Navvye/Olympiad-Problems-CSV huggingface dataset:
2012 Peru IMO TST
An infinite triangular lattice is given, such that the distance between any two adjacent points is always equal to $1$.

Points $A$, $B$, and $C$ are chosen on the lattice such that they are the vertices of an equilateral triangle of side length $L$, and the sides of $ABC$ contain no points from the lattice. Prove that, inside triangle $ABC$, there are exactly $\frac{L^2-1}{2}$ points from the lattice.
2012 Peru IMO TST
Let $ABCD$ be a parallelogram such that $\angle{ABC} > 90^{\circ}$, and $\mathcal{L}$ the line perpendicular to $BC$ that passes through $B$. Suppose that the segment $CD$ does not intersect $\mathcal{L}$. Of all the circumferences that pass through $C$ and $D$, there is one that is tangent to $\mathcal{L}$ at $P$, and there is another one that is tangent to $\mathcal{L}$ at $Q$ (where $P \neq Q$). If $M$ is the midpoint of $AB$, prove that $\angle{PMD} = \angle{QMD}$.
2012 Peru IMO TST
Let $p$ be an odd prime number. For every integer $a,$ define the number $S_a = \sum^{p-1}_{j=1} \frac{a^j}{j}.$ Let $m,n \in \mathbb{Z},$ such that $S_3 + S_4 - 3S_2 = \frac{m}{n}.$ Prove that $p$ divides $m.$



Proposed by Romeo Meštrović, Montenegro
2011 Peru IMO TST
Let $\Bbb{Z}^+$ denote the set of positive integers. Find all functions $f:\Bbb{Z}^+\to \Bbb{Z}^+$ that satisfy the following condition: for each positive integer $n,$ there exists a positive integer $k$ such that $$\sum_{i=1}^k f_i(n)=kn,$$where $f_1(n)=f(n)$ and $f_{i+1}(n)=f(f_i(n)),$ for $i\geq 1. $
2011 Peru IMO TST
Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively,

$$\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.$$

Proposed by Nairi Sedrakyan, Armenia
2011 Peru IMO TST
Let $ABC$ be an acute triangle, and $AA_1$, $BB_1$, and $CC_1$ its altitudes. Let $A_2$ be a point on segment $AA_1$ such that $\angle{BA_2C} = 90^{\circ}$. The points $B_2$ and $C_2$ are defined similarly. Let $A_3$ be the intersection point of segments $B_2C$ and $BC_2$. The points $B_3$ and $C_3$ are defined similarly. Prove that the segments $A_2A_3$, $B_2B_3$, and $C_2C_3$ are concurrent.
2011 Peru IMO TST
On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set.



Proposed by Tonći Kokan, Croatia
2011 Peru IMO TST
Let $a_1, a_2, \cdots , a_n$ be real numbers, with $n\geq  3,$ such that $a_1 + a_2 +\cdots +a_n = 0$ and  $$ 2a_k\leq a_{k-1} + a_{k+1} \ \  \ \text{for} \ \  \  k = 2, 3, \cdots , n-1.$$Find the least number $\lambda(n),$ such that for all $k\in \{ 1, 2, \cdots, n\} $ it is satisfied that $|a_k|\leq \lambda (n)\cdot \max \{|a_1|, |a_n|\} .$
2010 Peru IMO TST
Let $ABC$ be an acute-angled triangle and $F$ a point in its interior such that $$ \angle AFB = \angle BFC = \angle CFA = 120^{\circ}.$$Prove that the Euler lines of the triangles $AFB, BFC$ and $CFA$ are concurrent.
2010 Peru IMO TST
Let $ \displaystyle{a,b,c}$ be positive real numbers such that $\displaystyle{a+b+c=1.}$ Prove that $$ \displaystyle{\frac{1+ab}{a+b}+\frac{1+bc}{b+c}+\frac{1+ca}{c+a}\geq 5.}$$
2010 Peru IMO TST
Let $\Bbb{N}$ be the set of positive integers. For each subset $\mathcal{X}$ of $\Bbb{N}$ we define the set $\Delta(\mathcal{X})$ as the set of all numbers $| m - n |,$ where $m$ and $n$ are elements of $\mathcal{X}$, ie: $$\Delta (\mathcal{X}) = \{ |m-n| \ | \  m, n \in \mathcal{X} \}$$Let $\mathcal A$ and $\mathcal B$ be two infinite, disjoint sets whose union is $\Bbb{N.}$

a) Prove that the set $\Delta (\mathcal A) \cap \Delta (\mathcal B)$ has infinitely many elements.

b) Prove that there exists  an infinite subset $\mathcal C$ of $\Bbb{N}$ such that $\Delta (\mathcal C)$  is a subset of $\Delta (\mathcal A) \cap \Delta (\mathcal B).$
2010 Peru IMO TST
Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1$ on $CD$, the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent.



