Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 Croatia Team Selection Test
Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.
2011 Croatia Team Selection Test
Find all pairs of integers $x,y$ for which

$$x^3+x^2+x=y^2+y.$$
2009 Croatia Team Selection Test
Prove for all positive reals a,b,c,d:





$ \frac{a-b}{b+c}+\frac{b-c}{c+d}+\frac{c-d}{d+a}+\frac{d-a}{a+b} \geq 0$
2009 Croatia Team Selection Test
On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.
2009 Croatia Team Selection Test
On sides $ AB$ and $ AC$ of triangle $ ABC$ there are given points $ D,E$ such that  $ DE$ is tangent of circle inscribed in triangle $ ABC$ and $ DE \parallel BC$. Prove

$ AB+BC+CA\geq 8DE$
2009 Croatia Team Selection Test
Prove that there are infinite many positive integers $ n$ such that

$ n^2+1\mid n!$, and infinite many of those for which $ n^2+1 \nmid n!$.
2009 Croatia Team Selection Test
Determine the lowest positive integer n such that following statement is true:

If polynomial with integer coefficients gets value 2 for n different integers,

then it can't take value 4 for any integer.
2009 Croatia Team Selection Test
In each field of 2009*2009 table you can write either 1 or -1.

Denote Ak multiple of all numbers in k-th row and Bj the multiple of all numbers in j-th column.

Is it possible to write the numbers in such a way that

$ \sum_{i=1}^{2009}{Ai}+	\sum_{i=1}^{2009}{Bi}=0$?
2009 Croatia Team Selection Test
It is given a convex quadrilateral $ ABCD$ in which  $ \angle B+\angle C < 180^0$.

Lines $ AB$ and $ CD$ intersect in point E. Prove that

$ CD*CE=AC^2+AB*AE \leftrightarrow \angle B= \angle D$
2009 Croatia Team Selection Test
Determine all triplets off positive integers $ (a,b,c)$ for which $ \mid2^a-b^c\mid=1$
2009 Croatia Team Selection Test
Solve in the set of real numbers:

$$ 3\left(x^2 + y^2 + z^2\right) = 1,

$$



$$ x^2y^2 + y^2z^2 + z^2x^2 = xyz\left(x + y + z\right)^3.

$$
2009 Croatia Team Selection Test
Every natural number is coloured in one of the $ k$ colors. Prove that there exist four distinct natural numbers $ a, b, c, d$, all coloured in the same colour, such that $ ad = bc$, $ \displaystyle \frac b a$ is power of 2 and $ \displaystyle \frac c a$ is power of 3.
2009 Croatia Team Selection Test
A triangle $ ABC$ is given with $ \left|AB\right| > \left|AC\right|$. Line $ l$ tangents in a point $ A$ the circumcirle of $ ABC$. A circle centered in $ A$ with radius $ \left|AC\right|$ cuts $ AB$ in the point $ D$ and the line $ l$ in points $ E, F$ (such that $ C$ and $ E$ are in the same halfplane with respect to $ AB$). Prove that the line $ DE$ passes through the incenter of $ ABC$.
2009 Croatia Team Selection Test
Determine all natural $ n$ for which there exists natural $ m$ divisible by all natural numbers from 1 to $ n$ but not divisible by any of the numbers $ n + 1$, $ n + 2$, $ n + 3$.
2008 Croatia Team Selection Test
Let $ x$, $ y$, $ z$ be positive numbers. Find the minimum value of:

