Olympiad problems

2009 China Team Selection Test, 2

Tags: ceiling function , pigeonhole principle , induction , graph theory , combinatorics proposed , combinatorics

Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.

2012 Iran MO (3rd Round), 5

Tags: conics , parabola , geometry proposed , geometry

Two fixed lines $l_1$ and $l_2$ are perpendicular to each other at a point $Y$. Points $X$ and $O$ are on $l_2$ and both are on one side of line $l_1$. We draw the circle $\omega$ with center $O$ and radius $OY$. A variable point $Z$ is on line $l_1$. Line $OZ$ cuts circle $\omega$ in $P$. Parallel to $XP$ from $O$ intersects $XZ$ in $S$. Find the locus of the point $S$. [i]Proposed by Nima Hamidi[/i]

2023 ISL, C7

Tags: combinatorics , graph theory , IMO Shortlist

The Imomi archipelago consists of $n\geq 2$ islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of $k$ companies. It is known that if any one of the $k$ companies closes all its ferry lines, then it becomes impossible for a traveller, no matter where the traveller starts at, to visit all the islands exactly once (in particular, not returning to the island the traveller started at). Determine the maximal possible value of $k$ in terms of $n$. [i]Anton Trygub, Ukraine[/i]

2017 Czech-Polish-Slovak Match, 3

Tags: winning positions , game strategy , Sequence , divisible , number theory

Let ${k}$ be a fixed positive integer. A finite sequence of integers ${x_1,x_2, ..., x_n}$ is written on a blackboard. Pepa and Geoff are playing a game that proceeds in rounds as follows. - In each round, Pepa first partitions the sequence that is currently on the blackboard into two or more contiguous subsequences (that is, consisting of numbers appearing consecutively). However, if the number of these subsequences is larger than ${2}$, then the sum of numbers in each of them has to be divisible by ${k}$. - Then Geoff selects one of the subsequences that Pepa has formed and wipes all the other subsequences from the blackboard. The game finishes once there is only one number left on the board. Prove that Pepa may choose his moves so that independently of the moves of Geoff, the game finishes after at most ${3k}$ rounds. (Poland)

2012 239 Open Mathematical Olympiad, 1

Tags: combinatorics

On a $10 \times 10$ chessboard, several knights are placed, and in any $2 \times 2$ square there is at least one knight. What is the smallest number of cells these knights can threat? (The knight does not threat the square on which it stands, but it does threat the squares on which other knights are standing.)

LMT Accuracy Rounds, 2021 F7

Tags:

Find the number of ways to tile a $12 \times 3$ board with $1 \times 4$ and $2 \times 2$ tiles with no overlap or uncovered space.

2000 IMO Shortlist, 3

Tags: geometry , circumcircle , orthocenter , Triangle , concurrency , IMO Shortlist

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.

1973 AMC 12/AHSME, 7

Tags:

The sum of all integers between 50 and 350 which end in 1 is $ \textbf{(A)}\ 5880 \qquad \textbf{(B)}\ 5539 \qquad \textbf{(C)}\ 5208 \qquad \textbf{(D)}\ 4877 \qquad \textbf{(E)}\ 4566$

2024 Canada National Olympiad, 3

Tags: algebra , polynomial

Let $N{}$ be the number of positive integers with $10$ digits $\overline{d_9d_8\cdots d_0}$ in base $10$ (where $0\le d_i\le9$ for all $i$ and $d_9>0$) such that the polynomial \[d_9x^9+d_8x^8+\cdots+d_1x+d_0\] is irreducible in $\Bbb Q$. Prove that $N$ is even. (A polynomial is irreducible in $\Bbb Q$ if it cannot be factored into two non-constant polynomials with rational coefficients.)

2016 Azerbaijan IMO TST First Round, 4

Tags: algebra , functional equation , function

Find the solution of the functional equation $P(x)+P(1-x)=1$ with power $2015$ P.S: $P(y)=y^{2015}$ is also a function with power $2015$

1971 Putnam, A4

Tags: college contests

Show that for $0 <\epsilon <1$ the expression $(x+y)^n(x^2-(2-\epsilon)xy+y^2)$ is a polynomial with positive coefficients for $n$ sufficiently large and integral. For $\epsilon =.002$ find the smallest admissible value of $n$.

2012 Romania National Olympiad, 1

Tags: function , limit , real analysis , real analysis unsolved

[color=darkred]Let $f,g\colon [0,1]\to [0,1]$ be two functions such that $g$ is monotonic, surjective and $|f(x)-f(y)|\le |g(x)-g(y)|$ , for any $x,y\in [0,1]$ . [list] [b]a)[/b] Prove that $f$ is continuous and that there exists some $x_0\in [0,1]$ with $f(x_0)=g(x_0)$ . [b]b)[/b] Prove that the set $\{x\in [0,1]\, |\, f(x)=g(x)\}$ is a closed interval. [/list][/color]

Novosibirsk Oral Geo Oly VIII, 2020.3

Tags: geometry , right triangle

Maria Ivanovna drew on the blackboard a right triangle $ABC$ with a right angle $B$. Three students looked at her and said: $\bullet$ Yura said: "The hypotenuse of this triangle is $10$ cm." $\bullet$ Roma said: "The altitude drawn from the vertex $B$ on the side $AC$ is $6$ cm." $\bullet$ Seva said: "The area of the triangle $ABC$ is $25$ cm$^2$." Determine which of the students was mistaken if it is known that there is exactly one such person.

