Navvye/Olympiad-Problems-CSV huggingface dataset:
1978 Germany Team Selection Test
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$  is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$  such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.
1978 Germany Team Selection Test
Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds:



(i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$



(ii) some plane contains exactly three points from $E.$
1978 Germany Team Selection Test
A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let $k$ be a circle with radius $r \geq 2$, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle $k$ that has a neighboring point lying outside $k$. Similarly, an exterior boundary point is a lattice point lying outside the circle $k$ that has a neighboring point lying inside $k$. Prove that there are four more exterior boundary points than interior boundary points.
1977 Germany Team Selection Test
We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$
1977 Germany Team Selection Test
Determine the polynomials P of two variables so that:



a.) for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$



b.) for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$



c.) $P(1,0) =1.$
1977 Germany Team Selection Test
Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$
1977 Germany Team Selection Test
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)
2019 Greece Team Selection Test
Given an equilateral triangle with sidelength $k$ cm. With lines parallel to it's sides, we split it into $k^2$ small equilateral  triangles with sidelength $1$ cm. This way, a triangular grid is created. In every small triangle of sidelength $1$ cm,  we place exactly one integer from $1$ to $k^2$ (included), such that there are no such triangles having the same numbers. With vertices the points of the grid, regular hexagons are defined of sidelengths $1$ cm. We shall name as value  of the hexagon, the sum of the numbers that lie on the $6$ small equilateral triangles that the hexagon consists of . Find (in terms of the integer $k>4$) the maximum and the minimum value of the sum of the values of all hexagons .
2019 Greece Team Selection Test
Let a triangle  $ABC$ inscribed in a circle   $\Gamma$  with center  $O$. Let  $I$ the incenter of triangle  $ABC$ and  $D, E, F$  the contact points of the incircle with sides  $BC, AC, AB$ of triangle $ABC$  respectively . Let also  $S$ the foot of the perpendicular line from $D$ to the line  $EF$.Prove that line  $SI$ passes from the antidiametric point $N$  of  $A$ in the circle  $\Gamma$.( $AN$ is a diametre of the circle  $\Gamma$).
2019 Greece Team Selection Test
Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied:



Each number in the table is congruent to $1$ modulo $n$.

The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$.



Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.
2019 Greece Team Selection Test
Find all functions  $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every  $x,y>0$.
2018 Greece Team Selection Test
If $x, y, z$ are positive real numbers such that $x + y + z = 9xyz.$ Prove that:

$$\frac {x} {\sqrt {x^2+2yz+2}}+\frac {y} {\sqrt {y^2+2zx+2}}+\frac {z} {\sqrt {z^2+2xy+2}}\ge 1.$$

Consider when equality applies.
2018 Greece Team Selection Test
A triangle $ABC$ is inscribed in a circle $(C)$ .Let $G$ the centroid  of $\triangle ABC$ .

We draw   the altitudes $AD,BE,CF$ of the given triangle .Rays $AG$ and $GD$ meet (C) at $M$ and $N$.Prove that points $ F,E,M,N $ are concyclic.
2018 Greece Team Selection Test
Find all functions $f:\mathbb{Z}_{>0}\mapsto\mathbb{Z}_{>0}$ such that



$$xf(x)+(f(y))^2+2xf(y)$$is perfect square for all positive integers $x,y$.



**This problem was proposed by me for the BMO 2017 and it was shortlisted. We then used it in our TST.
2017 Greece Team Selection Test
Let $ABC$ be an acute-angled triangle inscribed in circle $c(O,R)$ with $AB<AC<BC$,

and $c_1$ be the inscribed circle of $ABC$ which intersects $AB, AC, BC$ at

$F, E, D$ respectivelly. Let $A', B', C'$ be points which lie on $c$ such that the quadrilaterals

$AEFA', BDFB', CDEC'$ are inscribable.

(1) Prove that $DEA'B'$ is inscribable.

(2) Prove that $DA', EB', FC'$ are concurrent.
2017 Greece Team Selection Test
Prove that the number $A=\frac{(4n)!}{(2n)!n!}$ is an integer and divisible by $2^{n+1}$,

where $n$ is a positive integer.
2017 Greece Team Selection Test
Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that:

$f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$

and $g(1)=-8$
2017 Greece Team Selection Test
Some positive integers are initially written on a board, where each $2$ of them are different.