Proposed by Eugene Bilopitov, Ukraine
2010 Peru IMO TST
Let $a, b, c$ be positive real numbers such that $a + b + c = 1.$ Prove that $$ \displaystyle{\frac{1}{a + b}+\frac{1}{b + c}+\frac{1}{c + a}+ 3(ab + bc + ca) \geq \frac{11}{2}.}$$
2009 Peru IMO TST
Show that there are infinitely many triples $(x, y, z)$ of real numbers such that $$\displaystyle{x^2+y = y^2+z= z^2 + x}$$and $x\ne y\ne z \ne x.$
2009 Peru IMO TST
300 bureaucrats are split into three comissions of 100 people. Each two bureaucrats are either familiar to each other or non familiar to each other. Prove that there exists two bureaucrats from  two distinct commissions such that the third commission contains either 17 bureaucrats familiar to both of them, or 17 bureaucrats familiar to none of them.

_________________________________________



This problem is taken from Russian Olympiad 2007-2008 district round 9.8



$ Tipe$
2009 Peru IMO TST
Let $ ABCDEF$ be a convex hexagon that has no pair of parallel sides. It is known  that, for every point $ P$ inside the hexagon, the sum: $$ \text{Area}[ABP]+\text{Area}[CDP]+\text{Area}[EFP]$$ has a constant value. Prove that the triangles  $ ACE$ and $ BDF$ have the same barycentre.



_____________________________________



This problem was proposed by Israel Diaz.



$ Tipe$
2009 Peru IMO TST
Show that there exist $2009$ consecutive positive integers such that for each of them the ratio between the largest and the smallest prime divisor is more than $20.$
2009 Peru IMO TST
Let $\mathcal{C}$ be the circumference inscribed  in the triangle $ABC,$ which is tangent to sides $BC, AC, AB$ at the points $A' , B' , C' ,$ respectively. The distinct points $K$ and $L$  are taken on $\mathcal{C}$  such that $$\angle AKB'+\angle BKA' =\angle ALB'+\angle BLA'=180^{\circ}.$$Prove that the points $A', B', C'$ are equidistant from the line $KL.$
2008 Peru IMO TST
Let $ ABC$ be a triangle and let $ I$ be the incenter. $ Ia$ $ Ib$ and $ Ic$ are the excenters opposite to points $ A$ $ B$ and $ C$ respectively. Let $ La$ be the line joining the orthocenters of triangles $ IBC$ and $ IaBC$. Define $ Lb$ and $ Lc$ in the same way.

Prove that $ La$ $ Lb$ and $ Lc$ are concurrent.



Daniel
2008 Peru IMO TST
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that $$ f(2f(x) + y) = f(f(x) - f(y)) + 2y + x, $$for all $x,y \in \mathbb{R}.$
2008 Peru IMO TST
Given a positive integer $n$, consider the sequence $(a_i)$, $1 \leq i \leq 2n$, defined as follows:



$a_{2k-1} = -k, 1 \leq k \leq n$

$a_{2k} = n-k+1, 1 \leq k \leq n.$



We call a pair of numbers $(b,c)$ good if the following conditions are met:



$i) 1 \leq b < c \leq 2n,$



$ii) \sum_{j=b}^{c}a_j = 0$



If $B(n)$ is the number of good pairs corresponding to $n$, prove that there are infinitely many $n$ for which $B(n) = n$.
2008 Peru IMO TST
Let $\mathcal{S}_1$ and $\mathcal{S}_2$ be two non-concentric circumferences such that $\mathcal{S}_1$ is inside $\mathcal{S}_2$. Let $K$ be a variable point on $\mathcal{S}_1$. The line tangent to $\mathcal{S}_1$ at point $K$ intersects $\mathcal{S}_2$ at points $A$ and $B$. Let $M$ be the midpoint of arc $AB$ that is in the semiplane determined by $AB$ that does not contain $\mathcal{S}_1$. Determine the locus of the point symmetric to $M$ with respect to $K.$
2008 Peru IMO TST
When we cut a rope into two pieces, we say that the cut is special if both pieces have different lengths. We cut a chord of length $2008$ into two pieces with integer lengths and we write those lengths on the board. Afterwards, we cut one of the pieces into two new pieces with integer lengths and we write those lengths on the board. This process ends until all pieces have length $1$.



