Navvye/Olympiad-Problems-CSV huggingface dataset:
2001 Tuymaada Olympiad
Ten volleyball teams played a tournament; every two teams met exactly once. The winner of the play gets 1 point, the loser gets 0 (there are no draws in volleyball). If the team that scored $n$-th has $x_{n}$ points ($n=1, \dots, 10$), prove that $x_{1}+2x_{2}+\dots+10x_{10}\geq 165$.



Proposed by D. Teryoshin
2001 Tuymaada Olympiad
Solve the equation

$$(a^{2},b^{2})+(a,bc)+(b,ac)+(c,ab)=199.$$

in positive integers.

(Here $(x,y)$ denotes the greatest common divisor of $x$ and $y$.)



Proposed by S. Berlov
2001 Tuymaada Olympiad
Do there exist quadratic trinomials $P, \ \ Q, \ \ R$ such that for every integers $x$ and $y$ an integer $z$ exists satisfying $P(x)+Q(y)=R(z)?$



Proposed by A. Golovanov
2001 Tuymaada Olympiad
Unit square $ABCD$ is divided into $10^{12}$ smaller squares (not necessarily equal). Prove that the sum of perimeters of all the smaller squares having common points with diagonal $AC$ does not exceed 1500.



Proposed by A. Kanel-Belov
2001 Tuymaada Olympiad
All positive integers are distributed among two disjoint sets $N_{1}$ and $N_{2}$ such that no difference of two numbers belonging to the same set is a prime greater than 100.



Find all such distributions.



Proposed by N. Sedrakyan
2001 Tuymaada Olympiad
Non-zero numbers are arranged in $n \times n$ square ($n>2$). Every number is exactly $k$ times less than the sum of all the other numbers in the same cross (i.e., $2n-2$ numbers written in the same row or column with this number).

Find all possible $k$.



Proposed by D. Rostovsky, A. Khrabrov, S. Berlov
2001 Tuymaada Olympiad
$ABCD$ is a convex quadrilateral; half-lines $DA$ and $CB$ meet at point $Q$; half-lines $BA$ and $CD$ meet at point $P$. It is known that $\angle AQB=\angle APD$. The bisector of angle $\angle AQB$ meets the sides $AB$ and $CD$ of the quadrilateral at points $X$ and $Y$, respectively; the bisector of angle $\angle APD$ meets the sides $AD$ and $BC$ at points $Z$ and $T$, respectively.

The circumcircles of triangles $ZQT$ and $XPY$ meet at point $K$ inside the quadrilateral.

Prove that $K$ lies on the diagonal $AC$.



Proposed by S. Berlov
2001 Tuymaada Olympiad
Is it possible to colour all positive real numbers by 10 colours so that every two numbers with decimal representations differing in one place only are of different colours? (We suppose that there is no place in a decimal representations such that all digits starting from that place are 9's.)



Proposed by A. Golovanov
2001 Tuymaada Olympiad
$16$ chess players held a tournament among themselves: every two chess players played exactly one game. For victory in the party was given $1$ point, for a draw  $0.5$ points, for defeat  $0$ points. It turned out that exactly 15 chess players shared the first place. How many points could the sixteenth chess player score?
2001 Tuymaada Olympiad
Is it possible to arrange integers in the cells of the infinite chechered sheet so that every integer appears at least in one cell, and the sum of any $10$ numbers in a row vertically or horizontal, would be divisible by $101$?
2001 Tuymaada Olympiad
Let ABC be an acute isosceles triangle  ($AB=BC$) inscribed in a circle with center $O$ .  The line through the midpoint of the chord $AB$ and point $O$ intersects the line $AC$ at $L$ and the circle at the point $P$. Let the bisector of angle $BAC$ intersects the circle at point $K$.  Lines $AB$ and  $PK$ intersect at point $D$. Prove that the points $L,B,D$  and $P$ lie on the same circle.
2001 Tuymaada Olympiad
Natural numbers $1, 2, 3,.., 100$ are contained in the union of $N$ geometric progressions (not necessarily with integer denominations). Prove that $N \ge 31$
2001 Tuymaada Olympiad
On the side $AB$ of an isosceles triangle $AB$ ($AC=BC$) lie points $P$ and $Q$ such that  $\angle PCQ \le \frac{1}{2} \angle ACB$.  Prove that $PQ \le \frac{1}{2} AB$.
2001 Tuymaada Olympiad
Several rational numbers were written on the blackboard. Dima wrote off their fractional parts on paper. Then all the numbers on the board squared, and Dima wrote off another paper with fractional parts of the resulting numbers. It turned out that on Dima's papers were written the same sets of numbers (maybe in different order). Prove that the original numbers on the board were integers.

