Problem ID stringlengths 6 27 | Problem stringlengths 60 1.99k ⌀ | Lean Statement stringlengths 74 27.3k | Category stringclasses 1
value | Level stringclasses 4
values | Difficulty float64 6 9 ⌀ | Source stringclasses 128
values | Formalization float64 |
|---|---|---|---|---|---|---|---|
compfiles-Imo1963P6 | Five students A, B, C, D, E were placed 1 to 5 in a contest with no ties.
One prediction was that the result would be the order A, B, C, D, E.
But no student finished in the position predicted, and no two students
predicted to finish consecutively did so. For example, the outcome for
C and D was not 1, 2 (respectively)... | import Mathlib
/-!
# International Mathematical Olympiad 1963, Problem 6
Five students A, B, C, D, E were placed 1 to 5 in a contest with no ties.
One prediction was that the result would be the order A, B, C, D, E.
But no student finished in the position predicted, and no two students
predicted to finish consecutive... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1963, Problem 6 | null |
compfiles-Imo1964P4 | Seventeen people correspond by mail with one another -- each one with
all the rest. In their letters only three different topics are
discussed. Each pair of correspondents deals with only one of the topics.
Prove that there are at least three people who write to each other
about the same topic. | import Mathlib
/-!
# International Mathematical Olympiad 1964, Problem 4
Seventeen people correspond by mail with one another -- each one with
all the rest. In their letters only three different topics are
discussed. Each pair of correspondents deals with only one of the topics.
Prove that there are at least three pe... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1964, Problem 4 | null |
compfiles-Imo1966P1 | In a mathematical contest, three problems, A, B, C were posed. Among the
participants there were 25 students who solved at least one problem each.
Of all the contestants who did not solve problem A, the number who solved
B was twice the number who solved C. The number of students who solved
only problem A was one more ... | import Mathlib
/-!
# International Mathematical Olympiad 1966, Problem 1.
In a mathematical contest, three problems, A, B, C were posed. Among the
participants there were 25 students who solved at least one problem each.
Of all the contestants who did not solve problem A, the number who solved
B was twice the number ... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1966, Problem 1. | null |
compfiles-Imo1970P6 | In a plane there are 100 points, no three of which are collinear.
Consider all possible triangles having these points as vertices.
Prove that no more that 70% of these triangles are acute. | import Mathlib
/-!
# International Mathematical Olympiad 1970, Problem 6
In a plane there are 100 points, no three of which are collinear.
Consider all possible triangles having these points as vertices.
Prove that no more that 70% of these triangles are acute.
-/
namespace Imo1970P6
open scoped EuclideanGeometry R... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1970, Problem 6 | null |
compfiles-Imo1971P5 | Prove that for every natural number m there exists a nonempty finite
set S of points in the plane with the following property:
For every point s in S, there are exactly m points which are at a unit
distance from s. | import Mathlib
/-!
# International Mathematical Olympiad 1971, Problem 5
Prove that for every natural number m there exists a nonempty finite
set S of points in the plane with the following property:
For every point s in S, there are exactly m points which are at a unit
distance from s.
-/
namespace Imo1971P5
open ... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1971, Problem 5 | null |
compfiles-Imo1972P1 | Prove that from a set of ten distinct two-digit numbers (in
decimal), it is possible to select two disjoint subsets whose
members have the same sum. | import Mathlib
/-!
# International Mathematical Olympiad 1972, Problem 1
Prove that from a set of ten distinct two-digit numbers (in
decimal), it is possible to select two disjoint subsets whose
members have the same sum.
-/
namespace Imo1972P1
theorem imo1972_p1 (S : Finset ℕ)
(Scard : S.card = 10)
(Sdigit... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1972, Problem 1 | null |
compfiles-Imo1974P1 | Three players $A, B$ and $C$ play the following game:
* On each of three cards an integer is written.
* These three numbers $p, q, r$ satisfy $0 < p < q < r$.
* The three cards are shuffled and one is dealt to each player.
* Each then receives the number of counters indicated by the card he holds.
* Then the cards are... | import Mathlib
/-!
# International Mathematical Olympiad 1974, Problem 1
Three players $A, B$ and $C$ play the following game:
* On each of three cards an integer is written.
* These three numbers $p, q, r$ satisfy $0 < p < q < r$.
* The three cards are shuffled and one is dealt to each player.
* Each then receives... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1974, Problem 1 | null |
compfiles-Imo1974P4 | Consider decompositions of an 8 × 8 chessboard into p non-overlapping
rectangles, subject to the following conditions:
(i) Each rectangle has as many white squares as black squares.
(ii) If aᵢ is the number of white squares in the i-th rectangle, then
a₁ < a₂ < ... < aₚ.
Find the maximum value of p for which suc... | import Mathlib
/-!
# International Mathematical Olympiad 1974, Problem 4
Consider decompositions of an 8 × 8 chessboard into p non-overlapping
rectangles, subject to the following conditions:
(i) Each rectangle has as many white squares as black squares.
(ii) If aᵢ is the number of white squares in the i-th rectangl... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1974, Problem 4 | null |
compfiles-Imo1975P5 | Determine, with proof, whether or not one can find $1975$ points on the circumference of a circle
with unit radius such that the distance between any two of them is a rational number. | import Mathlib
/-!
# International Mathematical Olympiad 1975, Problem 5
Determine, with proof, whether or not one can find $1975$ points on the circumference of a circle
with unit radius such that the distance between any two of them is a rational number.
-/
namespace Imo1975P5
abbrev constructible : Bool := true
... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1975, Problem 5 | null |
compfiles-Imo1976P5 | n is a positive integer and m = 2n. aᵢⱼ = 0, 1 or -1 for 1 ≤ i ≤ n,
1 ≤ j ≤ m. The m unknowns x₁, x₂, ... , xₘ satisfy the n equations:
aᵢ₁x₁ + aᵢ₂x₂ + ... + aᵢₘxₘ = 0,
for i = 1, 2, ... , n. Prove that the system has a solution in integers
of absolute value at most m, not all zero. | import Mathlib
/-!
# International Mathematical Olympiad 1976, Problem 5
n is a positive integer and m = 2n. aᵢⱼ = 0, 1 or -1 for 1 ≤ i ≤ n,
1 ≤ j ≤ m. The m unknowns x₁, x₂, ... , xₘ satisfy the n equations:
aᵢ₁x₁ + aᵢ₂x₂ + ... + aᵢₘxₘ = 0,
for i = 1, 2, ... , n. Prove that the system has a solution in integers
... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1976, Problem 5 | null |
compfiles-Imo1977P2 | In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive.
Determine the maximum number of terms in the sequence. | import Mathlib
/-!
# International Mathematical Olympiad 1977, Problem 2
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive.
Determine the maximum number of terms in the sequence.
-/
namespace Imo1977P2
def sum_successive_te... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1977, Problem 2 | null |
compfiles-Imo1978P6 | An international society has its members from six different countries.
The list of members has 1978 names, numbered $1, 2, \ldots, 1978$.
Prove that there is at least one member whose number is
the sum of the numbers of two (not necessarily distinct) members from his own country. | import Mathlib
/-!
# International Mathematical Olympiad 1978, Problem 6
An international society has its members from six different countries.
The list of members has 1978 names, numbered $1, 2, \ldots, 1978$.
Prove that there is at least one member whose number is
the sum of the numbers of two (not necessarily dist... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1978, Problem 6 | null |
compfiles-Imo1979P2 | A prism with pentagons $A_1A_2A_3A_4A_5$ and $B_1B_2B_3B_4B_5$ as the top and
bottom faces is given. Each side of the two pentagons and each of the 25
segments $A_iB_j$ is colored red or green. Every triangle whose vertices are
vertices of the prism and whose sides have all been colored has two sides of
a different col... | import Mathlib
/-!
# International Mathematical Olympiad 1979, Problem 2
A prism with pentagons $A_1A_2A_3A_4A_5$ and $B_1B_2B_3B_4B_5$ as the top and
bottom faces is given. Each side of the two pentagons and each of the 25
segments $A_iB_j$ is colored red or green. Every triangle whose vertices are
vertices of the p... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1979, Problem 2 | null |
compfiles-Imo1979P6 | Let $A$ and $E$ be opposite vertices of an octagon.