$ (a)\quad \frac{x^2 + y^2 + z^2}{xy + yz}$



$ (b)\quad \frac{x^2 + y^2 + 2z^2}{xy + yz}$
2008 Croatia Team Selection Test
For which $ n\in \mathbb{N}$ do there exist rational numbers $ a,b$ which are not integers such that both $ a + b$ and $ a^n + b^n$ are integers?
2008 Croatia Team Selection Test
Point $ M$ is taken on side $ BC$ of a triangle $ ABC$ such that the centroid $ T_c$ of triangle $ ABM$ lies on the circumcircle of $ \triangle ACM$ and the centroid $ T_b$ of $ \triangle ACM$ lies on the circumcircle of $ \triangle ABM$. Prove that the medians of the triangles $ ABM$ and $ ACM$ from $ M$ are of the same length.
2008 Croatia Team Selection Test
Let $ S$ be the set of all odd positive integers less than $ 30m$ which are not multiples of $ 5$, where $ m$ is a given positive integer. Find the smallest positive integer $ k$ such that each $ k$-element subset of $ S$ contains two distinct numbers, one of which divides the other.
2007 Croatia Team Selection Test
Find integral solutions to the equation $$(m^{2}-n^{2})^{2}=16n+1.$$
2007 Croatia Team Selection Test
Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.
2007 Croatia Team Selection Test
Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.
2007 Croatia Team Selection Test
Given a finite string $S$ of symbols $X$ and $O$, we write $@(S)$ for the number of $X$'s in $S$ minus the number of $O$'s. (For example, $@(XOOXOOX) =-1$.) We call a string $S$ balanced if every substring $T$ of (consecutive symbols) $S$ has the property $-2 \leq @(T) \leq 2$. (Thus $XOOXOOX$ is not balanced since it contains the sub-string $OOXOO$ whose $@$-value is $-3$.) Find, with proof, the number of balanced strings of length $n$.
2007 Croatia Team Selection Test
Let there be two circles. Find all points $M$ such that there exist two points, one on each circle such that $M$ is their midpoint.
2007 Croatia Team Selection Test
Let $a,b,c>0$ such that $a+b+c=1$. Prove: $$\frac{a^{2}}b+\frac{b^{2}}c+\frac{c^{2}}a \ge 3(a^{2}+b^{2}+c^{2}) $$
2006 Croatia Team Selection Test
Assume that $a, b,$ and $c$ are positive real numbers for which $(a+b)(a+c)(b+c) = 1$. Prove that $ab+bc+ca \leq\frac{3 }{4}.$
2006 Croatia Team Selection Test
Let $ABC$ be a triangle for which $AB+BC = 3AC$. Let $D$ and $E$ be the points of tangency of the incircle with the sides $AB$ and $BC$ respectively, and let $K$ and $L$ be the other endpoints of the diameters originating from $D$ and $E.$ Prove that $C , A, L$, and $K$ lie on a circle.
2006 Croatia Team Selection Test
Find all natural solutions of  $3^{x}= 2^{x}y+1.$
2004 Croatia Team Selection Test
Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$
2004 Croatia Team Selection Test
Prove that if $a,b,c$ are positive numbers with $abc=1$, then

$$\frac{a}{b} +\frac{b}{c} + \frac{c}{a} \ge a + b + c. $$
2004 Croatia Team Selection Test
A line intersects a semicircle with diameter $AB$ and center $O$ at $C$ and $D$, and the line $AB$ at $M$, where $MB < MA$ and $MD < MC.$ If the circumcircles of the triangles $AOC$ and $DOB$ meet again at $K,$ prove that $\angle MKO$ is right.
2003 Croatia Team Selection Test
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.
2003 Croatia Team Selection Test
Let $B$ be a point on a circle $k_1, A \ne B$ be a point on the tangent to the circle at $B$, and $C$ a point not lying on $k_1$ for which the segment $AC$ meets $k_1$ at two distinct points. Circle $k_2$ is tangent to line $AC$ at $C$ and to $k_1$ at point $D$, and does not lie in the same half-plane as $B$. Prove that the circumcenter of triangle $BCD$ lies on the circumcircle of $\vartriangle ABC$
2003 Croatia Team Selection Test
For which $n \in N$ is it possible to arrange a tennis tournament for doubles with $n$ players such that each player has every other player as an opponent exactly once?
2002 Croatia Team Selection Test
A quadrilateral $ABCD$ is circumscribed about a circle. Lines $AC$ and $DC$ meet at point $E$ and lines $DA$ and $BC$ meet at $F$, where $B$ is between $A$ and $E$ and between $C$ and $F$. Let $I_1, I_2$ and $I_3$ be the incenters of triangles $AFB, BEC$ and $ABC$, respectively. The line $I_1I_3$ intersects $EA$ at $K$ and $ED$ at $L$, whereas the line $I_2I_3$ intersects $FC$ at $M$ and $FD$ at $N$. Prove that $EK = EL$ if and only if $FM = FN$
2002 Croatia Team Selection Test
Prove that if $n$ is a natural number such that $1 + 2^n + 4^n$ is prime then $n = 3^k$ for some $k \in N_0$.
2001 Croatia Team Selection Test
Consider $A = \{1, 2, ..., 16\}$. A partition of $A$ into nonempty sets $A_1, A_2,..., A_n$ is said to be good if none of the Ai contains elements $a, b, c$ (not necessarily distinct) such that $a = b + c$.