$a)$ Find the minimum possible number of special cuts.

$b)$ Prove that, for all processes that have the minimum possible number of special cuts, the number of different integers on the board is always the same.
2008 Peru IMO TST
We say that a positive integer is happy if can expressed in the form $ (a^{2}b)/(a - b)$ where $ a > b > 0$ are integers. We also say that a positive integer $ m$ is evil if it doesn't a happy integer $ n$ such that $ d(n) = m$. Prove that all integers happy and evil are a power of $ 4$.
2007 Peru IMO TST
Let $P$ be an interior point of the semicircle whose diameter is $AB$ ($\angle APB$ is obtuse). The incircle of $\triangle ABP$ touches $AP$ and $BP$ at $M$ and $N$ respectively. The line $MN$ intersects the semicircle in $X$ and $Y$. Prove that $\widehat{XY}= \angle APB$.
2007 Peru IMO TST
Let $a,b,c$ be positive real numbers, such that: $a+b+c \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$



Prove that:

$$a+b+c \geq \frac{3}{a+b+c}+\frac{2}{abc}. $$
2007 Peru IMO TST
Let $N$ be a natural number which can be expressed in the form $a^{2}+b^{2}+c^{2}$, where $a,b,c$ are integers divisible by 3.

Prove that $N$ can be expressed in the form $x^{2}+y^{2}+z^{2}$, where $x,y,z$ are integers and any of them are divisible by 3.
2007 Peru IMO TST
Let be a board with $2007 \times 2007$ cells, we colour with black $P$ cells such that:



$\bullet$ there are no 3 colored cells that form a L-trinomes in any of its 4 orientations



Find the minimum value of $P$, such that when you colour one cell more, this configuration can't keep the condition above.
2007 Peru IMO TST
Let $k$ be a positive number and $P$ a Polynomio with integer coeficients.

Prove that exists a $n$ positive integer such that:

$P(1)+P(2)+\dots+P(N)$ is divisible by $k$.
2007 Peru IMO TST
Let $ABC$ be a triangle such that $CA \neq CB$,

the points $A_{1}$  and $B_{1}$ are tangency points for the ex-circles relative to sides $CB$ and $CA$,

respectively, and $I$ the incircle.

The line $CI$ intersects the cincumcircle of the triangle $ABC$ in the point $P$.

The line that trough $P$ that is perpendicular to $CP$, intersects the line $AB$ in $Q$.

Prove that the lines $QI$ and $A_{1}B_{1}$ are parallels.
2007 Peru IMO TST
Let $T$ a set with 2007 points on the plane, without any 3 collinear points.

Let $P$ any point which belongs to $T$.

Prove that the number of triangles that contains the point $P$ inside and

its vertices are from $T$, is even.
2007 Peru IMO TST
Let $a,b$ and $c$ be sides of a triangle. Prove that:

$\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3$
2006 Peru IMO TST
PERU TST IMO - 2006

Saturday, may 20.



Question 01



Find all $(x,y,z)$ positive integers, such that:



$\sqrt{\frac{2006}{x+y}} + \sqrt{\frac{2006}{y+z}} + \sqrt{\frac{2006}{z+x}},$



is an integer.



---

Spanish version



$\text{\LaTeX}{}$ed by carlosbr
2006 Peru IMO TST
PERU TST IMO - 2006

Saturday, may 20.



Question 02

Find all pairs $(a,b)$ real positive numbers $a$ and $b$ such that:



$[a[bn]]= n-1,$



for all $n$ positive integer.

Note: [x] denotes the integer part of $x$.



Spanish version



$\text{\LaTeX}{}$ed by carlosbr
2006 Peru IMO TST
PERU TST IMO - 2006

Saturday, may 20.



Question 03

In each square of a board drawn into squares of $2^n$   rows and

$n$ columns $(n\geq 1)$ are written a 1 or a -1, in such a way

that the rows of the board constitute all the possible sequences

of length $n$ that they are possible to be formed with numbers 1

and -1.

Next, some of the numbers are replaced by zeros.

Prove that it is possible to choose some of the rows of the board

(It could be a row only) so that in the chosen rows, is fulfilled that the

sum of the numbers in each column is zero.



