Olympiad problems

2018 Estonia Team Selection Test, 8

Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors $1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$.

2022 Polish MO Finals, 3

Tags: combinatorics

One has marked $n$ points on a circle and has drawn a certain number of chords whose endpoints are the marked points. It turned out that the following property is satisfied: whenever any $2021$ drawn chords are removed one can join any two marked points by a broken line composed of some of the remaining drawn chords. Prove that one can remove some of the drawn chords so that at most $2022n$ chords remain and the property described above is preserved.

ICMC 2, 1

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This questions comprises two independent parts. (i) Let $g:\mathbb{R}\to\mathbb{R}$ be continuous and such that $g(0)=0$ and $g(x)g(-x)>0$ for any $x > 0$. Find all solutions $f : \mathbb{R}\to\mathbb{R}$ to the functional equation \[g(f(x+y))=g(f(x))+g(f(y)),\ x,y\in\mathbb{R}\] (ii) Find all continuously differentiable functions $\phi : [a, \infty) \to \mathbb{R}$, where $a > 0$, that satisfies the equation \[(\phi(x))^2=\int_a^x \left(\left|\phi(y)\right|\right)^2+\left(\left|\phi'(y)\right|\right)^2\mathrm{d}y -(x-a)^3,\ \forall x\geq a.\]

2019 Iran MO (3rd Round), 2

Tags: combinatorics

Let $T$ be a triangulation of a $100$-gon.We construct $P(T)$ by copying the same $100$-gon and drawing a diagonal if it was not drawn in $T$ an there is a quadrilateral with this diagonal and two other vertices so that all the sides and diagonals(Except the one we are going to draw) are present in $T$.Let $f(T)$ be the number of intersections of diagonals in $P(T)$.Find the minimum and maximum of $f(T)$.

2002 Miklós Schweitzer, 1

Tags: function

For an arbitrary ordinal number $\alpha$ let $H(\alpha)$ denote the set of functions $f\colon \alpha \rightarrow \{ -1,0,1\}$ that map all but finitely many elements of $\alpha$ to $0$. Order $H(\alpha)$ according to the last difference, that is, for $f, g\in H(\alpha)$ let $f\prec g$ if $f(\beta) < g(\beta)$ holds for the maximum ordinal number $\beta < \alpha$ with $f(\beta) \neq g(\beta)$. Prove that the ordered set $(H(\alpha), \prec)$ is scattered (i.e. it doesn't contain a subset isomorphic to the set of rational numbers with the usual order), and that any scattered order type can be embedded into some $(H(\alpha), \prec)$.

2020 LMT Spring, 3

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Let $LMT$ represent a 3-digit positive integer where $L$ and $M$ are nonzero digits. Suppose that the 2-digit number $MT$ divides $LMT$. Compute the difference between the maximum and minimum possible values of $LMT$.

2018 USA TSTST, 5

Tags: geometry

Let $ABC$ be an acute triangle with circumcircle $\omega$, and let $H$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be the points on $\omega$ with $PA = PH$ and $QA = QH$. The tangent to $\omega$ at $P$ intersects lines $AC$ and $AB$ at $E_1$ and $F_1$ respectively; the tangent to $\omega$ at $Q$ intersects lines $AC$ and $AB$ at $E_2$ and $F_2$ respectively. Show that the circumcircles of $\triangle AE_1F_1$ and $\triangle AE_2F_2$ are congruent, and the line through their centers is parallel to the tangent to $\omega$ at $A$. [i]Ankan Bhattacharya and Evan Chen[/i]

2005 Cuba MO, 7

Tags: number theory , lcm , gcd , gcd and lcm

Determine all triples of positive integers $(x, y, z)$ that satisfy $$x < y < z, \ \ gcd(x, y) = 6, \ \ gcd(y, z) = 10, \ \ gcd(z, x) = 8 \ \ and \ \ lcm(x, y, z) = 2400.$$

1986 IMO Shortlist, 8

Tags: combinatorics , graph theory , permutation , extremal graph theory

From a collection of $n$ persons $q$ distinct two-member teams are selected and ranked $1, \cdots, q$ (no ties). Let $m$ be the least integer larger than or equal to $2q/n$. Show that there are $m$ distinct teams that may be listed so that : [b](i)[/b] each pair of consecutive teams on the list have one member in common and [b](ii)[/b] the chain of teams on the list are in rank order. [i]Alternative formulation.[/i] Given a graph with $n$ vertices and $q$ edges numbered $1, \cdots , q$, show that there exists a chain of $m$ edges, $m \geq \frac{2q}{n}$ , each two consecutive edges having a common vertex, arranged monotonically with respect to the numbering.

MIPT student olimpiad spring 2023, 3

Tags: geometry , 3d geometry , sphere

Prove that if a set $X\subset S^n$ takes up more than half a Riemannian volume of a unit sphere $S^n$, then the set of all possible geodesic segments length less than $\pi$ with endpoints in the set $X$ covers the entire sphere. Geodetic on sphere $S^n$ is a curve lying on some circle of intersection of the sphere $S^n\subset R^{n+1}$ two-dimensional linear subspace $L \subset R^{n+1}$

1999 National High School Mathematics League, 4

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Statement 1: Line $a\in\alpha$, line $b\in\beta$, and $a,b$ are skew lines. If $c=\alpha\cap\beta$, then $c$ intersects at most one of $a,b$. Statement 2: It's impossible to find infintely many lines, any two of them are skew lines. $\text{(A)}$ Statement 1 is true, Statement 2 is false. $\text{(B)}$ Statement 2 is true, Statement 1 is false. $\text{(C)}$ Both are true. $\text{(D)}$ Neither is true.