A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices.
When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$.
Prove that: \[a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} -... | import Mathlib
/-!
# International Mathematical Olympiad 1979, Problem 6
Let $A$ and $E$ be opposite vertices of an octagon.
A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices.
When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jump... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1979, Problem 6 | null |
compfiles-Imo1981P2 | Let $1 \le r \le n$ and consider all subsets of $r$ elements of the set $\{ 1, 2, \ldots , n \}$.
Each of these subsets has a smallest member. Let $F(n,r)$ denote the arithmetic mean of these smallest numbers; prove that
\[ F(n,r) = \frac{n+1}{r+1}. \] | import Mathlib
namespace Imo1981P2
/-!
# International Mathematical Olympiad 1981, Problem 2
Let $1 \le r \le n$ and consider all subsets of $r$ elements of the set $\{ 1, 2, \ldots , n \}$.
Each of these subsets has a smallest member. Let $F(n,r)$ denote the arithmetic mean of these smallest numbers; prove that
\[ ... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1981, Problem 2 | null |
compfiles-Imo1983P5 | Is it possible to choose $1983$ distinct positive integers,
all less than or equal to $10^5$,
no three of which are consecutive terms of an arithmetic progression?
Justify your answer. | import Mathlib
/-!
# International Mathematical Olympiad 1983, Problem 5
Is it possible to choose $1983$ distinct positive integers,
all less than or equal to $10^5$,
no three of which are consecutive terms of an arithmetic progression?
Justify your answer.
-/
namespace Imo1983P5
theorem imo1983_p5 :
∃ S : Finse... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1983, Problem 5 | null |
compfiles-Imo1984P3 | In the plane two different points O and A are given. For each point X of the plane,
other than O, denote by a(X) the measure of the angle between OA and OX in radians,
counterclockwise from OA (0 ≤ a(X) < 2π). Let C(X) be the circle with center O and
radius of length OX + a(X)/OX. Each point of the plane is colored by ... | import Mathlib
/-!
# International Mathematical Olympiad 1984, Problem 3
In the plane two different points O and A are given. For each point X of the plane,
other than O, denote by a(X) the measure of the angle between OA and OX in radians,
counterclockwise from OA (0 ≤ a(X) < 2π). Let C(X) be the circle with center ... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1984, Problem 3 | null |
compfiles-Imo1985P2 | Fix a natural number $n ≥ 3$ and define $N=\{1, 2, 3, \dots, n-1\}$.
Fix another natural number $j ∈ N$ coprime to $n$. Each number in
$N$ is now colored with one of two colors, say red or black, so that:
1. $i$ and $n-i$ always receive the same color, and
2. $i$ and $|j-i|$ receive the same color for all $i ∈ N, i ≠ ... | import Mathlib
/-!
# International Mathematical Olympiad 1985, Problem 2
Fix a natural number $n ≥ 3$ and define $N=\{1, 2, 3, \dots, n-1\}$.
Fix another natural number $j ∈ N$ coprime to $n$. Each number in
$N$ is now colored with one of two colors, say red or black, so that:
1. $i$ and $n-i$ always receive the sam... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1985, Problem 2 | null |
compfiles-Imo1986P3 | To each vertex of a regular pentagon, an integer is assigned,
in such a way that the sum of all five numbers is positive.
If three consecutive vertices are assigned the numbers $x, y, z$ respectively and $y < 0$,
then the following operation is allowed: $x, y, z$ are replaced by $x+y, -y, z+y$ respectively.
Such an ope... | import Mathlib
/-!
# International Mathematical Olympiad 1986, Problem 3
To each vertex of a regular pentagon, an integer is assigned,
in such a way that the sum of all five numbers is positive.
If three consecutive vertices are assigned the numbers $x, y, z$ respectively and $y < 0$,
then the following operation is ... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1986, Problem 3 | null |
compfiles-Imo1986P6 | Given a finite set of points in the plane, each with integer coordinates,
is it always possible to color the points red or white so that for any
straight line L parallel to one of the coordinate axes the difference
(in absolute value) between the numbers of white and red points on L
is not greater than 1? | import Mathlib
/-!
# International Mathematical Olympiad 1986, Problem 6
Given a finite set of points in the plane, each with integer coordinates,
is it always possible to color the points red or white so that for any
straight line L parallel to one of the coordinate axes the difference
(in absolute value) between th... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1986, Problem 6 | null |
compfiles-Imo1987P1 | Let $p_{n, k}$ be the number of permutations of a set of cardinality `n ≥ 1`
that fix exactly `k` elements. Prove that $∑_{k=0}^n k p_{n,k}=n!$. | import Mathlib
/-!
# International Mathematical Olympiad 1987, Problem 1
Let $p_{n, k}$ be the number of permutations of a set of cardinality `n ≥ 1`
that fix exactly `k` elements. Prove that $∑_{k=0}^n k p_{n,k}=n!$.
-/
namespace Imo1987P1
/-- Given `α : Type*` and `k : ℕ`, `fiber α k` is the set of permutations o... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1987, Problem 1 | null |
compfiles-Imo1988P2 | Let n be a positive integer and let A₁, A₂, ..., A₂ₙ₊₁ be subsets
of a set B. Suppose that
(i) each Aᵢ has exactly 2n elements,
(ii) each intersection Aᵢ ∩ Aⱼ (i ≠ j) contains exactly one element, and
(iii) every element of B belongs to at least two of the Aᵢ.
For which values of n can one assign to every element of ... | import Mathlib
/-!
# International Mathematical Olympiad 1988, Problem 2
Let n be a positive integer and let A₁, A₂, ..., A₂ₙ₊₁ be subsets
of a set B. Suppose that
(i) each Aᵢ has exactly 2n elements,
(ii) each intersection Aᵢ ∩ Aⱼ (i ≠ j) contains exactly one element, and
(iii) every element of B belongs to at leas... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1988, Problem 2 | null |
compfiles-Imo1989P1 | Prove that the integers from 1 to 1989 can be partitioned in 117 sets
of 17 elements each, all with the same sum of elements. | import Mathlib
/-!
# International Mathematical Olympiad 1989, Problem 1
Prove that the integers from 1 to 1989 can be partitioned in 117 sets
of 17 elements each, all with the same sum of elements.
-/
namespace Imo1989P1
abbrev S := Finset.Icc 1 1989
theorem imo1989_p1 : ∃ (P : Finpartition S),
P.parts.card ... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1989, Problem 1 | null |
compfiles-Imo1989P3 | Let $n$ and $k$ be positive integers and let $S$ be a set of $n$ points in
the plane such that
(i) no three points of $S$ are collinear, and
(ii) for each point $P$ of $S$ there are at least $k$ points of $S$
equidistant from $P$.
Prove that $k < \frac{1}{2} + \sqrt{2n}$. | import Mathlib
/-!
# International Mathematical Olympiad 1989, Problem 3
Let $n$ and $k$ be positive integers and let $S$ be a set of $n$ points in
the plane such that
(i) no three points of $S$ are collinear, and
(ii) for each point $P$ of $S$ there are at least $k$ points of $S$
equidistant from $P$.
Pro... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1989, Problem 3 | null |
compfiles-Imo1990P2 | Take n ≥ 3 and consider a set E of 2n - 1 distinct points on a circle.
Suppose that exactly k of these points are to be colored black. Such a
coloring is said to be "good" if there is at least one pair of black points
such that the interior of one of the arcs between them contains exactly n
points from E. Find the smal... | import Mathlib
/-!
# International Mathematical Olympiad 1990, Problem 2
Take n ≥ 3 and consider a set E of 2n - 1 distinct points on a circle.
Suppose that exactly k of these points are to be colored black. Such a
coloring is said to be "good" if there is at least one pair of black points
such that the interior of o... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1990, Problem 2 | null |
compfiles-Imo1990P5 | Given an initial integer n₀ > 1, two players A and B choose integers n₁, n₂, n₃, ...
alternately according to the following rules:
Knowing n₂ₖ, A chooses any integer n₂ₖ₊₁ such that n₂ₖ ≤ n₂ₖ₊₁ ≤ n₂ₖ².
Knowing n₂ₖ₊₁, B chooses any integer n₂ₖ₊₂ such that n₂ₖ₊₁/n₂ₖ₊₂ = p^r for some prime p
and integer r ≥ 1.