----

Spanish version



$\text{\LaTeX}{}$ed by carlosbr
2006 Peru IMO TST
PERU TST IMO - 2006

Saturday, may 20.



Question 04

In an actue-angled triangle $ABC$ draws up: its circumcircle $w$

with center $O$, the circumcircle $w_1$ of the triangle $AOC$ and

the diameter $OQ$ of $w_1$. The points are chosen $M$ and $N$ on

the straight lines $AQ$ and $AC$, respectively, in such a way that

the quadrilateral $AMBN$ is a parallelogram.



Prove that the intersection point of the straight lines $MN$ and $BQ$ belongs the

circumference $w_1.$



---

Spanish version



$\text{\LaTeX}{}$ed by carlosbr
2014 Puerto Rico Team Selection Test
Let $ABCD$ be a parallelogram with $AB>BC$ and $\angle DAB$ less than $\angle ABC$. The perpendicular bisectors of sides $AB$ and $BC$ intersect at the point $M$ lying on the extension of $AD$. If $\angle MCD=15^{\circ}$, find the measure of $\angle ABC$
2014 Puerto Rico Team Selection Test
We have shortened the usual notation indicating with a sub-index the number of times that a digit is conseutively repeated. For example, $1119900009$ is denoted $1_3 9_2 0_4 9_1$.



Find $(x, y, z)$ if  $2_x 3_y 5_z + 3_z 5_x 2_y = 5_3 7_2 8_3 5_1 7_3$
2014 Puerto Rico Team Selection Test
Is it possible to tile an $8\times8$ board with dominoes ($2\times1$ tiles) so that no two dominoes form a $2\times2$ square?
2014 Puerto Rico Team Selection Test
Let $S$ be the set of natural numbers whose digits are different and belong to the set $\{1, 3, 5, 7\}$. Calculate the sum of the elements of $S$.
2014 Puerto Rico Team Selection Test
In a cycling competition with $14$ stages, one each day, and $100$ participants, a competitor was characterized by finishing $93^{\text{rd}}$ each day.What is the best place he could have finished in the overall standings? (Overall standings take into account the total cycling time over all stages.)
2014 Puerto Rico Team Selection Test
Natural numbers are written in the cells of of a $2014\times2014$ regular square grid such that every number is the average of the numbers in the adjacent cells. Describe and prove how the number distribution in the grid can be.
2014 Puerto Rico Team Selection Test
Consider $N$ points in the plane such that the area of a triangle formed by any three of the points does not exceed $1$. Prove that there is a triangle of area not more than $4$ that contains all $N$ points.
2013 Puerto Rico Team Selection Test
Claudia and Adela are betting to see which one of them will ask the boy they like for his telephone number. To decide they roll dice. If none of the dice are a multiple of 3, Claudia will do it. If exactly one die is a multiple of 3, Adela will do it. If 2 or more of the dice are a multiple of 3 neither one of them will do it. How many dice should be rolled so that the risk is the same for both Claudia and Adela?
2013 Puerto Rico Team Selection Test
How many 3-digit numbers have the property that the sum of their digits is even?
2013 Puerto Rico Team Selection Test
Find all pairs of natural numbers n and prime numbers p such that $\sqrt{n+\frac{p}{n}}$ is a natural number.
2013 Puerto Rico Team Selection Test
If $x_0=x_1=1$, and for $n\geq1$



$x_{n+1}=\frac{x_n^2}{x_{n-1}+2x_n}$,



find a formula for $x_n$ as a function of $n$.
2013 Puerto Rico Team Selection Test
Given an equilateral triangle we select an arbitrary point on its interior. We draw theperpendiculars from that point to the three sides of the triangle. Show that the sum of the lengths of these perpendiculars is equal to the height of the triangle.
2013 Puerto Rico Team Selection Test
A $9\times9$ checkerboard is colored with 2 colors. If we choose any $3\times1$ region on the checkerboard we can paint  all of the squares in that region with the color that is in the majority in that region. Show that with a finite number of these operations, we can paint the checkerboard all in the same color.
2013 Puerto Rico Team Selection Test
Show that if $\sqrt{x}-\sqrt{y}=10$, then $x-2y\leq200$.
2012 Puerto Rico Team Selection Test
Let $x, y$ and $z$ be consecutive integers such that

$$\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.$$

Find the maximum value of $x + y + z$.
2012 Puerto Rico Team Selection Test
A cone is constructed with a semicircular piece of paper, with radius 10. Find the

height of the cone.
2012 Puerto Rico Team Selection Test
$ABC$ is a triangle that is inscribed in a circle. The angle bisectors of $A, B, C$ meet the circle at $D,

