Olympiad problems

Azerbaijan Al-Khwarizmi IJMO TST 2025, 4

Tags: combinatorics

The numbers $\frac{50}{1},\frac{50}{2},...\frac{50}{97},\frac{50}{98}$ are written on the board.In each step,two random numbers $a$ and $b$ are chosen and deleted.Then,the number $2ab-a-b+1$ is written instead.What will be the number remained on the board after the last step.

2005 Indonesia MO, 1

Tags: integration , calculus , inequalities , combinatorics

Let $ n$ be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is $ n$.

2022 Latvia Baltic Way TST, P6

Tags: combinatorics

The numbers $1,2,3,\ldots ,n$ are written in a row. Two players, Maris and Filips, take turns making moves with Maris starting. A move consists of crossing out a number from the row which has not yet been crossed out. The game ends when there are exactly two uncrossed numbers left in the row. If the two remaining uncrossed numbers are coprime, Maris wins, otherwise Filips is the winner. For each positive integer $n\ge 4$ determine which player can guarantee a win.

2009 Thailand Mathematical Olympiad, 3

Tags: combinatorics

Teeradet is a student in a class with $19$ people. He and his classmates form clubs, so that each club must have at least one student, and each student can be in more than one club. Suppose that any two clubs differ by at least one student, and all clubs Teeradet is in have an odd number of students. What is the maximum possible number of clubs?

2005 Poland - Second Round, 3

Tags: combinatorics , graph theory , extremal combinatorics , inequalities

In space are given $n\ge 2$ points, no four of which are coplanar. Some of these points are connected by segments. Let $K$ be the number of segments $(K>1)$ and $T$ be the number of formed triangles. Prove that $9T^2<2K^3$.

2024 Miklos Schweitzer, 4

Tags: combinatorics , permutation

Let $\pi$ be a given permutation of the set $\{1, 2, \dots, n\}$. Determine the smallest possible value of \[ \sum_{i=1}^n |\pi(i) - \sigma(i)|, \] where $\sigma$ is a permutation chosen from the set of all $n$-cycles. Express the result in terms of the number and lengths of the cycles in the disjoint cycle decomposition of $\pi$, including the fixed points.

1990 IMO Longlists, 63

Tags: tetrahedron , combinatorics , 3d geometry , geometry

Let $ P$ be a point inside a regular tetrahedron $ T$ of unit volume. The four planes passing through $ P$ and parallel to the faces of $ T$ partition $ T$ into 14 pieces. Let $ f(P)$ be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for $ f(P)$ as $ P$ varies over $ T.$

2016 Iran Team Selection Test, 2

Tags: combinatorics , greatest common divisor

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

1977 IMO Longlists, 16

Tags: combinatorics , counting , inequality system , algebra

Let $n$ be a positive integer. How many integer solutions $(i, j, k, l) , \ 1 \leq i, j, k, l \leq n$, does the following system of inequalities have: \[1 \leq -j + k + l \leq n\]\[1 \leq i - k + l \leq n\]\[1 \leq i - j + l \leq n\]\[1 \leq i + j - k \leq n \ ?\]

2001 Estonia National Olympiad, 5

Tags: combinatorics , magic square , difference , sum , square table

A $3\times 3$ table is filled with real numbers in such a way that each number in the table is equal to the absolute value of the difference of the sum of numbers in its row and the sum of numbers in its column. (a) Show that any number in this table can be expressed as a sum or a difference of some two numbers in the table. (b) Show that there is such a table not all of whose entries are $0$.

1974 IMO Longlists, 52

Tags: combinatorics , function , analytic geometry , trigonometry

A fox stands in the centre of the field which has the form of an equilateral triangle, and a rabbit stands at one of its vertices. The fox can move through the whole field, while the rabbit can move only along the border of the field. The maximal speeds of the fox and rabbit are equal to $u$ and $v$, respectively. Prove that: (a) If $2u>v$, the fox can catch the rabbit, no matter how the rabbit moves. (b) If $2u\le v$, the rabbit can always run away from the fox.

2000 Estonia National Olympiad, 1

Tags: combinatorics

There are three candidates in the Hundilaane forest governor elections: $A, B$ and $C$. For each of the $20$ forest dwellers, the names of all three candidates were written on the ballot paper in the order of their preference. Examination of the ballots revealed that $11$ forest dwellers prefer $A$, $12$ $B$ and $14$ $C$. Which of the candidates will be marked first on the largest number of ballot papers when it is known that each possible the order of the candidates appears on at least one ballot?