Player A... | import Mathlib
/-!
# International Mathematical Olympiad 1990, Problem 5
Given an initial integer n₀ > 1, two players A and B choose integers n₁, n₂, n₃, ...
alternately according to the following rules:
Knowing n₂ₖ, A chooses any integer n₂ₖ₊₁ such that n₂ₖ ≤ n₂ₖ₊₁ ≤ n₂ₖ².
Knowing n₂ₖ₊₁, B chooses any integer n₂ₖ₊... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1990, Problem 5 | null |
compfiles-Imo1991P3 | Let S = {1, 2, ..., 280}. Find the smallest natural number n such
that every n-element subset of S contains five numbers which are
pairwise relatively prime. | import Mathlib
/-!
# International Mathematical Olympiad 1991, Problem 3
Let S = {1, 2, ..., 280}. Find the smallest natural number n such
that every n-element subset of S contains five numbers which are
pairwise relatively prime.
-/
namespace Imo1991P3
abbrev solution : ℕ := 217
theorem imo1991_p3 :
IsLeast
... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1991, Problem 3 | null |
compfiles-Imo1991P4 | Suppose G is a connected graph with k edges. Prove that it is possible to
label the edges 1, 2, ... , k in such a way that at each vertex which belongs
to two or more edges, the greatest common divisor of the integers labeling
those edges is equal to 1. | import Mathlib
/-!
# International Mathematical Olympiad 1991, Problem 4
Suppose G is a connected graph with k edges. Prove that it is possible to
label the edges 1, 2, ... , k in such a way that at each vertex which belongs
to two or more edges, the greatest common divisor of the integers labeling
those edges is equ... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1991, Problem 4 | null |
compfiles-Imo1992P3 | Consider 9 points in space, no 4 coplanar. Each pair of points is joined by a
line segment which is colored either blue or red or left uncolored. Find the
smallest value of n such that whenever exactly n edges are colored, the set of
colored edges necessarily contains a triangle all of whose edges have the same
color. | import Mathlib
/-!
# International Mathematical Olympiad 1992, Problem 3
Consider 9 points in space, no 4 coplanar. Each pair of points is joined by a
line segment which is colored either blue or red or left uncolored. Find the
smallest value of n such that whenever exactly n edges are colored, the set of
colored edg... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1992, Problem 3 | null |
compfiles-Imo1992P5 | Let S be a finite set of points in three-dimensional space.
Let Sx,Sy and Sz be the sets consisting of the orthogonal projections of the points of S onto the yz-plane, zx-plane and xy-plane, respectively.
Prove that |S|²≤|Sx|·|Sy|·|Sz|.
where |A| denotes the number of elements in the finite set |A|.
(Note: The orthogon... | import Mathlib
/-!
# International Mathematical Olympiad 1992, Problem 5
Let S be a finite set of points in three-dimensional space.
Let Sx,Sy and Sz be the sets consisting of the orthogonal projections of the points of S onto the yz-plane, zx-plane and xy-plane, respectively.
Prove that |S|²≤|Sx|·|Sy|·|Sz|.
where |A... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1992, Problem 5 | null |
compfiles-Imo1993P3 | On an infinite chessboard a game is played as follows. At the start n² pieces are arranged
in an n × n block of adjoining squares, one piece on each square. A move in the game is a jump
in a horizontal or vertical direction over an adjacent occupied square to an unoccupied square
immediately beyond. The piece which has... | import Mathlib
/-!
# International Mathematical Olympiad 1993, Problem 3
On an infinite chessboard a game is played as follows. At the start n² pieces are arranged
in an n × n block of adjoining squares, one piece on each square. A move in the game is a jump
in a horizontal or vertical direction over an adjacent occu... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1993, Problem 3 | null |
compfiles-Imo1994P1 | Let `m` and `n` be two positive integers.
Let `a₁, a₂, ..., aₘ` be `m` different numbers from the set `{1, 2, ..., n}`
such that for any two indices `i` and `j` with `1 ≤ i ≤ j ≤ m` and `aᵢ + aⱼ ≤ n`,
there exists an index `k` such that `aᵢ + aⱼ = aₖ`.
Show that `(a₁+a₂+...+aₘ)/m ≥ (n+1)/2` | import Mathlib
/-!
# International Mathmatical Olympiad 1994, Problem 1
Let `m` and `n` be two positive integers.
Let `a₁, a₂, ..., aₘ` be `m` different numbers from the set `{1, 2, ..., n}`
such that for any two indices `i` and `j` with `1 ≤ i ≤ j ≤ m` and `aᵢ + aⱼ ≤ n`,
there exists an index `k` such that `aᵢ + aⱼ ... | Combinatorics | 1. Named contest | null | Compfiles: International Mathmatical Olympiad 1994, Problem 1 | null |
compfiles-Imo1995P6 | Let p be an odd prime number. How many p-element subsets A of
{1, 2, ..., 2p} are there, the sum of whose elements is
divisible by p? | import Mathlib
/-!
# International Mathematical Olympiad 1995, Problem 6
Let p be an odd prime number. How many p-element subsets A of
{1, 2, ..., 2p} are there, the sum of whose elements is
divisible by p?
-/
namespace Imo1995P6
abbrev solution (p : ℕ) : ℕ := 2 + (Nat.choose (2 * p) p - 2) / p
theorem imo1995_p6... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1995, Problem 6 | null |
compfiles-Imo1997P4 | An $n \times n$ matrix whose entries come from the set
$S = \{1, 2, \ldots, 2n - 1\}$ is called a *silver matrix* if, for each
$i = 1, \ldots, n$, the $i$-th row and the $i$-th column together contain
all elements of $S$. Show that:
(a) there is no silver matrix for $n = 1997$;
(b) silver matrices exist for infin... | import Mathlib
/-!
# International Mathematical Olympiad 1997, Problem 4
An $n \times n$ matrix whose entries come from the set
$S = \{1, 2, \ldots, 2n - 1\}$ is called a *silver matrix* if, for each
$i = 1, \ldots, n$, the $i$-th row and the $i$-th column together contain
all elements of $S$. Show that:
(a) there... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1997, Problem 4 | null |
compfiles-Imo1997P6 | For each positive integer n, let f(n) denote the number of ways of representing n
as a sum of powers of 2 with non-negative integer exponents. Representations
which differ only in the ordering of their summands are considered to be the
same. For example, f(4) = 4, because 4 can be represented as 4, 2 + 2, 2 + 1 + 1
or ... | import Mathlib
/-!
# International Mathematical Olympiad 1997, Problem 6
For each positive integer n, let f(n) denote the number of ways of representing n
as a sum of powers of 2 with non-negative integer exponents. Representations
which differ only in the ordering of their summands are considered to be the
same. For... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1997, Problem 6 | null |
compfiles-Imo1998P2 | In a competition, there are `a` contestants and `b` judges, where `b ≥ 3` is
an odd integer. Each judge rates each contestant as either "pass" or "fail".
Suppose `k` is a number such that, for any two judges, their ratings
coincide for at most `k` contestants.
Prove that `k / a ≥ (b - 1) / (2b)`. | import Mathlib
/-!
# International Mathematical Olympiad 1998, Problem 2
In a competition, there are `a` contestants and `b` judges, where `b ≥ 3` is
an odd integer. Each judge rates each contestant as either "pass" or "fail".
Suppose `k` is a number such that, for any two judges, their ratings
coincide for at most `... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1998, Problem 2 | null |
compfiles-Imo1999P3 | Consider an `n × n` square board, where `n` is a fixed even positive integer.
The board is divided into `n^2` unit squares. We say that two different squares on the
board are adjacent if they have a common side.
`N` unit squares on the board are marked in such a way that every square (marked
or unmarked) on the board i... | import Mathlib
/-!
# International Mathematical Olympiad 1999, Problem 3
Consider an `n × n` square board, where `n` is a fixed even positive integer.
The board is divided into `n^2` unit squares. We say that two different squares on the
board are adjacent if they have a common side.
`N` unit squares on the board are... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 1999, Problem 3 | null |
compfiles-Imo2000P3 | Let n ≥ 2 be a positive integer and λ a positive real number. Initially there are n fleas
on a horizontal line, not all at the same point. We define a move as choosing two fleas at
some points A and B, with A to the left of B, and letting the flea from A jump over the
flea from B to the point C so that BC/AB = λ.