E, F$, respectively. Show that $AD$ is perpendicular to $EF$.
2012 Puerto Rico Team Selection Test
Let $a, b, c, d$ be digits such that $d > c > b > a \geq 0$. How many numbers of the form $1a1b1c1d1$ are

multiples of $33$?
2012 Puerto Rico Team Selection Test
A point $P$ is outside of a circle and the distance to the center is $13$. A secant line from $P$ meets the circle at $Q$ and $R$ so that the exterior segment of the secant, $PQ$, is $9$ and $QR$ is $7$. Find the radius of the circle.
2012 Puerto Rico Team Selection Test
The increasing sequence $1; 3; 4; 9; 10; 12; 13; 27; 28; 30; 31, \ldots$ is formed with positive integers which are powers of $3$ or sums of different powers of $3$. Which number is in the $100^{th}$ position?
2012 Puerto Rico Team Selection Test
Let $f$ be a function with the following properties:



1) $f(n)$ is defined for every positive integer $n$;

2) $f(n)$ is an integer;

3) $f(2)=2$;

4) $f(mn)=f(m)f(n)$ for all $m$ and $n$;

5) $f(m)>f(n)$ whenever $m>n$.



Prove that $f(n)=n$.
2021 Romania Team Selection Test
Let $k>1$ be a positive integer. A set $S$ is called good if there exists a colouring of the positive integers with $k$ colours, such that no element from  $S$ can be written as the sum of two distinct positive integers having the same colour. Find the greatest positive integer $t$ (in terms of $k$) for which the set $$S=\{a+1,a+2,...,a+t\}$$is good, for any positive integer $a$.
2021 Romania Team Selection Test
For any positive integer $n>1$, let $p(n)$ be the greatest prime factor of $n$. Find all the triplets of distinct positive integers $(x,y,z)$ which satisfy the following properties: $x,y$ and $z$ form an arithmetic progression, and $p(xyz)\leq 3.$
2021 Romania Team Selection Test
The external bisectors of the angles of the convex quadrilateral $ABCD$ intersect each other in $E,F,G$ and $H$ such that $A\in EH, \ B\in EF, \ C\in FG, \ D\in GH$. We know that the perpendiculars from $E$ to $AB$, from $F$ to $BC$ and from $G$ to $CD$ are concurrent. Prove that $ABCD$ is cyclic.
2021 Romania Team Selection Test
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the relationship $$f(xf(y)-f(x))=2f(x)+xy,\text{ for any }x,y\in\mathbb{R}.$$
2021 Romania Team Selection Test
Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$
2021 Romania Team Selection Test
Let $\mathcal{P}$ be a convex quadrilateral. Consider a point $X$ inside $\mathcal{P}.$ Let $M,N,P,Q$ be the projections of $X$ on the sides of $\mathcal{P}.$ We know that $M,N,P,Q$ all sit on a circle of center $L.$ Let $J$ and $K$ be the midpoints of the diagonals of $\mathcal{P}.$ Prove that $J,K$ and $L$ lie on a line.
2021 Romania Team Selection Test
Consider a fixed triangle $ABC$ such that $AB=AC.$ Let $M$ be the midpoint of $BC.$ Let $P$ be a variable point inside $\triangle ABC,$ such that $\angle PBC=\angle PCA.$ Prove that the sum of the measures of $\angle BPM$ and $\angle APC$ is constant.
2021 Romania Team Selection Test
Let $N\geq 4$ be a fixed positive integer. Two players, $A$ and $B$ are forming an ordered set $\{x_1,x_2,...\},$ adding elements alternatively. $A$ chooses $x_1$ to be $1$ or $-1,$ then $B$ chooses $x_2$ to be $2$ or $-2,$ then $A$ chooses $x_3$ to be $3$ or $-3,$ and so on. (at the $k^{th}$ step, the chosen number must always be $k$ or $-k$)