2017 USAJMO, 5

Tags: geometry

Let $O$ and $H$ be the circumcenter and the orthocenter of an acute triangle $ABC$. Points $M$ and $D$ lie on side $BC$ such that $BM=CM$ and $\angle BAD = \angle CAD$. Ray $MO$ intersects the circumcircle of triangle $BHC$ in point $N$. Prove that $\angle ADO = \angle HAN$.

2013 Danube Mathematical Competition, 1

Tags: algebra , sum , reciprocal sum

Determine the natural numbers $n\ge 2$ for which exist $x_1,x_2,...,x_n \in R^*$, such that $$x_1+x_2+...+x_n=\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}=0$$

2007 Grigore Moisil Intercounty, 4

Tags: inequalities , solve inequality , algebra

Solve in the set of real numbers the fractional part inequality $ \{ x \}\le\{ nx \} , $ where $ n $ is a fixed natural number.

Russian TST 2018, P3

Tags: algebra , function

A function $f:\mathbb{R} \to \mathbb{R}$ has the following property: $$\text{For every } x,y \in \mathbb{R} \text{ such that }(f(x)+y)(f(y)+x) > 0, \text{ we have } f(x)+y = f(y)+x.$$ Prove that $f(x)+y \leq f(y)+x$ whenever $x>y$.

1990 IMO Longlists, 11

Tags: extremal combinatorics , extremal graph theory , graph theory , vertex degree , combinatorics

In a group of mathematicians, every mathematician has some friends (the relation of friend is reciprocal). Prove that there exists a mathematician, such that the average of the number of friends of all his friends is no less than the average of the number of friends of all these mathematicians.

1976 USAMO, 1

Tags: geometry , rectangle

(a) Suppose that each square of a 4 x 7 chessboard is colored either black or white. Prove that with [i]any[/i] such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board) whose four distinct unit corner squares are all of the same color. (b) Exhibit a black-white coloring of a 4 x6 board in which the four corner squares of every rectangle, as described above, are not all of the same color.

2010 District Olympiad, 1

Tags: superior algebra

Let $ S$ be the sum of the inversible elements of a finite ring. Prove that $ S^2\equal{}S$ or $ S^2\equal{}0$.

1965 All Russian Mathematical Olympiad, 060

Tags: algebra

There is a lighthouse on a small island. Its lamp enlights a segment of a sea to the distance $a$. The light is turning uniformly, and the end of the segment moves with the speed $v$. Prove that a ship, whose speed doesn't exceed $v/8$ cannot arrive to the island without being enlightened.

2013 Thailand Mathematical Olympiad, 7

Tags: combinatorial geometry , combinatorics

Let $P_1, ... , P_{2556}$ be distinct points in a regular hexagon $ABCDEF$ with unit side length. Suppose that no three points in the set $S = \{A, B, C, D, E, F, P_1, ... , P_{2556}\}$ are collinear. Show that there is a triangle whose vertices are in $S$ and whose area is less than $\frac{1}{1700}$ .

VMEO IV 2015, 10.2

Tags: circumcircle , geometry

Given triangle $ABC$ and $P,Q$ are two isogonal conjugate points in $\triangle ABC$. $AP,AQ$ intersects $(QBC)$ and $(PBC)$ at $M,N$, respectively ( $M,N$ be inside triangle $ABC$) 1. Prove that $M,N,P,Q$ locate on a circle - named $(I)$ 2. $MN\cap PQ$ at $J$. Prove that $IJ$ passed through a fixed line when $P,Q$ changed

1997 Estonia National Olympiad, 4

Tags: combinatorics , grid , square table , coloring

In a $3n \times 3n$ grid, each square is either black or red. Each red square not on the edge of the grid has exactly five black squares among its eight neighbor squares.. On every black square that not at the edge of the grid, there are exactly four reds in the adjacent squares box. How many black and how many red squares are in the grid?

II Soros Olympiad 1995 - 96 (Russia), 10.10

Tags: combinatorics

Each deputy of the Academic Duma quarreled with exactly three other deputies. The President ordered the Speaker to divide the deputies into n factions so that agreement reigned within one faction. For what smallest $n$ is this always possible? (This means that there is such $n$ that deputies could always be divided into $n$ factions, but not always into $(n- 1)$ factions.)

2014 Contests, 2

Tags: function , limit , functional equation , algebra

Find all functions $f:R\rightarrow R$ such that \[ f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy)) \] for all $x,y\in R$.

2004 Flanders Junior Olympiad, 3

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A car has a 4-digit integer price, which is written digitally. (so in digital numbers, like on your watch probably) While the salesmen isn't watching, the buyer turns the price upside down and gets the car for 1626 less. How much did the car initially cost?