Dete... | import Mathlib
/-!
# International Mathematical Olympiad 2000, Problem 3
Let n ≥ 2 be a positive integer and λ a positive real number. Initially there are n fleas
on a horizontal line, not all at the same point. We define a move as choosing two fleas at
some points A and B, with A to the left of B, and letting the fl... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 2000, Problem 3 | null |
compfiles-Imo2014P5 | For every positive integer n, the Bank of Cape Town issues coins of
denomination 1/n. Given a finite collection of such coins (of not
necessarily different denominations) with total value at most 99 + 1/2,
prove that it is possible to split this collection into 100 or fewer
groups, such that each group has total value ... | import Mathlib
/-!
# International Mathematical Olympiad 2014, Problem 5
For every positive integer n, the Bank of Cape Town issues coins of
denomination 1/n. Given a finite collection of such coins (of not
necessarily different denominations) with total value at most 99 + 1/2,
prove that it is possible to split this... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 2014, Problem 5 | null |
compfiles-Imo2020P6 | Consider an integer n > 1, and a set S of n points in the plane
such that the distance between any two points in S is at least 1.
Prove that there is a line l separating S such that the distance
from any point of S to l is at least Ω(n ^ (-1/3)).
(A line l separates a set of points S if some segment jointing
two point... | import Mathlib
/-!
# International Mathematical Olympiad 2020, Problem 6
Consider an integer n > 1, and a set S of n points in the plane
such that the distance between any two points in S is at least 1.
Prove that there is a line l separating S such that the distance
from any point of S to l is at least Ω(n ^ (-1/3))... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 2020, Problem 6 | null |
compfiles-Imo2025P6 | Consider a 2025 × 2025 grid of unit squares. Matilda wishes to place
on the grid some rectangular tiles, possibly of different sizes,
such that each side of every tile lies on a grid line and
every unit square is covered by at most one tile.
Determine the minimum number of tiles Matilda needs to place
so that each row ... | import Mathlib
open Finset Function
/-!
# International Mathematical Olympiad 2025, Problem 6
Consider a 2025 × 2025 grid of unit squares. Matilda wishes to place
on the grid some rectangular tiles, possibly of different sizes,
such that each side of every tile lies on a grid line and
every unit square is covered b... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 2025, Problem 6 | null |
compfiles-Imo2026P3 | Let n be a positive integer. Liu Bang and Xiang Yu have a stick of length 1
and want to divide it between themselves. Liu marks at most n points on the
stick, and then Xiang marks at most n points on the stick. The marked points
are distinct. Then, the stick is cut at all marked points, creating a number
of pieces. Aft... | import Mathlib
/-!
# International Mathematical Olympiad 2026, Problem 3
Let n be a positive integer. Liu Bang and Xiang Yu have a stick of length 1
and want to divide it between themselves. Liu marks at most n points on the
stick, and then Xiang marks at most n points on the stick. The marked points
are distinct. Th... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 2026, Problem 3 | null |
compfiles-Imo2026P4 | Shan-Yu and Mulan are playing a game. Let θ be an angle with 0° < θ < 180°
known to both players. Initially, Shan-Yu makes a paper triangle T with
measurements of his choice. Then, they repeatedly perform the following steps:
* If T has at least one angle measuring exactly θ, then the game stops and
Mulan wins.
* Ot... | import Mathlib
/-!
# International Mathematical Olympiad 2026, Problem 4
Shan-Yu and Mulan are playing a game. Let θ be an angle with 0° < θ < 180°
known to both players. Initially, Shan-Yu makes a paper triangle T with
measurements of his choice. Then, they repeatedly perform the following steps:
* If T has at leas... | Combinatorics | 1. Named contest | null | Compfiles: International Mathematical Olympiad 2026, Problem 4 | null |
compfiles-IntegersInACircle | null | import Mathlib
/-!
There are 101 positive integers arranged in a circle.
Suppose that the integers sum to 300.
Prove that there exists a contiguous subarray that sums to 200.
https://mathstodon.xyz/@alexdbolton/110292738044661739
https://math.stackexchange.com/questions/282589/101-positive-integers-placed-on-a-circle... | Combinatorics | 1. Named contest | null | Compfiles: IntegersInACircle | null |
compfiles-KolmogorovStreams | null | import Mathlib
/-!
Puzzle referenced from this tweet: https://twitter.com/sigfpe/status/1474173467016589323
From the book _Out of their Minds: The Lives and Discoveries of 15 Great Computer Scientists_
by Dennis Shasha and Cathy Lazere.
Problem: Suppose each (finite) word is either "decent" or "indecent". Given an... | Combinatorics | 1. Named contest | null | Compfiles: KolmogorovStreams | null |
compfiles-UK2024R1P1 | An unreliable typist can guarantee that when they try to type a word with
different letters, every letter of the word will appear exactly once in what
they type, and each letter will occur at most one letter late (though it may
occur more than one letter early). Thus, when trying to type MATHS, the
typist may type MATH... | import Mathlib
/-!
# British Mathematical Olympiad 2024, Round 1, Problem 1
An unreliable typist can guarantee that when they try to type a word with
different letters, every letter of the word will appear exactly once in what
they type, and each letter will occur at most one letter late (though it may
occur more tha... | Combinatorics | 1. Named contest | null | Compfiles: British Mathematical Olympiad 2024, Round 1, Problem 1 | null |
compfiles-Usa1974P4 | A, B, C play a series of games. Each game is between two players.
The next game is between the winner and the person who was not playing.
The series continues until one player has won two games. He wins the series.
A is the weakest player, C the strongest. Each player has a fixed probability
of winning against a given ... | import Mathlib
/-!
# USA Mathematical Olympiad 1974, Problem 4
A, B, C play a series of games. Each game is between two players.
The next game is between the winner and the person who was not playing.
The series continues until one player has won two games. He wins the series.
A is the weakest player, C the strongest... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1974, Problem 4 | null |
compfiles-Usa1975P5 | A pack of n cards, including three aces, is well shuffled. Cards are turned
over in turn. Show that the expected number of cards that must be turned over
to reach the second ace is (n+1)/2.
## Formalization notes
We model the well-shuffled pack by the set of positions occupied by the three
aces, which is uniformly di... | import Mathlib
/-!
# USA Mathematical Olympiad 1975, Problem 5
A pack of n cards, including three aces, is well shuffled. Cards are turned
over in turn. Show that the expected number of cards that must be turned over
to reach the second ace is (n+1)/2.
## Formalization notes
We model the well-shuffled pack by the s... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1975, Problem 5 | null |
compfiles-Usa1978P5 | There are 9 delegates at a conference, each speaking at most three languages.
Given any three delegates, at least 2 speak a common language.
Show that there are three delegates with a common language. | import Mathlib
/-!
# USA Mathematical Olympiad 1978, Problem 5
There are 9 delegates at a conference, each speaking at most three languages.
Given any three delegates, at least 2 speak a common language.
Show that there are three delegates with a common language.
-/
namespace Usa1978P5
/-- Delegate `a` and delegate... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1978, Problem 5 | null |
compfiles-Usa1979P5 | X has n members. Given n+1 subsets of X, each with 3 members,
show that we can always find two which have just one member in common. | import Mathlib
/-!
# USA Mathematical Olympiad 1979, Problem 5
X has n members. Given n+1 subsets of X, each with 3 members,
show that we can always find two which have just one member in common.
-/
namespace Usa1979P5
open Finset
theorem usa1979_p5 {α : Type*} [DecidableEq α] (n : ℕ) (X : Finset α) (hX : X.card ... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1979, Problem 5 | null |
compfiles-Usa1981P2 | What is the largest number of towns that can meet the following criteria?
Each pair is directly linked by just one of air, bus or train.
At least one pair is linked by air, at least one pair by bus and at least
one pair by train. No town has an air link, a bus link and a train link.
No three towns, A, B, C are such tha... | import Mathlib
/-!
# USA Mathematical Olympiad 1981, Problem 2
What is the largest number of towns that can meet the following criteria?
Each pair is directly linked by just one of air, bus or train.