The winner is the first player to make the sequence sum up to a multiple of $N.$ Depending on $N,$ find out, with proof, which player has a winning strategy.
2021 Romania Team Selection Test
Let $\alpha$ be a real number in the interval $(0,1).$ Prove that there exists a sequence $(\varepsilon_n)_{n\geq 1}$ where each term is either $0$ or $1$ such that the sequence $(s_n)_{n\geq 1}$ $$s_n=\frac{\varepsilon_1}{(n)(n+1)}+\frac{\varepsilon_2}{(n+1)(n+2)}+...+\frac{\varepsilon_n}{(2n-1)2n}$$verifies the inequality $$0\leq \alpha-2ns_n\leq\frac{2}{n+1}$$for any $n\geq 2.$
2019 Romania Team Selection Test
Let $k\geq 2$,$n_1,n_2,\cdots ,n_k\in \mathbb{N}_+$,satisfied $n_2|2^{n_1}-1,n_3|2^{n_2}-1,\cdots ,n_k|2^{n_{k-1}}-1,n_1|2^{n_k}-1$.

Prove:$n_ 1=n_ 2=\cdots=n_k=1$.
2019 Romania Team Selection Test
Let be three positive integers $ a,b,c $ and a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ defined as

$$ f(n)=\left\{ \begin{matrix} n-a, & n>c\\ f\left( f(n+b) \right) ,& n\le c \end{matrix} \right. . $$Determine the number of fixed points this function has.
2019 Romania Team Selection Test
Let be two natural numbers $ m,n, $ and $ m $ pairwise disjoint sets of natural numbers $ A_0,A_1,\ldots ,A_{m-1}, $ each having $ n $ elements, such that no element of $ A_{i\pmod m} $ is divisible by an element of $ A_{i+1\pmod m} , $ for any natural number $ i. $

Determine the number of ordered pairs

$$ (a,b)\in\bigcup_{0\le j < m} A_j\times\bigcup_{0\le j < m} A_j $$such that $ a|b $ and such that $ \{ a,b \}\not\in A_k, $ for any $ k\in\{ 0,1,\ldots ,m-1 \} . $





Radu Bumbăcea
2019 Romania Team Selection Test
Let be a natural number $ n\ge 3. $ Find

$$ \inf_{\stackrel{ x_1,x_2,\ldots ,x_n\in\mathbb{R}_{>0}}{1=P\left( x_1,x_2,\ldots ,x_n\right)}}\sum_{i=1}^n\left( \frac{1}{x_i} -x_i \right) , $$where $ P\left( x_1,x_2,\ldots ,x_n\right) :=\sum_{i=1}^n \frac{1}{x_i+n-1} , $ and find in which circumstances this infimum is attained.
2019 Romania Team Selection Test
Determine the largest natural number $ N $ having the following property: every $ 5\times 5 $ array consisting of pairwise distinct natural numbers from $ 1 $ to $ 25 $ contains a $ 2\times 2 $ subarray of numbers whose sum is, at least, $ N. $





Demetres Christofides and Silouan Brazitikos
2019 Romania Team Selection Test
Prove that there exists an integer $n$, $n\geq 2002$, and $n$ distinct positive integers $a_1,a_2,\ldots,a_n$ such that the number $N= a_1^2a_2^2\cdots a_n^2 - 4(a_1^2+a_2^2+\cdots + a_n^2) $ is a perfect square.
2019 Romania Team Selection Test
Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.
2019 Romania Team Selection Test
For a natural number $ n, $ a string $ s $ of $ n $ binary digits and a natural number $ k\le n, $ define an $ n,s,k$ -block as a string of $ k $ consecutive elements from $ s. $ We say that two $ n,s,k\text{-blocks} , $ namely, $ a_1a_2\ldots a_k,b_1b_2\ldots b_k, $ are incompatible if there exists an $ i\in\{1,2,\ldots ,k\} $ such that $ a_i\neq b_i. $ Also, for two natural numbers $ r\le n, l, $ we say that $ s $ is $ r,l $ -typed if there are, at most, $ l $ pairwise incompatible $ n,s,r\text{-blocks} . $

Let be a $ 3,7\text{-typed} $ string $ t $ consisting of $ 10000 $ binary digits. Determine the maximum number $ M $ that satisfies the condition that $ t $ is $ 10,M\text{-typed} . $





Cătălin Gherghe
2019 Romania Team Selection Test
Let $ I,O $ denote the incenter, respectively, the circumcenter of a triangle $ ABC. $ The $ A\text{-excircle} $ touches the lines $ AB,AC,BC $ at $ K,L, $ respectively, $ M. $ The midpoint of $ KL $ lies on the circumcircle of $ ABC. $ Show that the points $ I,M,O $ are collinear.





Павел Кожевников
2019 Romania Team Selection Test
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.