2013 Romania Team Selection Test, 3

Tags: combinatorics

Determine the largest natural number $r$ with the property that among any five subsets with $500$ elements of the set $\{1,2,\ldots,1000\}$ there exist two of them which share at least $r$ elements.

2001 Croatia Team Selection Test, 1

Tags: combinatorics , partition , subset

Consider $A = \{1, 2, ..., 16\}$. A partition of $A$ into nonempty sets $A_1, A_2,..., A_n$ is said to be good if none of the Ai contains elements $a, b, c$ (not necessarily distinct) such that $a = b + c$. (a) Find a good partition $\{A_1, A_2, A_3, A_4\}$ of $A$. (b) Prove that no partition $\{A_1, A_2, A_3\}$ of $A$ is good

2021 Saint Petersburg Mathematical Olympiad, 4

Tags: combinatorics

Stierlitz wants to send an encryption to the Center, which is a code containing $100$ characters, each a "dot" or a "dash". The instruction he received from the Center the day before about conspiracy reads: i) when transmitting encryption over the radio, exactly $49$ characters should be replaced with their opposites; ii) the location of the "wrong" characters is decided by the transmitting side and the Center is not informed of it. Prove that Stierlitz can send $10$ encryptions, each time choosing some $49$ characters to flip, such that when the Center receives these $10$ ciphers, it may unambiguously restore the original code.

2010 Contests, 1

Tags: combinatorial geometry , combinatorics , point , collinear

$3n$ points are given ($n\ge 1$) in the plane, each $3$ of them are not collinear. Prove that there are $n$ distinct triangles with the vertices those points.

2022 LMT Spring, 5

Tags: combinatorics

A bag contains $5$ identical blue marbles and $5$ identical green marbles. In how many ways can $5$ marbles from the bag be arranged in a row if each blue marble must be adjacent to at least $1$ green marble?

1977 All Soviet Union Mathematical Olympiad, 236

Tags: combinatorial geometry , combinatorics , algebra

Given several points, not all lying on one straight line. Some number is assigned to every point. It is known, that if a straight line contains two or more points, than the sum of the assigned to those points equals zero. Prove that all the numbers equal to zero.

2020 Argentina National Olympiad Level 2, 1

Tags: combinatorics

Fede chooses $50$ distinct integers from the set $\{1, 2, 3, \ldots, 100\}$ such that their sum equals $2900$. Determine the minimum number of even numbers that can be among the $50$ numbers chosen by Fede.

2004 Germany Team Selection Test, 3

Tags: function , counting , decimal representation , combinatorics , modular arithmetic

Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

1974 IMO, 4

Tags: combinatorics , rectangle , chessboard , dissection

Consider decompositions of an $8\times 8$ chessboard into $p$ non-overlapping rectangles subject to the following conditions: (i) Each rectangle has as many white squares as black squares. (ii) If $a_i$ is the number of white squares in the $i$-th rectangle, then $a_1<a_2<\ldots <a_p$. Find the maximum value of $p$ for which such a decomposition is possible. For this value of $p$, determine all possible sequences $a_1,a_2,\ldots ,a_p$.

1998 Kurschak Competition, 3

Tags: combinatorial geometry , combinatorics

For which integers $N\ge 3$ can we find $N$ points on the plane such that no three are collinear, and for any triangle formed by three vertices of the points’ convex hull, there is exactly one point within that triangle?

Russian TST 2020, P2

Tags: combinatorics , graph theory

There are 10,000 vertices in a graph, with at least one edge coming out of each vertex. Call a set $S{}$ of vertices [i]delightful[/i] if no two of its vertices are connected by an edge, but any vertex not from $S{}$ is connected to at least one vertex from $S{}$. For which smallest $m$ is there necessarily a delightful set of at most $m$ vertices?

2000 Argentina National Olympiad, 3

Tags: combinatorics

There is a board with 32 rows and 10 columns. Pablo writes 1 or -1 in each box. Matías, with Pablo's board in view, chooses one or more columns, and in each of the chosen columns, changes all of Pablo's numbers to their opposites (where there is 1 he puts -1 and where there is -1 he puts 1) . In the other columns, leave Pablo's numbers. Matías wins if he manages to make his board have each of the rows different from all the rows on Pablo's board. Otherwise, that is, if any row on Matías's board is equal to any row on Pablo's board, Pablo wins. If both play perfectly, determine which of the two is assured of victory.

2002 HKIMO Preliminary Selection Contest, 4

Tags: combinatorics

A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get $-1$ mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)