(The fractional part of a number $x$ is such a number $\{x\}, 0 \le \{x\} <1$, that $x-\{x\}$ is an integer.)
2001 Tuymaada Olympiad
Can three persons, having one double motorcycle, overcome the distance of $70$ km in $3$ hours? Pedestrian speed is $5$ km / h and  motorcycle speed is $50$ km / h.
2000 Tuymaada Olympiad
Let $d(n)$ denote the number of positive divisors of $n$ and let

$e(n)=\left[2000\over n\right]$ for positive integer $n$. Prove that

$$d(1)+d(2)+\dots+d(2000)=e(1)+e(2)+\dots+e(2000).$$
2000 Tuymaada Olympiad
A tangent $l$ to the circle inscribed in a rhombus meets its sides $AB$ and $BC$ at points $E$ and $F$ respectively.

Prove that the product $AE\cdot CF$ is independent of the choice of $l$.
2000 Tuymaada Olympiad
Can the 'brick wall' (infinite in all directions) drawn at the picture be made of wires of length $1, 2, 3, \dots$ (each positive integral length  occurs exactly once)? (Wires can be bent but should not overlap; size of a 'brick' is $1\times 2$).



[asy]

unitsize(0.5 cm);



for(int i = 1; i <= 9; ++i) {

  draw((0,i)--(10,i));

}



for(int i = 0; i <= 4; ++i) {

  for(int j = 0; j <= 4; ++j) {

    draw((2*i + 1,2*j)--(2*i + 1,2*j + 1));

  }

}



for(int i = 0; i <= 3; ++i) {

  for(int j = 0; j <= 4; ++j) {

    draw((2*i + 2,2*j + 1)--(2*i + 2,2*j + 2));

  }

}

[/asy]
2000 Tuymaada Olympiad
Prove for real $x_1$, $x_2$, ....., $x_n$,

$0 < x_k \leq {1\over 2}$, the inequality

$$ \left( {n \over x_1 + \dots + x_n} - 1 \right)^n \leq \left( {1 \over x_1} - 1 \right) \dots \left( {1 \over x_n} - 1 \right). $$
2000 Tuymaada Olympiad
Can the plane be coloured in 2000 colours so that any nondegenerate circle contains points of all 2000 colors?
2000 Tuymaada Olympiad
There are 2000 cities in Graphland; some of them are connected by roads.

For every city the number of roads going from it is counted. It is known that there are exactly two equal numbers among all the numbers obtained. What can be these numbers?
2000 Tuymaada Olympiad
Polynomial $ P(t)$ is such that for all real $ x$,

$$ P(\sin x) + P(\cos x) = 1.

$$

What can be the degree of this polynomial?
2000 Tuymaada Olympiad
Prove that no number of the form $10^{-n}$, $n\geq 1,$ can be represented as the sum of reciprocals of factorials of different positive integers.
2000 Tuymaada Olympiad
Given the number $188188...188$ (number $188$ is written $101$ times). Some digits of this number are crossed out. What is the largest multiple of $7$, that could happen?
2000 Tuymaada Olympiad
Is it possible to paint the plane in $4$ colors so that inside any circle are the dots of all four colors?
2000 Tuymaada Olympiad
Prove that if the product of positive numbers $a,b$ and  $c$  equals one, then $\frac{1}{a(a+1)}+\frac{1}{b(b+1)}+\frac{1}{c(c+1)}\ge \frac{3}{2}$
2000 Tuymaada Olympiad
Are there prime $p$ and  $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and  $q^2-1$  divided by $p$?
2000 Tuymaada Olympiad
Let $O$ be the center of the circle circumscribed around the the triangle $ABC$.  The centers of the circles circumscribed around the squares $OAB,OBC,OCA$ lie at the vertices of a regular triangle. Prove that the triangle $ABC$ is right.
2000 Tuymaada Olympiad
Every two of five regular pentagons on the plane have a common point.

Is it true that some of these pentagons have a common point?
2000 Tuymaada Olympiad
There are $2000$ cities in the country, each of which has exactly three roads to other cities. Prove that you can close $1000$ roads, so that there is not a single closed route in the country, consisting of an odd number of roads.
1999 Tuymaada Olympiad
Find all polynomials $P(x)$ such that

$$

P(x^3+1)=P(x^3)+P(x^2).

$$



Proposed by A. Golovanov
1999 Tuymaada Olympiad
What maximum number of elements can be selected from the set $\{1, 2, 3, \dots, 100\}$ so that no sum of any three selected numbers is equal to a selected number?