1998 Tournament Of Towns, 5

Tags: combinatorial geometry , combinatorics , geometry , isosceles

Pinocchio claims that he can divide an isoceles triangle into three triangles, any two of which can be put together to form a new isosceles triangle. Is Pinocchio lying? (A Shapovalov)

1991 Tournament Of Towns, (297) 4

Tags: 3D geometry , geometry , combinatorics , combinatorial geometry , sphere

Five points are chosen on the sphere, no three of them lying on a great circle (a great circle is the intersection of the sphere with some plane passing through the sphere’s centre). Two great circles not containing any of the chosen points are called equivalent if one of them can be moved to the other without passing through any chosen points. (a) How many nonequivalent great circles not containing any chosen points can be drawn on the sphere? (b) Answer the same problem, but with $n$ chosen points.

2007 Junior Tuymaada Olympiad, 1

Tags: number theory proposed , number theory

Positive integers $ a<b$ are given. Prove that among every $ b$ consecutive positive integers there are two numbers whose product is divisible by $ ab$.

Mathematical Minds 2023, P5

Tags: combinatorics , graph theory

At a company, there are several workers, some of which are enemies. They go to their job with 100 buses, in such a way that there aren't any enemies in either bus. Having arrived at the job, their chief wants to assign them to brigades of at least two people, without assigning two enemies to the same brigade. Prove that the chief can split the workers in at most 100 brigades, or he cannot split them at all in any number of brigades.

2023 Brazil Team Selection Test, 5

Tags: combinatorics , APMO , APMO 2023

There are $n$ line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his $2n - 1$ friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows: First, he chooses an endpoint of each segment as a “sink”. Then he places the present at the endpoint of the segment he is at. The present moves as follows : $\bullet$ If it is on a line segment, it moves towards the sink. $\bullet$ When it reaches an intersection of two segments, it changes the line segment it travels on and starts moving towards the new sink. If the present reaches an endpoint, the friend on that endpoint can receive their present. Prove that Tony can send presents to exactly $n$ of his $2n - 1$ friends.

2015 Saint Petersburg Mathematical Olympiad, 1

Tags: number theory

There is child camp with some rooms. Call room as $4-$room, if $4$ children live here. Not less then half of all rooms are $4-$rooms , other rooms are $3-$rooms. Not less than $2/3$ girls live in $3-$rooms. Prove that not less than $35\%$ of all children are boys.

1952 Poland - Second Round, 3

Tags: geometry , rectangle , inscribed , square , rhombus

Are the following statements true? a) if the four vertices of a rectangle lie on the four sides of a rhombus, then the sides of the rectangle are parallel to the diagonals of the rhombus; b) if the four vertices of a square lie on the four sides of a rhombus that is not a square, then the sides of the square are parallel to the diagonals of the rhombus.

2013 Mexico National Olympiad, 5

Tags: inequalities , number theory unsolved , number theory

A pair of integers is special if it is of the form $(n, n-1)$ or $(n-1, n)$ for some positive integer $n$. Let $n$ and $m$ be positive integers such that pair $(n, m)$ is not special. Show $(n, m)$ can be expressed as a sum of two or more different special pairs if and only if $n$ and $m$ satisfy the inequality $ n+m\geq (n-m)^2 $. Note: The sum of two pairs is defined as $ (a, b)+(c, d) = (a+c, b+d) $.

2013 Singapore Junior Math Olympiad, 5

Tags: combinatorics

$6$ musicians gathered at a chamber music festival. At each scheduled concert, some of the musicians played while the others listened as members of the audience. What is the least number of such concerts which would need to be scheduled so that every $2$ musicians each must play for the other in some concert?

2015 Online Math Open Problems, 30

Tags: Online Math Open

Ryan is learning number theory. He reads about the [i]Möbius function[/i] $\mu : \mathbb N \to \mathbb Z$, defined by $\mu(1)=1$ and \[ \mu(n) = -\sum_{\substack{d\mid n \\ d \neq n}} \mu(d) \] for $n>1$ (here $\mathbb N$ is the set of positive integers). However, Ryan doesn't like negative numbers, so he invents his own function: the [i]dubious function[/i] $\delta : \mathbb N \to \mathbb N$, defined by the relations $\delta(1)=1$ and \[ \delta(n) = \sum_{\substack{d\mid n \\ d \neq n}} \delta(d) \] for $n > 1$. Help Ryan determine the value of $1000p+q$, where $p,q$ are relatively prime positive integers satisfying \[ \frac{p}{q}=\sum_{k=0}^{\infty} \frac{\delta(15^k)}{15^k}. \] [i]Proposed by Michael Kural[/i]

1991 Tournament Of Towns, (287) 3

Tags: number theory , Digits

We are looking for numbers ending with the digit $5$ such that in their decimal expansion each digit beginning with the second digit is no less than the previous one. Moreover the squares of these numbers must also possess the same property. (a) Find four such numbers. (b) Prove that there are infinitely many. (A. Andjans, Riga)

2013 Turkey MO (2nd round), 1

Tags: geometry , circumcircle , geometric transformation , reflection , symmetry , cyclic quadrilateral , similar triangles

The circle $\omega_1$ with diameter $[AB]$ and the circle $\omega_2$ with center $A$ intersects at points $C$ and $D$. Let $E$ be a point on the circle $\omega_2$, which is outside $\omega_1$ and at the same side as $C$ with respect to the line $AB$. Let the second point of intersection of the line $BE$ with $\omega_2$ be $F$. For a point $K$ on the circle $\omega_1$ which is on the same side as $A$ with respect to the diameter of $\omega_1$ passing through $C$ we have $2\cdot CK \cdot AC = CE \cdot AB$. Let the second point of intersection of the line $KF$ with $\omega_1$ be $L$. Show that the symmetric of the point $D$ with respect to the line $BE$ is on the circumcircle of the triangle $LFC$.