Each time we can do the following moves:

(1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$

(2) If there are 2 numbers (written in the board) in the form $n, n+4$ we can erase them and write down $n-1$

After some moves, there might appear negative numbers. Find the maximum value of the integer $c$ such that:

Independetly of the starting numbers, each number which appears in any move is greater or equal to $c$
2016 Greece Team Selection Test
Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$.



Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.
2015 Greece Team Selection Test
Solve in positive integers the following equation; $xy(x+y-10)-3x^2-2y^2+21x+16y=60$
2015 Greece Team Selection Test
Consider $111$ distinct points which lie on or in the internal of a circle with radius 1.Prove that there are at least $1998$ segments formed by these points with length $\leq \sqrt{3}$
2015 Greece Team Selection Test
Let $ABC$ be an acute triangle with $\displaystyle{AB<AC<BC}$ inscribed in circle $ \displaystyle{c(O,R)}$.The excircle $\displaystyle{(c_A)}$ has center $\displaystyle{I}$ and touches the sides $\displaystyle{BC,AC,AB}$ of the triangle $ABC$ at $\displaystyle{D,E,Z} $ respectively.$ \displaystyle{AI}$ cuts $\displaystyle{(c)}$ at point $M$ and the circumcircle $\displaystyle{(c_1)}$ of triangle $\displaystyle{AZE}$ cuts $\displaystyle{(c)}$ at $K$.The circumcircle $\displaystyle{(c_2)}$ of the triangle $\displaystyle{OKM}$ cuts $\displaystyle{(c_1)} $ at point $N$.Prove that the point of intersection of the lines $AN,KI$ lies on $ \displaystyle{(c)}$.
2015 Greece Team Selection Test
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy $yf(x)+f(y) \geq f(xy)$
2014 Greece Team Selection Test
Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$



a) Prove that the sequence consists only of natural numbers.

b) Check if there are terms of the sequence divisible by $2011$.
2014 Greece Team Selection Test
Find all real non-zero polynomials satisfying $P(x)^3+3P(x)^2=P(x^{3})-3P(-x)$ for all $x\in\mathbb{R}$.
2014 Greece Team Selection Test
Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.
2014 Greece Team Selection Test
Square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides.A spider starts from $A$ and moving only rightward or upwards,tries to reach $C$.Every "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards(it can follow any possible order for the steps of each "movement").The spider completes $l$ "movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If $n=m\cdot l$,find the number of the possible paths the spider can follow to reach $C$.Note that $n,m,k,l\in \mathbb{N^{*}}$ with $k<m$.
2013 Greece Team Selection Test
Find all pairs of non-negative integers $(m,n)$ satisfying $\frac{n(n+2)}{4}=m^4+m^2-m+1$
2013 Greece Team Selection Test
Let $ABC$ be a non-isosceles,aqute triangle with $AB<AC$ inscribed in circle $c(O,R)$.The circle $c_{1}(B,AB)$ crosses $AC$ at $K$ and $c$ at $E$.

$KE$ crosses $c$ at $F$ and $BO$ crosses $KE$ at $L$ and $AC$ at $M$ while $AE$ crosses $BF$ at $D$.Prove that:

i)$D,L,M,F$ are concyclic.

ii)$B,D,K,M,E$ are concyclic.
2013 Greece Team Selection Test
Find the largest possible value of $M$ for which $\frac{x}{1+\frac{yz}{x}}+\frac{y}{1+\frac{zx}{y}}+\frac{z}{1+\frac{xy}{z}}\geq M$ for all $x,y,z>0$ with $xy+yz+zx=1$
2013 Greece Team Selection Test
Given are $n$ different concentric circles on the plane.Inside the disk with the smallest radius (strictly inside it),we consider two distinct points $A,B$.We consider $k$ distinct lines passing through $A$ and $m$ distinct lines passing through $B$.There is no line passing through both $A$ and $B$ and all the lines passing through $k$ intersect with all the lines passing through $B$.The intersections do not lie on some of the circles.Determine the maximum and the minimum number of regions formed by the lines and the circles and are inside the circles.
2013 Greece Team Selection Test
Determine whether the polynomial $P(x)=(x^2-2x+5)(x^2-4x+20)+1$ is irreducible over $\mathbb{Z}[X]$.
2013 Greece Team Selection Test
For the several values of the parameter $m\in \mathbb{N^{*}}$,find the pairs of integers $(a,b)$ that satisfy the relation



$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{[a,m]+[b,m]}{(a+b)m}=\frac{10}{11}$,



and,moreover,on the Cartesian plane $Oxy$ the lie in the square $D=\{(x,y):1\leq x\leq 36,1\leq y\leq 36\}$.