Proposed by A. Golovanov
1999 Tuymaada Olympiad
Prove the inequality

$$

{x\over y^2-z}+{y\over z^2-x}+{z\over x^2-y} > 1,

$$

where $2 < x, y, z < 4.$



Proposed by A. Golovanov
1999 Tuymaada Olympiad
In the triangle $ABC$ we have $\angle ABC=100^\circ$, $\angle ACB=65^\circ$, $M\in AB$, $N\in AC$, and $\angle MCB=55^\circ$, $\angle NBC=80^\circ$. Find $\angle NMC$.



St.Petersburg folklore
1999 Tuymaada Olympiad
Can the graphs of a polynomial of degree 20 and the function $\displaystyle y={1\over x^{40}}$ have exactly 30 points of intersection?



Proposed by K. Kokhas
1999 Tuymaada Olympiad
A sequence of integers $a_0,\ a_1,\dots a_n \dots $ is defined by the following rules: $a_0=0,\ a_1=1,\ a_{n+1} > a_n$ for each $n\in \mathbb{N}$, and $a_{n+1}$ is the minimum number such that no three numbers among $a_0,\ a_1,\dots a_{n+1}$ form an arithmetical progression. Prove that $a_{2^n}=3^n$ for each $n \in \mathbb{N}.$
1999 Tuymaada Olympiad
A right parallelepiped (i.e. a parallelepiped one of whose edges is perpendicular to a face) is given. Its vertices have integral coordinates, and no other points with integral coordinates lie on its faces or edges. Prove that the volume of this parallelepiped is a sum of three perfect squares.



Proposed by A. Golovanov
1998 Tuymaada Olympiad
Write the number $\frac{1997}{1998}$  as a sum of different numbers, inverse to naturals.
1998 Tuymaada Olympiad
Solve the equation $(x^3-1000)^{1/2}=(x^2+100)^{1/3}$
1998 Tuymaada Olympiad
The segment of length  $\ell$ with the ends on the border of a triangle divides the area of that triangle in half. Prove that $\ell >r\sqrt2$, where  $r$ is the radius of the inscribed circle of the  triangle.
1998 Tuymaada Olympiad
Given the tetrahedron $ABCD$, whose opposite edges are equal, that is, $AB=CD, AC=BD$ and $BC=AD$. Prove that exist exactly $6$ planes intersecting the triangular angles of the tetrahedron and dividing the total surface and volume of this tetrahedron in half.
1998 Tuymaada Olympiad
A right triangle is inscribed in  parabola  $y=x^2$. Prove that it's hypotenuse is not less than $2$.
1998 Tuymaada Olympiad
Prove that the sequence of the first digits of the numbers in the form $2^n+3^n$ is nonperiodic.
1998 Tuymaada Olympiad
All possible sequences of numbers $-1$ and $+1$  of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values.
1998 Tuymaada Olympiad
Given the pyramid $ABCD$. Let $O$ be the middle of  edge $AC$. Given that $DO$ is the height of the pyramid, $AB=BC=2DO$ and the angle $ABC$ is right. Cut this pyramid into $8$ equal and similar to it pyramids.
1997 Tuymaada Olympiad
The product of any three of these four natural numbers is a perfect square.

Prove that these numbers themselves are perfect squares.
1997 Tuymaada Olympiad
Solve in natural numbers the system of equations $3x^2+6y^2+5z^2=1997$ and $3x+6y+5z=161$ .
1997 Tuymaada Olympiad
Is it possible to paint all natural numbers in $6$ colors, for each one color to be used and the sum of any five numbers

of different color to be painted in the sixth color?
1997 Tuymaada Olympiad
Using only angle with angle $\frac{\pi}{7}$ and a ruler, constuct  angle $\frac{\pi}{14}$
1997 Tuymaada Olympiad
Prove the inequality $\left(1+\frac{1}{q}\right)\left(1+\frac{1}{q^2}\right)...\left(1+\frac{1}{q^n}\right)<\frac{q-1}{q-2}$