(a) Find a good partition $\{A_1, A_2, A_3, A_4\}$ of $A$.

(b) Prove that no partition $\{A_1, A_2, A_3\}$ of $A$ is good
2001 Croatia Team Selection Test
Circles $k_1$ and $k_2$ intersect at $P$ and $Q$, and $A$ and $B$ are the tangency points of their common tangent that is closer to $P$ (where $A$ is on $k_1$ and $B$ on $k_2$). The tangent to $k_1$ at $P$ intersects $k_2$ again at $C$. The lines $AP$ and $BC$ meet at $R$. Show that the lines $BP$ and $BC$ are tangent to the circumcircle of triangle $PQR$.
2001 Croatia Team Selection Test
Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.
2018 Cyprus IMO TST
Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square.
2018 Cyprus IMO TST
Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$.
2018 Cyprus IMO TST
Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression

$$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$obtains its minimum value.
2018 Cyprus IMO TST
Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$.



(a) Prove that

$$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$(b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence:

$$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$
2018 Cyprus IMO TST
Taken from Cyprus national math olympiad webpage.
2018 Dutch BxMO TST
We have $1000$ balls in $40$ different colours, $25$ balls of each colour. Determine the smallest $n$ for which the following holds: if you place the $1000$ balls in a circle, in any arbitrary way, then there are always $n$ adjacent balls which have at least $20$ different colours.
2018 Dutch BxMO TST
Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.
2018 Dutch BxMO TST
Let $p$ be a prime number.

Prove that it is possible to choose a permutation $a_1, a_2,...,a_p$ of $1,2,...,p$ such that the numbers $a_1, a_1a_2, a_1a_2a_3,..., a_1a_2a_3...a_p$ all have different remainder upon division by $p$.
2018 Dutch BxMO TST
In a non-isosceles triangle $\vartriangle ABC$ we have $\angle BAC = 60^o$. Let $D$ be the intersection of the angular bisector of $\angle BAC$ with side $BC, O$ the centre of the circumcircle of $\vartriangle ABC$ and $E$ the intersection of $AO$ and $BC$. Prove that $\angle AED + \angle ADO = 90^o$.
2018 Dutch BxMO TST
Let $n$ be a positive integer. Determine all positive real numbers $x$ satisfying

$nx^2 +\frac{2^2}{x + 1}+\frac{3^2}{x + 2}+...+\frac{(n + 1)^2}{x + n}= nx + \frac{n(n + 3)}{2}$
2017 Dutch BxMO TST
Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq  m \leq  n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine

the minimum number of integers in a complete sequence of $n$ numbers.
2017 Dutch BxMO TST
Let  define a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that :

$i)$$f(p)=1$ for all prime numbers $p$.

$ii)$$f(xy)=xf(y)+yf(x)$ for all  positive integers $x,y$

find  the smallest $n \geq 2016$ such that $f(n)=n$
2017 Dutch BxMO TST
Let $ABC$ be a triangle with $\angle A  = 90$ and let $D$ be the orthogonal projection of $A$ onto $BC$. The midpoints of $AD$ and $AC$ are called $E$ and $F$, respectively. Let $M$ be the circumcentre of $BEF$. Prove that

$AC$ and $ BM$ are parallel.
2017 Dutch BxMO TST
A quadruple $(a; b; c; d)$ of positive integers with $a \leq b \leq c \leq d$ is called good if we can colour each integer red, blue, green or purple, in such a way that

$i$ of each $a$ consecutive integers at least one is coloured red;

$ii$ of each $b$ consecutive integers at least one is coloured blue;

$iii$ of each $c$ consecutive integers at least one is coloured green;

$iiii$ of each $d$ consecutive integers at least one is coloured purple.

Determine all good quadruples with $a = 2.$
2017 Dutch BxMO TST
Determine all pairs of prime numbers $(p; q)$ such that $p^2 + 5pq + 4q^2$ is the square of an integer.
2016 Dutch BxMO TST
For a positive integer $n$ that is not a power of two, we define $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.
2016 Dutch BxMO TST
Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations

$\begin{cases} x^2 - y = (z - 1)^2\\
y^2 - z = (x - 1)^2 \\
z^2 - x = (y -1)^2 \end{cases}$.
2016 Dutch BxMO TST
Let $\vartriangle ABC$ be a right-angled triangle with $\angle A = 90^o$  and circumcircle $\Gamma$. The inscribed circle is tangent to $BC$ in point $D$. Let $E$ be the midpoint of the arc $AB$ of $\Gamma$ not containing $C$ and let $F$ be the midpoint of the arc $AC$ of $\Gamma$ not containing $B$.