Note:$[k,l]$ denotes the least common multiple of the positive integers $k,l$.
2013 Greece Team Selection Test
Let $n$ be a positive integer. An equilateral triangle with side $n$ will be denoted by $T_n$ and is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two).

Let also $m$ be a positive integer with $m<n$ and suppose that $T_n$ and $T_m$ can be tiled with "trapezoids".

Prove that, if from $T_n$ we remove a $T_m$ with the same orientation, then the rest can be tiled with "trapezoids".
2012 Greece Team Selection Test
Find all triples $(p,m,n)$ satisfying the equation $p^m-n^3=8$ where $p$ is a prime number and $m,n$ are nonnegative integers.
2012 Greece Team Selection Test
Let $a,b,c$ be positive real numbers satisfying $a+b+c=3$.Prove that $\sum_{sym} \frac{a^{2}}{(b+c)^{3}}\geq \frac{3}{8}$
2012 Greece Team Selection Test
Let $n=3k$ be a positive integer (with $k\geq 2$). An equilateral triangle is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). We colour the points of the grid with three colours (red, blue and green) such that each two neighboring points have different colour.

Finally, the colour of a "trapezoid" will be the colour of the midpoint of its big base.

Find the number of all "trapezoids" in the grid (not necessarily disjoint)  and determine the number of red, blue and green "trapezoids".
2011 Greece Team Selection Test
Find all prime numbers $p,q$ such that:

$$p^4+p^3+p^2+p=q^2+q$$
2011 Greece Team Selection Test
What is the maximal number of crosses than can fit in a $10\times 11$ board without overlapping?

Is this problem well-known?

[asy]

size(4.58cm); 

real labelscalefactor = 0.5; /* changes label-to-point distance */

pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ 

pen dotstyle = black; /* point style */ 

real xmin = -3.18, xmax = 1.4, ymin = -0.22, ymax = 3.38;  /* image dimensions */



 /* draw figures */

draw((-3.,2.)--(1.,2.)); 

draw((-2.,3.)--(-2.,0.)); 

draw((-2.,0.)--(-1.,0.)); 

draw((-1.,0.)--(-1.,3.)); 

draw((-1.,3.)--(-2.,3.)); 

draw((-3.,1.)--(1.,1.)); 

draw((1.,1.)--(1.,2.)); 

draw((-3.,2.)--(-3.,1.)); 

draw((0.,2.)--(0.,1.)); 

draw((-1.,2.)--(-1.,1.)); 

draw((-2.,2.)--(-2.,1.)); 

 /* dots and labels */

clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); 

 /* end of picture */

[/asy]
2011 Greece Team Selection Test
Find all functions $f,g: \mathbb{Q}\to \mathbb{Q}$ such that the following two conditions hold:



$$f(g(x)-g(y))=f(g(x))-y \ \  (1)$$

$$g(f(x)-f(y))=g(f(x))-y\ \ (2)$$

for all $x,y \in \mathbb{Q}$.
2011 Greece Team Selection Test
Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively.  Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.
2010 Greece Team Selection Test
Solve in positive reals the system:

$x+y+z+w=4$

$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}=5-\frac{1}{xyzw}$
2010 Greece Team Selection Test
In a blackboard there are $K$ circles in a row such that one of the numbers $1,...,K$ is assigned to each circle from the left to the right.

Change of situation of a circle is to write in it or erase the number which is assigned to it.At the beginning no number is written in its own circle.

For every positive divisor $d$ of $K$ ,$1\leq d\leq K$ we change the situation of the circles in which their assigned numbers are divisible by $d$,performing for each divisor $d$ $K$ changes of situation.

Determine the value of $K$ for which the following holds;when this procedure is applied once for all positive divisors of $K$ ,then all numbers $1,2,3,...,K$ are written in the circles they were assigned in.
2010 Greece Team Selection Test
Let $ABC$ be a triangle,$O$ its circumcenter and $R$ the radius of its circumcircle.Denote by $O_{1}$ the symmetric of $O$ with respect to $BC$,$O_{2}$ the symmetric of $O$ with respect to $AC$ and by $O_{3}$ the symmetric of $O$ with respect to $AB$.

(a)Prove that the circles $C_{1}(O_{1},R)$, $C_{2}(O_{2},R)$, $C_{3}(O_{3},R)$ have a common point.