for $n\in N, q>2$
1997 Tuymaada Olympiad
Are there $14$ consecutive positive integers, each of which has a divisor other than $1$ and not exceeding $11$?
1997 Tuymaada Olympiad
It is known that every student of the class for Sunday once visited the rink, and every boy met there with every girl. Prove that there was a point in time when all the boys, or all the girls of the class were simultaneously on the rink.
1997 Tuymaada Olympiad
Find a right triangle that can be cut into $365$ equal triangles.
1996 Tuymaada Olympiad
Prove the inequality $x_1y_1+x_2y_2+x_2y_1+2x_2y_2\le 1996$

if $x_1^2+2x_1x_2+2x_2^2\le 998$  and $y_1^2+2y_1y_2+2y_2^2\le 3992$.
1996 Tuymaada Olympiad
Given a finite set of real numbers $A$, not containing $0$ and $1$ and possessing the property: if the number a belongs to  $A$, then numbers $\frac{1}{a}$ and  $1-a$ also belong to $A$. How many numbers are in the set $A$?
1996 Tuymaada Olympiad
Nine points of the plane, located at the vertices of a regular nonagon, are pairwise connected by segments, each of which is colored either red or blue. It is known that in any triangle with vertices at the vertices of the nonagon at least

one side is red. Prove that there are four points, any two of which are connected by red lines.
1996 Tuymaada Olympiad
Given a segment of length $7\sqrt3$ .

Is it possible to use only compass  to construct a segment of length $\sqrt7$?
1996 Tuymaada Olympiad
Solve the equation $\sqrt{1981-\sqrt{1996+x}}=x+15$
1996 Tuymaada Olympiad
Given the sequence

$f_1(a)=sin(0,5\pi a)$

$f_2(a)=sin(0,5\pi (sin(0,5\pi a)))$

$...$

$f_n(a)=sin(0,5\pi (sin(...(sin(0,5\pi a))...)))$ , where  $a$ is any real number.

What limit aspire the members of this sequence   as  $n \to \infty$?
1996 Tuymaada Olympiad
In the set of all positive real numbers define the  operation $a * b =  a^b$ .

Find all positive rational numbers for which $a * b = b * a$.
1996 Tuymaada Olympiad
Given a tetrahedron $ABCD$, in which $AB=CD= 13 , AC=BD=14$ and $AD=BC=15$.

Show that the centers of the inscribed sphere and sphere around it coincide, and find the radii of these spheres.
1995 Tuymaada Olympiad
Give a geometric proof of the statement that the fold line on a sheet of paper is straight.
1995 Tuymaada Olympiad
Let $x_1=a, x_2=a^{x_1}, ...,  x_n=a^{x_{n-1}}$ where $a>1$. What is the maximum value of  $a$ for which lim exists $\lim_{n\to \infty} x_n$ and what is this limit?
1995 Tuymaada Olympiad
Prove that the equation  $(\sqrt5 +1)^{2x}+ (\sqrt5 -1)^{2x}=2^x(y^2+2)$ has an infinite number of solutions in natural numbers.
1995 Tuymaada Olympiad
It is known that the merchant’s  $n$ clients live in locations laid along the ring road. Of these,  $k$ customers have debts to the merchant for $a_1,a_2,...,a_k$ rubles, and the merchant owes the remaining  $n-k$ clients, whose debts are $b_1,b_2,...,b_{n-k}$ rubles, moreover, $a_1+a_2+...+a_k=b_1+b_2+...+b_{n-k}$. Prove that a merchant who has no money can pay all his debts and have paid all the customer debts, by starting a customer walk along the road from one of points and not missing any of their customers.
1995 Tuymaada Olympiad
Given a circle of radius $r= 1995$. Show that around it you can describe exactly $16$ primitive Pythagorean triangles. The primitive  Pythagorean triangle is a right-angled triangle, the lengths of the sides of which are expressed by coprime integers.
1995 Tuymaada Olympiad
A set consisting of  $n$ points of a plane is called an isosceles $n$-point if any three of its points are located in