(a) Prove that $\vartriangle  ABC \sim \vartriangle DEF$.

(b) Prove that $EF$ goes through the points of tangency of the incircle to $AB$ and $AC$.
2016 Dutch BxMO TST
The Facebook group Olympiad training has at least five members. There is a certain integer $k$ with following property: for each $k$-tuple of members there is at least one member of this $k$-tuple friends with each of the other $k - 1$.

(Friendship is mutual: if $A$ is friends with $B$, then also $B$ is friends with $A$.)

(a) Suppose $k = 4$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members?

(b) Suppose $k = 5$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members?
2016 Dutch BxMO TST
Determine all pairs $(m, n)$ of positive integers for which $(m + n)^3 /  2n (3m^2 + n^2) + 8$
2015 Dutch BxMO/EGMO TST
Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$.

Prove that $m$ is a divisor of $n$.
2015 Dutch BxMO/EGMO TST
Given are positive integers $r$ and $k$ and an infinite sequence of positive integers $a_1 \le a_2 \le  ...$ such that $\frac{r}{a_r}= k + 1$. Prove that there is a $t$  satisfying  $\frac{t}{a_t}=k$.
2015 Dutch BxMO/EGMO TST
Let $n \ge 2$ be a positive integer. Each square of an $n\times n$ board is coloured red or blue. We put dominoes on the board, each covering two squares of the board. A domino is called even  if it lies on two red or two blue squares and colourful  if it lies on a red and a blue square. Find the largest positive integer $k$ having the following property: regardless of how the red/blue-colouring of the board is done, it is always possible to put $k$ non-overlapping dominoes on the board that are either all even  or all colourful.
2015 Dutch BxMO/EGMO TST
In a triangle $ABC$ the point $D$ is the intersection of the interior angle bisector of  $\angle BAC$ and side $BC$. Let $P$ be the second intersection point of the exterior angle bisector of $\angle BAC$ with the circumcircle of $\angle ABC$. A circle through $A$ and $P$ intersects line segment $BP$ internally in $E$ and line segment $CP$ internally in $F$. Prove that $\angle DEP = \angle  DFP$.
2015 Dutch BxMO/EGMO TST
Find all functions $f : R \to R$ satisfying $(x^2 + y^2)f(xy) = f(x)f(y)f(x^2 + y^2)$ for all real numbers $x$ and $y$.
2014 Dutch BxMO/EGMO TST
Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$
2014 Dutch BxMO/EGMO TST
Find all functions $f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$ for which $xf(xy) + f(-y) = xf(x)$ for all non-zero real numbers $x, y$.
2014 Dutch BxMO/EGMO TST
In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent

to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more

in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$.

Prove that $|AR|\cdot |BQ|=|P I|^2$
2014 Dutch BxMO/EGMO TST
Let $m\ge 3$ and $n$ be positive integers such that $n>m(m-2)$. Find the largest positive integer $d$ such that $d\mid n!$ and $k\nmid d$ for all $k\in\{m,m+1,\ldots,n\}$.
2014 Dutch BxMO/EGMO TST
Let $n$ be a positive integer. Daniel and Merlijn are playing a game. Daniel

has $k$ sheets of paper lying next to each other on a table, where $k$ is a

positive integer. On each of the sheets, he writes some of the numbers

from $1$ up to $n$ (he is allowed to write no number at all, or all numbers).

On the back of each of the sheets, he writes down the remaining numbers.

Once Daniel is ﬁnished, Merlijn can ﬂip some of the sheets of paper (he is

allowed to ﬂip no sheet at all, or all sheets). If Merlijn succeeds in making

all of the numbers from $1$ up to n visible at least once, then he wins.