At least one pair is linked by air, at least one pair by bus and at least
one pair by train. No town has an air link, ... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1981, Problem 2 | null |
compfiles-Usa1982P1 | A graph has 1982 points. Given any four points, there is at least one
joined to the other three. What is the smallest number of points which
are joined to 1981 points? | import Mathlib
/-!
# USA Mathematical Olympiad 1982, Problem 1
A graph has 1982 points. Given any four points, there is at least one
joined to the other three. What is the smallest number of points which
are joined to 1981 points?
-/
namespace Usa1982P1
open Classical
/-- The hypothesis of the problem: among any f... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1982, Problem 1 | null |
compfiles-Usa1983P3 | S₁, S₂, ..., Sₙ are subsets of the real line. Each Sᵢ is the union of two
closed intervals. Any three Sᵢ have a point in common. Show that there is a
point which belongs to at least half the Sᵢ. | import Mathlib
/-!
# USA Mathematical Olympiad 1983, Problem 3
S₁, S₂, ..., Sₙ are subsets of the real line. Each Sᵢ is the union of two
closed intervals. Any three Sᵢ have a point in common. Show that there is a
point which belongs to at least half the Sᵢ.
-/
namespace Usa1983P3
theorem usa1983_p3 {n : ℕ} (hn : 0... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1983, Problem 3 | null |
compfiles-Usa1984P4 | A maths exam has two papers, each with at least one question and 28 questions
in total. Each pupil attempted 7 questions. Each pair of questions was
attempted by just two pupils. Show that one pupil attempted either nil or at
least 4 questions in the first paper. | import Mathlib
/-!
# USA Mathematical Olympiad 1984, Problem 4
A maths exam has two papers, each with at least one question and 28 questions
in total. Each pupil attempted 7 questions. Each pair of questions was
attempted by just two pupils. Show that one pupil attempted either nil or at
least 4 questions in the firs... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1984, Problem 4 | null |
compfiles-Usa1985P4 | A graph has n > 2 points. Show that we can find two points A and B such that
at least ⌊n/2⌋ - 1 of the remaining points are joined to either both or
neither of A and B. | import Mathlib
/-!
# USA Mathematical Olympiad 1985, Problem 4
A graph has n > 2 points. Show that we can find two points A and B such that
at least ⌊n/2⌋ - 1 of the remaining points are joined to either both or
neither of A and B.
-/
namespace Usa1985P4
open Finset
theorem usa1985_p4 (n : ℕ) (hn : 2 < n) (G : Si... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1985, Problem 4 | null |
compfiles-Usa1985P5 | 0 < a₁ ≤ a₂ ≤ a₃ ≤ ⋯ is an unbounded sequence of integers. Let bₙ = m if aₘ is the
first member of the sequence to equal or exceed n. Given that a₁₉ = 85, what is the
maximum possible value of a₁ + a₂ + ⋯ + a₁₉ + b₁ + b₂ + ⋯ + b₈₅? | import Mathlib
/-!
# USA Mathematical Olympiad 1985, Problem 5
0 < a₁ ≤ a₂ ≤ a₃ ≤ ⋯ is an unbounded sequence of integers. Let bₙ = m if aₘ is the
first member of the sequence to equal or exceed n. Given that a₁₉ = 85, what is the
maximum possible value of a₁ + a₂ + ⋯ + a₁₉ + b₁ + b₂ + ⋯ + b₈₅?
-/
namespace Usa1985P5... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1985, Problem 5 | null |
compfiles-Usa1986P2 | Five professors attended a lecture. Each fell asleep just twice. For each pair
there was a moment when both were asleep. Show that there was a moment when
three of them were asleep. | import Mathlib
/-!
# USA Mathematical Olympiad 1986, Problem 2
Five professors attended a lecture. Each fell asleep just twice. For each pair
there was a moment when both were asleep. Show that there was a moment when
three of them were asleep.
-/
namespace Usa1986P2
/-- Each professor `p : Fin 5` takes two naps, a... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1986, Problem 2 | null |
compfiles-Usa1986P5 | A partition of n is an increasing sequence of integers with sum n.
For example, the partitions of 5 are: 1, 1, 1, 1, 1; 1, 1, 1, 2; 1, 1, 3;
1, 4; 5; 1, 2, 2; and 2, 3. If p is a partition, f(p) = the number of 1s
in p, and g(p) = the number of distinct integers in the partition.
Show that ∑ f(p) = ∑ g(p), where the su... | import Mathlib
/-!
# USA Mathematical Olympiad 1986, Problem 5
A partition of n is an increasing sequence of integers with sum n.
For example, the partitions of 5 are: 1, 1, 1, 1, 1; 1, 1, 1, 2; 1, 1, 3;
1, 4; 5; 1, 2, 2; and 2, 3. If p is a partition, f(p) = the number of 1s
in p, and g(p) = the number of distinct i... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1986, Problem 5 | null |
compfiles-Usa1988P3 | Let X be the set {1, 2, ... , 20} and let P be the set of all 9-element
subsets of X. Show that for any map f : P → X we can find a 10-element
subset Y of X, such that f(Y - {k}) ≠ k for any k in Y. | import Mathlib
/-!
# USA Mathematical Olympiad 1988, Problem 3
Let X be the set {1, 2, ... , 20} and let P be the set of all 9-element
subsets of X. Show that for any map f : P → X we can find a 10-element
subset Y of X, such that f(Y - {k}) ≠ k for any k in Y.
-/
namespace Usa1988P3
open Finset
theorem usa1988_p... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1988, Problem 3 | null |
compfiles-Usa1989P2 | The 20 members of a local tennis club have scheduled exactly 14 two-person
games among themselves, with each member playing in at least one game.
Prove that within this schedule can be found a set of six games with
12 distinct players. | import Mathlib
/-!
# USA Mathematical Olympiad 1989, Problem 2
The 20 members of a local tennis club have scheduled exactly 14 two-person
games among themselves, with each member playing in at least one game.
Prove that within this schedule can be found a set of six games with
12 distinct players.
-/
namespace Usa19... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1989, Problem 2 | null |
compfiles-Usa1990P1 | A license plate has six digits from 0 to 9 and may have leading zeros.
If two plates must always differ in at least two places, what is the
largest number of plates that is possible? | import Mathlib
/-!
# USA Mathematical Olympiad 1990, Problem 1
A license plate has six digits from 0 to 9 and may have leading zeros.
If two plates must always differ in at least two places, what is the
largest number of plates that is possible?
-/
namespace Usa1990P1
/-- A license plate: six digits, each from 0 to... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1990, Problem 1 | null |
compfiles-Usa1990P4 | How many positive integers can be written in base n so that
(1) the integer has no two digits the same, and
(2) each digit after the first differs by one from an earlier digit?
For example, in base 3, the possible numbers are 1, 2, 10, 12, 21, 102, 120, 210. | import Mathlib
/-!
# USA Mathematical Olympiad 1990, Problem 4
How many positive integers can be written in base n so that
(1) the integer has no two digits the same, and
(2) each digit after the first differs by one from an earlier digit?
For example, in base 3, the possible numbers are 1, 2, 10, 12, 21, 102, 120, 2... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1990, Problem 4 | null |
compfiles-Usa1992P3 | A set of 11 distinct positive integers has the property that we can find a
subset with sum n for any n between 1 and 1500 inclusive. What is the smallest
possible value for the second largest element? | import Mathlib
/-!
# USA Mathematical Olympiad 1992, Problem 3
A set of 11 distinct positive integers has the property that we can find a
subset with sum n for any n between 1 and 1500 inclusive. What is the smallest
possible value for the second largest element?
-/
namespace Usa1992P3
/--
Formalization of "a set ... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1992, Problem 3 | null |
compfiles-Usa1994P2 | The sequence a₁, a₂, ... , a₉₉ has a₁ = a₃ = a₅ = ... = a₉₇ = 1,
a₂ = a₄ = a₆ = ... = a₉₈ = 2, and a₉₉ = 3. We interpret subscripts greater
than 99 by subtracting 99, so that a₁₀₀ means a₁ etc. An allowed move is to
change the value of any one of the aₙ to another member of {1, 2, 3}
different from its two neighbors, a... | import Mathlib
/-!