(b)Denote by $T$ this point.Let $l$ be an arbitary line passing through $T$ which intersects $C_{1}$ at $L$, $C_{2}$ at $M$ and $C_{3}$ at $K$.From $K,L,M$ drop perpendiculars to $AB,BC,AC$ respectively.Prove that these perpendiculars pass through a point.
2010 Greece Team Selection Test
Find all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$

and are strictly monotone in $(0,+\infty  )$
2009 Greece Team Selection Test
Suppose that $a$ is an even positive integer and $A=a^{n}+a^{n-1}+\ldots +a+1,n\in \mathbb{N^{*}}$ is a perfect square.Prove that $8\mid a$.
2009 Greece Team Selection Test
Given is a triangle $ABC$ with barycenter $G$ and circumcenter $O$.The perpendicular bisectors of $GA,GB,GC$ intersect at $A_1,B_1,C_1$.Show that $O$ is the barycenter of $\triangle{A_1B_1C_1}$.
2009 Greece Team Selection Test
Find all triples $(x,y,z)\in \mathbb{R}^{3}$ such that $x,y,z>3$ and $\frac{(x+2)^2}{y+z-2}+\frac{(y+4)^2}{z+x-4}+\frac{(z+6)^2}{x+y-6}=36$
2009 Greece Team Selection Test
Given are $N$ points on the plane such that no three of them are collinear,which are coloured red,green and black.We consider all the segments between these points and give to each segment a "value" according to the following conditions:

i.If at least one of the endpoints of a segment is black then the segment's "value" is $0$.

ii.If the endpoints of the segment have the same colour,re or green,then the segment's "value" is $1$.

iii.If the endpoints of the segment have different colours but none of them is black,then the segment's "value" is $-1$.



Determine the minimum possible sum of the "values" of the segments.
2008 Greece Team Selection Test
Find all possible values of $a\in \mathbb{R}$ and $n\in \mathbb{N^*}$ such that $f(x)=(x-1)^n+(x-2)^{2n+1}+(1-x^2)^{2n+1}+a$



is divisible by $\phi (x)=x^2-x+1$
2008 Greece Team Selection Test
In a village $X_0$ there are $80$ tourists who are about to visit $5$ nearby villages $X_1,X_2,X_3,X_4,X_5$.Each of them has chosen to visit only one of them.However,there are cases when the visit in a village forces the visitor to visit other villages among $X_1,X_2,X_3,X_4,X_5$.Each tourist visits only the village he has chosen and the villages he is forced to.If $X_1,X_2,X_3,X_4,X_5$ are totally visited by $40,60,65,70,75$ tourists respectively,then find how many tourists had chosen each one of them and determine all the ordered pairs $(X_i,X_j):i,j\in \{1,2,3,4,5\}$ which are such that,the visit in $X_i$ forces the visitor to visit $X_j$ as well.
2008 Greece Team Selection Test
Given is the equation $x^2+y^2-axy+2=0$ where $a$ is a positive integral parameter.



$i.$Show that,for $a\neq 4$ there exist no pairs $(x,y)$ of positive integers satisfying the equation.