vertices of an isosceles triangle. Find all natural the numbers for which there exist isosceles $n$-points.
1995 Tuymaada Olympiad
Find a continuous function $f(x)$ satisfying the identity $f(x)-f(ax)=x^n-x^m$, where $n,m\in N , 0<a<1$
1995 Tuymaada Olympiad
Inside the triangle $ABC$ a point $M$ is given . Find the points  $P,Q$ and $R$ lying on the sides $AB,BC$ and  $AC$ respectively and such so that the sum  $MP+PQ+QR+RM$ is the smallest.
1994 Tuymaada Olympiad
World Cup in America introduced a new point system. For a victory $3$ points are given, for a draw  $1$ point and for defeat  $0$ points. In the preliminary games, the teams are divided into groups of $4$ teams. In groups, teams play with each other, once, then in accordance with the points scored $a,b,c$ and $d$ ($a>b>c>d$) teams take the first, second, third and fourth place in their groups. Give all possible options for the distribution points $a,b,c$ and $d$
1994 Tuymaada Olympiad
The set of numbers $M=\{4k-3 | k\in N\}$ is considered. A number of of this set is called “simple” if it is impossible to put in the form of a product of numbers from $M$ other than $1$. Show that in this set, the decomposition of numbers in the product of "simple" factors is ambiguous.
1994 Tuymaada Olympiad
Point $M$ lies inside triangle $ABC$. Prove that for any other point $N$ lying inside the triangle $ABC$,  at least one of the following three inequalities is fulfilled: $AN>AM, BN>BM, CN>CM$.
1994 Tuymaada Olympiad
Let a convex polyhedron be given with volume $V$ and full surface $S$.

Prove that inside a polyhedron it is possible to arrange a ball of radius $\frac{V}{S}$.
1994 Tuymaada Olympiad
Find the smallest natural number $n$ for which  $sin \Big(\frac{1}{n+1934}\Big)<\frac{1}{1994}$ .
1994 Tuymaada Olympiad
In three houses $A,B$ and $C$, forming a right triangle with the legs $AC=30$ and $CB=40$, live three beetles $a,b$ and $c$, capable of moving at speeds of $2, 3$ and $4$, respectively. Suppose that you simultaneously release these bugs from point $M$ and mark the time after which beetles reach their homes. Find on the plane such a point $M$, where is the last time to reach the house a bug would be minimal.
1994 Tuymaada Olympiad
Prove that there are infinitely many natural numbers $a,b,c,u$ and $v$  with greatest common divisor $1$ satisfying the system of equations:  $a+b+c=u+v$ and $a^2+b^2+c^2=u^2+v^2$
1994 Tuymaada Olympiad
Prove that in space there is a sphere containing exactly $1994$ points with integer coordinates.
2021 Francophone Mathematical Olympiad
Let $R$ and $S$ be the numbers defined by

$$R = \dfrac{1}{2} \times \dfrac{3}{4} \times \dfrac{5}{6} \times \cdots \times \dfrac{223}{224} \text{ and } S = \dfrac{2}{3} \times \dfrac{4}{5} \times \dfrac{6}{7} \times \cdots \times \dfrac{224}{225}.$$Prove that $R < \dfrac{1}{15} < S$.
2021 Francophone Mathematical Olympiad
Evariste has drawn twelve triangles as follows, so that two consecutive triangles share exactly one edge.



Sophie colors every triangle side in red, green or blue. Among the $3^{24}$ possible colorings, how many have the property that every triangle has one edge of each color?
2021 Francophone Mathematical Olympiad
Every point in the plane was colored in red or blue.

Prove that one the two following statements is true:

$\bullet$ there exist two red points at distance $1$ from each other;

$\bullet$   there exist four blue points $B_1$, $B_2$, $B_3$, $B_4$ such that the points $B_i$ and $B_j$ are at distance $|i-j|$ from each other, for all integers $i$ and $j$ such as $1 \le i \le 4$ and $1 \le  j  \le  4$.
2021 Francophone Mathematical Olympiad
Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers.

Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$:

$$\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) =

\mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).$$

Note: if $a$ and $b$ are positive integers, $\mathrm{GCD}(a,b)$ is the largest positive integer that divides both $a$ and $b$, and $\mathrm{LCM}(a,b)$ is the smallest positive integer that is a multiple of both $a$ and $b$.
2021 Francophone Mathematical Olympiad
Let $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ be positive integers such that $a_{n+2} = a_n + a_{n+1}$ and $b_{n+2} = b_n + b_{n+1}$ for all $n \ge 1$. Assume that $a_n$ divides $b_n$ for infinitely many values of $n$. Prove that there exists an integer $c$ such that $b_n = c a_n$ for all $n \ge 1$.
2021 Francophone Mathematical Olympiad
Albert and Beatrice play a game. $2021$ stones lie on a table. Starting with Albert, they alternatively remove stones from the table, while obeying the following rule. At the $n$-th turn, the active player (Albert if $n$ is odd, Beatrice if $n$ is even) can remove from $1$ to $n$ stones. Thus, Albert first removes $1$ stone; then, Beatrice can remove $1$ or $2$ stones, as she wishes; then, Albert can remove from $1$ to $3$ stones, and so on.