# USA Mathematical Olympiad 1994, Problem 2
The sequence a₁, a₂, ... , a₉₉ has a₁ = a₃ = a₅ = ... = a₉₇ = 1,
a₂ = a₄ = a₆ = ... = a₉₈ = 2, and a₉₉ = 3. We interpret subscripts greater
than 99 by subtracting 99, so that a₁₀₀ means a₁ etc. An allowed move is to
change the value of any one of the aₙ t... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1994, Problem 2 | null |
compfiles-Usa1994P5 | Let |U|, σ(U) and π(U) denote the number of elements, the sum, and the
product, respectively, of a finite set U of positive integers. (If U is the
empty set, |U| = 0, σ(U) = 0, π(U) = 1.) Let S be a finite set of positive
integers. As usual, we define (n k) = n! / (k! (n-k)!) for 0 ≤ k ≤ n and
(n k) = 0 otherwise. Prov... | import Mathlib
/-!
# USA Mathematical Olympiad 1994, Problem 5
Let |U|, σ(U) and π(U) denote the number of elements, the sum, and the
product, respectively, of a finite set U of positive integers. (If U is the
empty set, |U| = 0, σ(U) = 0, π(U) = 1.) Let S be a finite set of positive
integers. As usual, we define (n ... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1994, Problem 5 | null |
compfiles-Usa1995P5 | A graph with n points and k edges has no triangles. Show that it has a
point P such that there are at most k(1 - 4k/n²) edges between points
not joined to P (by an edge). | import Mathlib
/-!
# USA Mathematical Olympiad 1995, Problem 5
A graph with n points and k edges has no triangles. Show that it has a
point P such that there are at most k(1 - 4k/n²) edges between points
not joined to P (by an edge).
-/
namespace Usa1995P5
open Finset SimpleGraph
variable {n : ℕ} (G : SimpleGraph ... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1995, Problem 5 | null |
compfiles-Usa1996P2 | Let S be a set of n positive integers. Let P be the set of all integers which
are the sum of one or more distinct elements of S. Show that we can find n
subsets of P whose union is P such that if a, b belong to the same subset,
then a ≤ 2b. | import Mathlib
/-!
# USA Mathematical Olympiad 1996, Problem 2
Let S be a set of n positive integers. Let P be the set of all integers which
are the sum of one or more distinct elements of S. Show that we can find n
subsets of P whose union is P such that if a, b belong to the same subset,
then a ≤ 2b.
-/
namespace ... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1996, Problem 2 | null |
compfiles-Usa1996P4 | An n-term sequence (x₁, x₂, …, xₙ) in which each term is either 0 or 1 is called a
binary sequence of length n. Let aₙ be the number of binary sequences of length n
containing no three consecutive terms equal to 0, 1, 0 in that order. Let bₙ be the
number of binary sequences of length n that contain no four consecutive... | import Mathlib
/-!
# USA Mathematical Olympiad 1996, Problem 4
An n-term sequence (x₁, x₂, …, xₙ) in which each term is either 0 or 1 is called a
binary sequence of length n. Let aₙ be the number of binary sequences of length n
containing no three consecutive terms equal to 0, 1, 0 in that order. Let bₙ be the
number... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1996, Problem 4 | null |
compfiles-Usa1996P6 | Determine, with proof, whether there is a subset X of the integers with
the following property: for any integer n there is exactly one pair (a, b)
of elements of X satisfying a + 2b = n. | import Mathlib
/-!
# USA Mathematical Olympiad 1996, Problem 6
Determine, with proof, whether there is a subset X of the integers with
the following property: for any integer n there is exactly one pair (a, b)
of elements of X satisfying a + 2b = n.
-/
namespace Usa1996P6
def UniqueRepresentationSet (X : Set ℤ) : P... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1996, Problem 6 | null |
compfiles-Usa1998P4 | A computer screen shows a 98 × 98 chessboard, colored in the usual way.
One can select with a mouse any rectangle with sides on the lines of the
chessboard and click the mouse button: as a result, the colors in the
selected rectangle switch (black becomes white, white becomes black).
Find, with proof, the minimum numbe... | import Mathlib
/-!
# USA Mathematical Olympiad 1998, Problem 4
A computer screen shows a 98 × 98 chessboard, colored in the usual way.
One can select with a mouse any rectangle with sides on the lines of the
chessboard and click the mouse button: as a result, the colors in the
selected rectangle switch (black becomes... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 1998, Problem 4 | null |
compfiles-Usa1999P1 | null | import Mathlib
/-!
USA Mathematical Olympiad 1999, Problem 1
Some checkers placed on an n × n checkerboard satisfy the following conditions:
a. every square that does not contain a checker shares a side with one that does;
b. given any pair of squares that contain checkers, there is a sequence of squares
contai... | Combinatorics | 1. Named contest | null | Compfiles: Usa1999P1 | null |
compfiles-Usa2000P3 | A player starts with A blue cards, B red cards and C white cards. He scores
points as he plays each card. If he plays a blue card, his score is the number
of white cards remaining in his hand. If he plays a red card it is three times
the number of blue cards remaining in his hand. If he plays a white card, it is
twice ... | import Mathlib
/-!
# USA Mathematical Olympiad 2000, Problem 3
A player starts with A blue cards, B red cards and C white cards. He scores
points as he plays each card. If he plays a blue card, his score is the number
of white cards remaining in his hand. If he plays a red card it is three times
the number of blue ca... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2000, Problem 3 | null |
compfiles-Usa2001P1 | Each of eight boxes contains six balls.
Each ball has been colored with one of n colors, such that no two balls
in the same box are the same color, and no two colors occur together in
more than one box. Determine, with justification, the smallest integer n
for which this is possible. | import Mathlib
/-!
# USA Mathematical Olympiad 2001, Problem 1
Each of eight boxes contains six balls.
Each ball has been colored with one of n colors, such that no two balls
in the same box are the same color, and no two colors occur together in
more than one box. Determine, with justification, the smallest integer ... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2001, Problem 1 | null |
compfiles-Usa2002P1 | Let S be a set with 2002 elements, and let N be an integer with
0 ≤ N ≤ 2^2002.
Prove that it is possible to color every subset of S either blue or
red so that the following conditions hold:
a) the union of any two red subsets is red;
b) the union of any two blue subsets is blue;
c) there are exactly N red subs... | import Mathlib
/-!
# USA Mathematical Olympiad 2002, Problem 1
Let S be a set with 2002 elements, and let N be an integer with
0 ≤ N ≤ 2^2002.
Prove that it is possible to color every subset of S either blue or
red so that the following conditions hold:
a) the union of any two red subsets is red;
b) the union ... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2002, Problem 1 | null |
compfiles-Usa2002P6 | I have an n×n sheet of stamps, from which I've been asked to tear out blocks
of three adjacent stamps in a single row or column. (I can only tear along the
perforations separating adjacent stamps, and each block must come out of the
sheet in one piece.) Let b(n) be the smallest number of blocks I can tear out
and make ... | import Mathlib
/-!
# USA Mathematical Olympiad 2002, Problem 6
I have an n×n sheet of stamps, from which I've been asked to tear out blocks
of three adjacent stamps in a single row or column. (I can only tear along the
perforations separating adjacent stamps, and each block must come out of the
sheet in one piece.) L... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2002, Problem 6 | null |
compfiles-Usa2003P3 | Let n be a positive integer. For every sequence of integers
A = (a₀, a₁, a₂, . . . , aₙ) satisfying 0 ≤ aᵢ ≤ i, for i = 0, . . . , n,
we define another sequence t(A) = (t(a₀), t(a₁), t(a₂), . . . , t(aₙ))
by setting t(aᵢ) to be the number of terms in the sequence A that precede
the term aᵢ and are different from aᵢ. Sh... | import Mathlib
/-!
# USA Mathematical Olympiad 2003, Problem 3
Let n be a positive integer. For every sequence of integers
A = (a₀, a₁, a₂, . . . , aₙ) satisfying 0 ≤ aᵢ ≤ i, for i = 0, . . . , n,
we define another sequence t(A) = (t(a₀), t(a₁), t(a₂), . . . , t(aₙ))
by setting t(aᵢ) to be the number of terms in the ... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2003, Problem 3 | null |
compfiles-Usa2003P6 | At the vertices of a regular hexagon are written six nonnegative integers
whose sum is 2003^2003. Bert is allowed to make moves of the following form:
he may pick a vertex and replace the number written there by the absolute
value of the difference between the numbers written at the two neighboring
vertices. Prove that... | import Mathlib
/-!