$ii.$ Show that,for $a=4$ there exist infinite pairs $(x,y)$ of positive integers satisfying the equation,and determine those pairs.
2005 Greece Team Selection Test
Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than  three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$  are divisible by $p$.
2020 Hong Kong TST
Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for every positive integer $n$ the following is valid: If $d_1,d_2,\ldots,d_s$ are all the positive divisors of $n$, then $$f(d_1)f(d_2)\ldots f(d_s)=n.$$
2020 Hong Kong TST
Let D be an arbitrary point inside $\Delta ABC$. Let $\Gamma$ be the circumcircle of $\Delta BCD$. The external angle bisector of $\angle ABC$ meets $\Gamma$ again at $E$. The external angle bisector of $\angle ACB$ meets $\Gamma$ again at $F$. The line $EF$ meets the extension of $AB$ and $AC$ at $P$ and $Q$ respectively. Prove that the circumcircles of $\Delta BFP$ and $\Delta CEQ$ always pass through the same fixed point regardless of the position of $D$. (Assume all the labelled points are distinct.)
2020 Hong Kong TST
Given a list of integers $2^1+1, 2^2+1, \ldots, 2^{2019}+1$, Adam chooses two different integers from the list and computes their greatest common divisor. Find the sum of all possible values of this greatest common divisor.
2020 Hong Kong TST
Find the total number of primes $p<100$ such that $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$ is divisible by $p$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
2020 Hong Kong TST
In $\Delta ABC$, let $D$ be a point on side $BC$. Suppose the incircle $\omega_1$ of $\Delta ABD$ touches sides $AB$ and $AD$ at $E,F$ respectively, and the incircle $\omega_2$ of $\Delta ACD$ touches sides $AD$ and $AC$ at $F,G$ respectively. Suppose the segment $EG$ intersects $\omega_1$ and $\omega_2$ again at $P$ and $Q$ respectively. Show that line $AD$, tangent of $\omega_1$ at $P$ and tangent of $\omega_2$ at $Q$ are concurrent.
2020 Hong Kong TST
For a sequence with some ones and zeros, we count the number of continuous runs of equal digits in it. (For example the sequence $011001010$ has $7$ continuous runs: $0,11,00,1,0,1,0$.) Find the sum of the number of all continuous runs for all possible sequences with $2019$ ones and $2019$ zeros.
2020 Hong Kong TST
Let $\Delta ABC$ be an acute triangle with incenter $I$ and orthocenter $H$. $AI$ meets the circumcircle of $\Delta ABC$ again at $M$. Suppose the length $IM$ is exactly the circumradius of $\Delta ABC$. Show that $AH\geq AI$.
2020 Hong Kong TST
Suppose there are $2019$ distinct points in a plane and the distances between pairs of them attain $k$ different values. Prove that $k$ is at least $44$.
2020 Hong Kong TST
Two circles $\Gamma$ and $\Omega$ intersect at two distinct points $A$ and $B$. Let $P$ be a point on $\Gamma$. The tangent at $P$ to $\Gamma$ meets $\Omega$ at the points $C$ and $D$, where $D$ lies between $P$ and $C$, and $ABCD$ is a convex quadrilateral. The lines $CA$ and $CB$ meet $\Gamma$ again at $E$ and $F$ respectively. The lines $DA$ and $DB$ meet $\Gamma$ again at $S$ and $T$ respectively. Suppose the points $P,E,S,F,B,T,A$ lie on $\Gamma$ in this order. Prove that $PC,ET,SF$ are parallel.
2020 Hong Kong TST
Find all real-valued functions $f$ defined on the set of real numbers such that $$f(f(x)+y)+f(x+f(y))=2f(xf(y))$$for any real numbers $x$ and $y$.
2019 Hong Kong TST
Let $a$ be a real number. Suppose the function $f(x) = \frac{a}{x-1} + \frac{1}{x-2} + \frac{1}{x-6}$ defined in the interval $3 < x < 5$ attains its maximum at $x=4$. Find the value of $a.$
2019 Hong Kong TST
A circle is circumscribed around an isosceles triangle whose two base angles are equal to $x^{\circ}$. Two points are chosen independently and randomly on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}.$ Find the sum of the largest and smallest possible value of $x$.
2019 Hong Kong TST
Find an integral solution of the equation

$$ \left \lfloor \frac{x}{1!} \right \rfloor + \left \lfloor \frac{x}{2!} \right \rfloor + \left \lfloor \frac{x}{3!} \right \rfloor + \dots + \left \lfloor \frac{x}{10!} \right \rfloor = 2019. $$(Note $\lfloor u \rfloor$ stands for the greatest integer less than or equal to $u$.)
2019 Hong Kong TST
Let $ABC$ be an acute-angled triangle such that $\angle{ACB} = 45^{\circ}$. Let $G$ be the point of intersection of the three medians and let $O$ be the circumcentre. Suppose $OG=1$ and $OG \parallel BC$. Determine the length of the segment $BC$.
2019 Hong Kong TST
Is it is possible to choose 24 distinct points in the space such that no three of them lie on the same line and choose 2019 distinct planes in a way that each plane passes through at least 3 of the chosen points and each triple belongs to one of the chosen planes?
2019 Hong Kong TST
If $57a + 88b + 125c \geq 1148$, where $a,b,c > 0$, what is the minimum value of

$$ a^3 + b^3 + c^3 + 5a^2 + 5b^2 + 5c^2? $$
2019 Hong Kong TST
Determine all sequences $p_1, p_2, \dots $ of prime numbers for which there exists an integer $k$ such that the recurrence relation

$$ p_{n+2} = p_{n+1} + p_n + k $$holds for all positive integers $n$.