The player who removes the last stone on the table loses, and the other one wins. Which player has a strategy to win regardless of the other player's moves?
2021 Francophone Mathematical Olympiad
Let $ABCD$ be a square with incircle $\Gamma$. Let $M$ be the midpoint of the segment $[CD]$. Let $P \neq B$ be a point on the segment $[AB]$. Let $E \neq M$ be the point on $\Gamma$ such that $(DP)$ and $(EM)$ are parallel. The lines $(CP)$ and $(AD)$ meet each other at $F$. Prove that the line $(EF)$ is tangent to $\Gamma$
2021 Francophone Mathematical Olympiad
Let $\mathbb{N}_{\ge 1}$ be the set of positive integers.

Find all functions $f \colon \mathbb{N}_{\ge 1} \to \mathbb{N}_{\ge 1}$ such that, for all positive integers $m$ and $n$:



(a) $n = \left(f(2n)-f(n)\right)\left(2 f(n) - f(2n)\right)$,

(b)$f(m)f(n) - f(mn) = \left(f(2m)-f(m)\right)\left(2 f(n) - f(2n)\right) + \left(f(2n)-f(n)\right)\left(2 f(m) - f(2m)\right)$,

(c) $m-n$ divides $f(2m)-f(2n)$ if $m$ and $n$ are distinct odd prime numbers.
2020 Francophone Mathematical Olympiad
Let $ABC$ be a triangle such that $AB <AC$, $\omega$ its inscribed circle and $\Gamma$ its circumscribed circle. Let also $\omega_b$ be the excircle relative to vertex $B$, then $B'$ is the point of tangency between $\omega_b$ and $(AC)$. Similarly, let the circle $\omega_c$ be the excircle exinscribed relative to vertex $C$, then $C'$ is the point of tangency between $\omega_c$ and $(AB)$. Finally, let $I$ be the center of $\omega$  and $X$ the point of $\Gamma$  such that $\angle XAI$ is a right angle. Prove that the triangles $XBC'$ and $XCB'$ are congruent.
2020 Francophone Mathematical Olympiad
Emperor Zorg wishes to found a colony on a new planet. Each of the $n$ cities that he will establish there will have to speak exactly one of the Empire's $2020$ official languages. Some towns in the colony will be connected by a direct air link, each link can be taken in both directions. The emperor fixed the cost of the ticket for each connection to $1$ galactic credit. He wishes that, given any two cities speaking the same language, it is always possible to travel from one to the other via these air links, and that the cheapest trip between these two cities costs exactly $2020$ galactic credits. For what values of $n$ can Emperor Zorg fulfill his dream?
2020 Francophone Mathematical Olympiad
Let $n$ be an integer greater than or equal to $1$. Find, as a function of $n$, the smallest integer $k\ge 2$ such that, among any $k$ real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than $1 / n$, either  strictly greater than $n$.
2020 Francophone Mathematical Olympiad
Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \cdot  7^y = z^3$
2020 Francophone Mathematical Olympiad
Let $ABC$ be an acute triangle with $AC>AB$, Let $DEF$ be the intouch triangle with $D \in (BC)$,$E \in (AC)$,$F \in (AB)$,, let $G$ be the intersecttion of the perpendicular from $D$ to $EF$ with $AB$, and $X=(ABC)\cap (AEF)$.

Prove that $B,D,G$ and $X$ are concylic
2020 Francophone Mathematical Olympiad
Let $a_1,a_2,\ldots,a_n$ be a finite sequence of non negative integers, its subsequences are the sequences of the form $a_i,a_{i+1},\ldots,a_j$ with $1\le i\le j \le n$. Two subsequences are said to be equal if they have the same length and have the same terms, that is, two subsequences $a_i,a_{i+1},\ldots,a_j$ and $a_u,a_{u+1},\ldots a_v$ are equal iff $j-i=u-v$ and $a_{i+k}=a_{u+k}$ forall integers $k$ such that $0\le k\le j-1$. Finally, we say that a subsequence $a_i,a_{i+1},\ldots,a_j$ is palindromic if $a_{i+k}=a_{j-k}$ forall integers $k$ such that $0\le k \le j-i$

What is the greatest number of different palindromic subsequences that can a palindromic sequence of length $n$ contain?
2020 Francophone Mathematical Olympiad
Let $(a_i)_{i\in \mathbb{N}}$ be a sequence with $a_1=\frac{3}2$ such that

$$a_{n+1}=1+\frac{n}{a_n}$$Find $n$ such that $2020\le a_n <2021$