reply

parmenides51

Jun 10, 2024

jasperE3

Mar 26, 2025

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