# USA Mathematical Olympiad 2003, Problem 6
At the vertices of a regular hexagon are written six nonnegative integers
whose sum is 2003^2003. Bert is allowed to make moves of the following form:
he may pick a vertex and replace the number written there by the absolute
value of the difference betwee... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2003, Problem 6 | null |
compfiles-Usa2004P4 | Alice and Bob play a game on a 6 by 6 grid. On his turn, a player chooses
a rational number not yet appearing in the grid and writes it in an empty square
of the grid. Alice goes first and then the players alternate. When all squares
have numbers written in them, in each row, the square with the greatest number in
that... | import Mathlib
/-!
# USA Mathematical Olympiad 2004, Problem 4
Alice and Bob play a game on a 6 by 6 grid. On his turn, a player chooses
a rational number not yet appearing in the grid and writes it in an empty square
of the grid. Alice goes first and then the players alternate. When all squares
have numbers written ... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2004, Problem 4 | null |
compfiles-Usa2005P4 | Legs L₁, L₂, L₃, L₄ of a square table each have length n, where n is a positive
integer. For how many ordered 4-tuples (k₁, k₂, k₃, k₄) of nonnegative integers
can we cut a piece of length kᵢ from the end of leg Lᵢ (i = 1, 2, 3, 4) and
still have a stable table? (The table is stable if it can be placed so that
all four... | import Mathlib
/-!
# USA Mathematical Olympiad 2005, Problem 4
Legs L₁, L₂, L₃, L₄ of a square table each have length n, where n is a positive
integer. For how many ordered 4-tuples (k₁, k₂, k₃, k₄) of nonnegative integers
can we cut a piece of length kᵢ from the end of leg Lᵢ (i = 1, 2, 3, 4) and
still have a stable... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2005, Problem 4 | null |
compfiles-Usa2005P6 | For a positive integer m, let s(m) denote the sum of the decimal digits of m.
A set S of positive integers is k-stable if s(∑_{x∈X} x) = k for any nonempty
subset X ⊆ S. For each integer n ≥ 2 let f(n) be the minimal k for which there
exists a k-stable set with n integers. Prove that there are constants
0 < C₁ < C₂ wit... | import Mathlib
/-!
# USA Mathematical Olympiad 2005, Problem 6
For a positive integer m, let s(m) denote the sum of the decimal digits of m.
A set S of positive integers is k-stable if s(∑_{x∈X} x) = k for any nonempty
subset X ⊆ S. For each integer n ≥ 2 let f(n) be the minimal k for which there
exists a k-stable se... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2005, Problem 6 | null |
compfiles-Usa2006P2 | For a given positive integer k find, in terms of k, the minimum value
of N for which there is a set of 2k + 1 distinct positive integers
that has sum greater than N but every subset of size k has sum at
most N / 2. | import Mathlib
/-!
# USA Mathematical Olympiad 2006, Problem 2
For a given positive integer k find, in terms of k, the minimum value
of N for which there is a set of 2k + 1 distinct positive integers
that has sum greater than N but every subset of size k has sum at
most N / 2.
-/
namespace Usa2006P2
abbrev solution... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2006, Problem 2 | null |
compfiles-Usa2006P5 | A mathematical frog jumps along the number line. The frog starts at 1,
and jumps according to the following rule: if the frog is at integer n,
then it can jump either to n + 1 or to n + 2 ^ (mₙ + 1), where 2 ^ mₙ is
the largest power of 2 that is a factor of n. Show that if k ≥ 2 is a
positive integer and i is a nonneg... | import Mathlib
/-!
# USA Mathematical Olympiad 2006, Problem 5
A mathematical frog jumps along the number line. The frog starts at 1,
and jumps according to the following rule: if the frog is at integer n,
then it can jump either to n + 1 or to n + 2 ^ (mₙ + 1), where 2 ^ mₙ is
the largest power of 2 that is a factor... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2006, Problem 5 | null |
compfiles-Usa2007P3 | Let S be a set containing n² + n − 1 elements. Suppose that the n-element
subsets of S are partitioned into two classes. Prove that there are at least
n pairwise disjoint sets in the same class. | import Mathlib
/-!
# USA Mathematical Olympiad 2007, Problem 3
Let S be a set containing n² + n − 1 elements. Suppose that the n-element
subsets of S are partitioned into two classes. Prove that there are at least
n pairwise disjoint sets in the same class.
-/
namespace Usa2007P3
theorem usa2007_p3 (n : ℕ) (hn : 0... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2007, Problem 3 | null |
compfiles-Usa2008P3 | Let n be a positive integer. Denote by Sₙ the set of points (x, y)
with integer coordinates such that
|x| + |y + 1/2| < n.
A path is a sequence of distinct points (x₁, y₁), (x₂, y₂), ..., (x_ℓ, y_ℓ)
in Sₙ such that, for i = 2, ..., ℓ, the distance between (xᵢ, yᵢ) and
(xᵢ₋₁, yᵢ₋₁) is 1. Prove that the points in Sₙ ... | import Mathlib
/-!
# USA Mathematical Olympiad 2008, Problem 3
Let n be a positive integer. Denote by Sₙ the set of points (x, y)
with integer coordinates such that
|x| + |y + 1/2| < n.
A path is a sequence of distinct points (x₁, y₁), (x₂, y₂), ..., (x_ℓ, y_ℓ)
in Sₙ such that, for i = 2, ..., ℓ, the distance bet... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2008, Problem 3 | null |
compfiles-Usa2008P5 | Three nonnegative real numbers r₁, r₂, r₃ are written on a blackboard.
These numbers have the property that there exist integers a₁, a₂, a₃,
not all zero, satisfying a₁r₁ + a₂r₂ + a₃r₃ = 0. We are permitted to
perform the following operation: find two numbers x, y on the
blackboard with x ≤ y, then erase y and write y ... | import Mathlib
/-!
# USA Mathematical Olympiad 2008, Problem 5
Three nonnegative real numbers r₁, r₂, r₃ are written on a blackboard.
These numbers have the property that there exist integers a₁, a₂, a₃,
not all zero, satisfying a₁r₁ + a₂r₂ + a₃r₃ = 0. We are permitted to
perform the following operation: find two num... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2008, Problem 5 | null |
compfiles-Usa2008P6 | At a certain mathematical conference, every pair of mathematicians are
either friends or strangers. At mealtime, every participant eats in one of
two large dining rooms. Each mathematician insists upon eating in a room
which contains an even number of his or her friends. Prove that the number
of ways that the mathemati... | import Mathlib
/-!
# USA Mathematical Olympiad 2008, Problem 6
At a certain mathematical conference, every pair of mathematicians are
either friends or strangers. At mealtime, every participant eats in one of
two large dining rooms. Each mathematician insists upon eating in a room
which contains an even number of his... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2008, Problem 6 | null |
compfiles-Usa2009P2 | Let n be a positive integer. Determine the size of the largest subset of
{−n, −n+1, ..., n−1, n} which does not contain three elements a, b, c
(not necessarily distinct) satisfying a + b + c = 0. | import Mathlib
/-!
# USA Mathematical Olympiad 2009, Problem 2
Let n be a positive integer. Determine the size of the largest subset of
{−n, −n+1, ..., n−1, n} which does not contain three elements a, b, c
(not necessarily distinct) satisfying a + b + c = 0.
-/
namespace Usa2009P2
/-- A set of integers is called *g... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2009, Problem 2 | null |
compfiles-Usa2010P2 | There are $n$ students standing in a circle, one behind the other. The students
have heights $h_1 < h_2 < \dots < h_n$. If a student with height $h_k$ is
standing directly behind a student with height $h_{k-2}$ or less, the two
students are permitted to switch places. Prove that it is not possible to make
more than $\b... | import Mathlib
/-!