Determine the smallest $k$ for which Merlijn can always win, regardless of

Daniel’s actions.
2013 Dutch BxMO/EGMO TST
In quadrilateral $ABCD$ the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of diagonal $AC$. Suppose that triangles $ABM$ and $ACD$ have equal area. Prove that $DM // BC$.
2013 Dutch BxMO/EGMO TST
Consider a triple $(a, b, c)$ of pairwise distinct positive integers satisfying $a + b + c = 2013$. A step consists of replacing the triple $(x, y, z)$ by the triple $(y + z - x,z + x - y,x + y - z)$. Prove that, starting from the given triple $(a, b,c)$, after $10$ steps we obtain a triple containing at least one negative number.
2013 Dutch BxMO/EGMO TST
Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$
2013 Dutch BxMO/EGMO TST
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying

$$f(x+yf(x))=f(xf(y))-x+f(y+f(x))$$
2013 Dutch BxMO/EGMO TST
Let $ABCD$ be a cyclic quadrilateral for which $|AD| =|BD|$. Let $M$ be the intersection of $AC$ and $BD$. Let $I$ be the incentre (centre of the inscribed circle) of $\triangle BCM$. Let $N$ be the second point of intersection of $AC$ and the circumscribed circle of $\triangle BMI$. Prove that $|AN| \cdot |NC| = |CD | \cdot  |BN|$.
2012 Dutch BxMO/EGMO TST
Do there exist quadratic polynomials $P(x)$ and $Q(x)$ with real coeffcients such that the polynomial $P(Q(x))$ has precisely the zeros $x = 2, x = 3, x =5$ and $x = 7$?
2012 Dutch BxMO/EGMO TST
Let  $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of  $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.
2012 Dutch BxMO/EGMO TST
Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$
2012 Dutch BxMO/EGMO TST
Let $ABCD$ a convex quadrilateral (this means that all interior angles are smaller than $180^o$), such that there exist a point $M$ on line segment $AB$ and a point $N$ on line segment $BC$ having the property that $AN$ cuts the quadrilateral in two parts of equal area, and such that the same property holds for $CM$.

Prove that $MN$ cuts the diagonal $BD$ in two segments of equal length.
2012 Dutch BxMO/EGMO TST
Let $A$ be a set of positive integers having the following property:

for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$.

Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.
2011 Dutch BxMO TST
All positive integers are coloured either red or green, such that the following conditions are satisfied:

- There are equally many red as green integers.

- The sum of three (not necessarily distinct) red integers is red.

- The sum of three (not necessarily distinct) green integers is green.

Find all colourings that satisfy these conditions.
2011 Dutch BxMO TST
In an acute triangle $ABC$ the angle $\angle C$ is greater than $\angle A$. Let $E$ be such that $AE$ is a diameter of the circumscribed circle $\Gamma$ of \vartriangle ABC. Let $K$ be the intersection of $AC$ and the tangent line at $B$ to $\Gamma$. Let $L$ be the orthogonal projection of $K$ on $AE$ and let $D$ be the intersection of $KL$ and $AB$. Prove that $CE$ is the bisector of $\angle BCD$.
2011 Dutch BxMO TST
Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.
2011 Dutch BxMO TST
Let $n \ge 2$ be an integer. Let $a$ be the greatest positive integer such that $2^a | 5^n - 3^n$.

Let $b$ be the greatest positive integer such that $2^b \le n$. Prove that $a \le b + 3$.
2011 Dutch BxMO TST
A trapezoid $ABCD$ is given with $BC // AD$. Assume that the bisectors of the angles $BAD$ and $CDA$ intersect on the perpendicular bisector of the line segment $BC$. Prove that $|AB|= |CD|$ or $|AB| +|CD| =|AD|$.
2010 Dutch BxMO TST
Let $ABCD$ be a trapezoid with $AB // CD$, $2|AB| = |CD|$ and $BD \perp BC$. Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$. Let $O$ be the intersection of $AM$ and $BD$. Let $N$ be the intersection of $OE$ and $AB$.

(a) Prove that $ABMD$ is a rhombus.

(b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$.
2010 Dutch BxMO TST
Find all functions $f : R \to R$ satisfying $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.
2010 Dutch BxMO TST
Let $N$ be the number of ordered 5-tuples $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ of positive integers satisfying



$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}=1$



Is $N$ even or odd?



Oh and HINTS ONLY, please do not give full solutions. Thanks.
2010 Dutch BxMO TST
The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.
2010 Dutch BxMO TST
For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.
2020 Dutch IMO TST
In acute-angled triangle $ABC, I$ is the center of the inscribed circle and holds $| AC | + | AI | = | BC |$. Prove that $\angle BAC = 2 \angle ABC$.
2020 Dutch IMO TST
Determine all polynomials $P (x)$ with real coefficients that apply $P (x^2) + 2P (x) = P (x)^2 + 2$.