# USA Mathematical Olympiad 2010, Problem 2
There are $n$ students standing in a circle, one behind the other. The students
have heights $h_1 < h_2 < \dots < h_n$. If a student with height $h_k$ is
standing directly behind a student with height $h_{k-2}$ or less, the two
students are permitted to s... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2010, Problem 2 | null |
compfiles-Usa2010P6 | There are 68 ordered pairs (not necessarily distinct) of nonzero integers
on a blackboard. It's known that for no integer k does both (k, k) and
(−k, −k) appear. A student erases some of the 136 integers such that no two
erased integers have sum zero, and scores one point for each ordered pair
with at least one erased ... | import Mathlib
/-!
# USA Mathematical Olympiad 2010, Problem 6
There are 68 ordered pairs (not necessarily distinct) of nonzero integers
on a blackboard. It's known that for no integer k does both (k, k) and
(−k, −k) appear. A student erases some of the 136 integers such that no two
erased integers have sum zero, and... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2010, Problem 6 | null |
compfiles-Usa2011P2 | An integer is assigned to each vertex of a regular pentagon so that the sum of
the five integers is 2011. A turn of a solitaire game consists of subtracting
an integer m (not necessarily positive) from each of the integers at two
neighboring vertices and adding 2m to the opposite vertex, which is not
adjacent to either... | import Mathlib
/-!
# USA Mathematical Olympiad 2011, Problem 2
An integer is assigned to each vertex of a regular pentagon so that the sum of
the five integers is 2011. A turn of a solitaire game consists of subtracting
an integer m (not necessarily positive) from each of the integers at two
neighboring vertices and ... | Combinatorics | 1. Named contest | null | Compfiles: USA Mathematical Olympiad 2011, Problem 2 | null |
End of preview. Expand in Data Studio
Olympiad Lean 4 Geometry and Combinatorics
Two problem sets of Lean 4 theorem statements for training provers on olympiad-level mathematics:
| File | Problems |
|---|---|
geometry.csv |
2,500 |
combinatorics.csv |
2,500 |
Each row is an informal problem paired with a Lean 4 statement whose proof is sorry. No proofs are included. The
sets are meant for reinforcement learning or trace collection, where a model writes the proof and Lean checks it.
Every statement compiles (with only the expected sorry warning) on Lean 4.27.0 with Mathlib. Rows are ordered
hardest first, in four tiers.
Tiers
| Tier | Geometry | Combinatorics | What it is |
|---|---|---|---|
| 1. Named contest | 8 | 114 | Expert formalizations of IMO, USAMO and national olympiad problems from Compfiles |
| 2. Difficulty 8–10 | 409 | 678 | FineLeanCorpus problems rated 8–10 |
| 3. Difficulty 7 | 2,065 | 1,663 | FineLeanCorpus problems rated 7 |
| 4. Difficulty 6 | 18 | 45 | FineLeanCorpus problems rated 6, used only to reach 2,500 |
Only tier 1 is certified contest material. Tiers 2–4 are autoformalized problems whose difficulty was rated by a model.
Calibrating the difficulty scale. FineLeanCorpus rates problems from 1 to 10. We compared those ratings with Omni-MATH's AoPS-standard ratings on the 129 problems the two datasets share:
| FineLeanCorpus rating | Mean Omni-MATH rating |
|---|---|
| 6 | 6.3 |
| 7 | 6.9 |
Shared problems that came from the IMO were rated 6–7. So 6 is roughly olympiad level, and 7 or above is roughly IMO level.
Columns
The columns match IMO-LeanProofBench, so existing loaders work unchanged. Three columns are added.
| Column | Meaning |
|---|---|
Problem ID |
compfiles-<File> or flc-<FineLeanCorpus id> |
Problem |
Informal problem statement |
Lean Statement |
Complete Lean 4 file: import Mathlib, then definitions and the theorem ending in sorry |
Category |
Geometry or Combinatorics (combinatorics includes FineLeanCorpus "Discrete Mathematics") |
Level |
Tier name, as in the table above |
Difficulty |
FineLeanCorpus rating, 1–10; blank for Compfiles rows |
Source |
Compfiles: <contest and problem> or FineLeanCorpus/<original source> |
Formalization |
Geometry only: synthetic, coordinates or real-arithmetic (see Known issues) |
How it was built
The build script, build_geo_comb.py, is included in this repository.
Candidates
Compfiles (commit
1745e6c): every problem file tagged.Geometryor.Combinatorics, 183 files. Each file was converted to a plain statement:- proof blocks (
snip begin … snip end) removed; problemchanged totheorem … := by sorry;determinechanged toabbrev, so the stated answer stays in the statement.
- proof blocks (
FineLeanCorpus:
- rows rated 6 or above with a Geometry, Combinatorics or Discrete Mathematics domain label;
- only rows whose recorded compile result had no errors;
- duplicates removed by normalized statement text.
This left about 14,700 candidates. A row with both labels was assigned by its first domain label.
Held-out benchmarks removed. Anything overlapping these benchmarks was dropped, so they can still be used for evaluation:
- IMO-ProofBench Basic and Advanced;
- CombiBench, commit
c67e421; - LeanGeo-Bench, commit
9212b89.
A candidate counted as overlapping if either:
- its contest, year and problem number matched (for example, IMO 2017 P5); or
- 50% or more of its 8-word sequences matched a benchmark problem's text.
This removed 40 candidates, including Compfiles' IMO combinatorics problems from 2000 on, which CombiBench covers.
Recompiled on Lean 4.27.0 with Mathlib.
- Compfiles targets Lean 4.35, and FineLeanCorpus was checked against an older Mathlib, so every candidate was compiled again, one file per statement.
- A statement passed only if it compiled with no errors and Lean reported
declaration uses 'sorry'. - Candidates were checked in priority order until each category had more than 2,500 passes. That took 5,662 checks; 92% of geometry and 89% of combinatorics candidates passed.
- The most common failure was
unexpected token 'in', from older Lean syntax.
Ordering. Rows are sorted by tier, then by difficulty, highest first. Within a tier, geometry statements in the
real-arithmeticstyle come last. The first 2,500 of each category were kept.
Source mix:
| Source | Geometry | Combinatorics |
|---|---|---|
| FineLeanCorpus: AoPS | 1,961 | 1,731 |
| FineLeanCorpus: DeepTheorem | 164 | 211 |
| FineLeanCorpus: DeepMath-103k | 136 | 145 |
| Compfiles | 8 | 113 |
| Other FineLeanCorpus sources (DeepScaleR, DAPO, BlueMO, NuminaMath-TIR, Multi-Source competitions) | 231 | 300 |
Known issues
Read these before training on the data.
Faithfulness is not verified.
- FineLeanCorpus statements are autoformalized. Its authors report 79% accuracy from manual sampling of the full corpus, and geometry is likely lower.
- Spot checks found statements that compile but don't match the problem. In one, the Lean builds a set whose only member is the claimed answer, then asks to prove that answer is the maximum, which is trivially true.
- Screen statements for faithfulness and triviality before use.
Topic labels are loose. Some "Geometry" rows are really calculus or probability over Euclidean space, such as an expected value over a random point in a cone.
Geometry formalization style varies:
Style Rows How the geometry is written synthetic902 Mathlib geometry objects such as EuclideanSpace,∠ordist, the style IMO-LeanProofBench usescoordinates307 Points as ℝ × ℝpairs or complex numbersreal-arithmetic1,291 No geometric objects: the problem is restated as equations over side lengths or angles, often a lossy translation The style labels come from a simple text-pattern match and are approximate.
Synthetic geometry is hard in Mathlib. Mathlib has little support for olympiad geometry: incenters, tangency, touchpoints and similar. Provers tend to stall building that support themselves, so consider supplying a library of proved helper lemmas.
No reference proofs. Difficulty ratings are a model's estimate, not measured prover success rates.
Tier 1 is small, especially for geometry, because Compfiles has few geometry problems.
Licenses and credits
Released under Apache-2.0, the license of both upstream sources. Please cite them:
- FineLeanCorpus, Apache-2.0. Peng et al., CriticLean: Critic-Guided Reinforcement Learning for Mathematical Formalization, 2025. https://huggingface.co/datasets/m-a-p/FineLeanCorpus
- Compfiles, Apache-2.0. David Renshaw and contributors. https://github.com/dwrensha/compfiles
- Omni-MATH was used only to calibrate difficulty and to find contest names for 1 row.
Decontamination used IMO-ProofBench, CombiBench and LeanGeo-Bench; none of their problems are included.
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