Navvye/Olympiad-Problems-CSV huggingface dataset:
1999 Croatia National Olympiad
In a triangle $ABC$, the inner and outer angle bisectors at $C$ intersect the line $AB$ at $L$ and $M$, respectively. Prove that if $CL=CM$ then $AC^2+BC^2=4R^2$, where $R$ is the circumradius of $\triangle ABC$.
1999 Croatia National Olympiad
For a real parameter $a$, solve the equation $x^4-2ax^2+x+a^2-a=0$. Find all $a$ for which all solutions are real.
1999 Croatia National Olympiad
Let $a,b,c$ be positive real numbers with $abc=1$. Prove the inequality

$$a^{b+c}b^{c+a}c^{a+b}\le1.$$
1999 Croatia National Olympiad
In a basketball competition, $n$ teams took part. Each pair of teams played exactly one match, and there were no draws. At the end of the competition the $i$-th team had $x_i$ wins and $y_i$ defeats $(i=1,\ldots,n)$. Prove that $x_1^2+x_2^2+\ldots+x_n^2=y_1^2+y_2^2+\ldots+y_n^2$.
1999 Croatia National Olympiad
A triangle $ABC$ is inscribed in a rectangle $APQR$ so that points $B$ and $C$ lie on segments $PQ$ and $QR$, respectively. If $\alpha,\beta,\gamma$ are the angles of the triangle, prove that

$$\cot\alpha\cdot S_{BCQ}=\cot\beta\cdot S_{ACR}+\cot\gamma\cdot S_{ABP}.$$
1999 Croatia National Olympiad
The base of a pyramid $ABCDV$ is a rectangle $ABCD$ with the sides $AB=a$ and $BC=b$, and all lateral edges of the pyramid have length $c$. Find the area of the intersection of the pyramid with a plane that contains the diagonal $BD$ and is parallel to $VA$.
1999 Croatia National Olympiad
The vertices of a triangle with sides $a\ge b\ge c$ are centers of three circles, such that no two of the circles have common interior points and none contains any other vertex of the triangle. Determine the maximum possible total area of these three circles.
1999 Croatia National Olympiad
Given nine positive integers, is it always possible to choose four different numbers $a,b,c,d$ such that $a+b$ and $c+d$ are congruent modulo $20$?
1999 Croatia National Olympiad
For every edge of a tetrahedron, we consider a plane through its midpoint that is perpendicular to the opposite edge. Prove that these six planes intersect in a point symmetric to the circumcenter of the tetrahedron with respect to its centroid.
1999 Croatia National Olympiad
Let $n>1$ be an integer. Find the number of permutations $(a_1,a_2,\ldots,a_n)$ of the numbers $1,2,\ldots,n$ such that $a_i>a_{i+1}$ holds for exactly one $i\in\{1,2,\ldots,n-1\}$.
1999 Croatia National Olympiad
Let $(a_n)$ be defined by $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2}$ for $n>2$. Compute the sum $\frac{a_1}2+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\ldots$.
1999 Croatia National Olympiad
On the coordinate plane is given the square with vertices $T_1(1,0),T_2(0,1),T_3(-1,0),T_4(0,-1)$. For every $n\in\mathbb N$, point $T_{n+4}$ is defined as the midpoint of the segment $T_nT_{n+1}$. Determine the coordinates of the limit point of $T_n$ as $n\to\infty$, if it exists.
2000 Croatia National Olympiad
Find all positive integer solutions $x,y,z$ such that $1/x +2/y - 3/z=1$
2000 Croatia National Olympiad
The incircle of a triangle $ABC$ touches $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Find the angles of $\triangle A_1B_1C_1$ in terms of the angles of $\triangle ABC$.
2000 Croatia National Olympiad
Let $m>1$ be an integer. Determine the number of positive integer solutions of the equation $\left\lfloor\frac xm\right\rfloor=\left\lfloor\frac x{m-1}\right\rfloor$.
2000 Croatia National Olympiad
We are given coins of $1,2,5,10,20,50$ lipas and of $1$ kuna (Croatian currency: $1$ kuna = $100$ lipas). Prove that if a bill of $M$ lipas can be paid by $N$ coins, then a bill of $N$ kunas can be paid by M coins.
2000 Croatia National Olympiad
Let $a>0$ and $x_1,x_2,x_3$ be real numbers with $x_1+x_2+x_3=0$. Prove that

$$\log_2\left(1+a^{x_1}\right)+\log_2\left(1+a^{x_2}\right)+\log_2\left(1+a^{x_3}\right)\ge3.$$
2000 Croatia National Olympiad
Two squares $ACXE$ and $CBDY$ are constructed in the exterior of an acute-angled triangle $ABC$. Prove that the intersection of the lines $AD$ and $BE$ lies on the altitude of the triangle from $C$.
2000 Croatia National Olympiad
Let $j$ and $k$ be integers. Prove that the inequality

$$\lfloor(j+k)\alpha\rfloor+\lfloor(j+k)\beta\rfloor\ge\lfloor j\alpha\rfloor+\lfloor j\beta\rfloor+\lfloor k(\alpha+\beta)\rfloor$$holds for all real numbers $\alpha,\beta$ if and only if $j=k$.
2000 Croatia National Olympiad
Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.
2000 Croatia National Olympiad
Let $B$ and $C$ be fixed points, and let $A$ be a variable point such that $\angle BAC$ is fixed. The midpoints of $AB$ and $AC$ are $D$ and $E$ respectively, and $F,G$ are points such that $DF\perp AB$, $EG\perp AC$ and $BF$ and $CG$ are perpendicular to $BC$. Prove that $BF\cdot CG$ remains constant as $A$ varies.
2000 Croatia National Olympiad
Find all $5$-tuples of different four-digit integers with the same initial digit such that the sum of the five numbers is divisible by four of them.
2000 Croatia National Olympiad
A plane intersects a rectangular parallelepiped in a regular hexagon. Prove that the rectangular parallelepiped is a cube.
2000 Croatia National Olympiad
If $n\ge2$ is an integer, prove the equality

$$\lfloor\log_2n\rfloor+\lfloor\log_3n\rfloor+\ldots+\lfloor\log_nn\rfloor=\left\lfloor\sqrt n\right\rfloor+\left\lfloor\sqrt[3]n\right\rfloor+\ldots+\left\lfloor\sqrt[n]n\right\rfloor.$$
2000 Croatia National Olympiad
Let $\mathcal P$ be the parabola $y^2=2px$, and let $T_0$ be a point on it. Point $T_0'$ is such that the midpoint of the segment $T_0T_0'$ lies on the axis of the parabola. For a variable point $T$ on $\mathcal P$, the perpendicular from $T_0'$ to the line $T_0T$ intersects the line through $T$ parallel to the axis of $\mathcal P$ at a point $T'$. Find the locus of $T'$.
2000 Croatia National Olympiad
Let $ABC$ be a triangle with $AB = AC$. With center in a point of the side $BC$, the circle $S$ is constructed that is tangent to the sides $AB$ and $AC$. Let $P$ and $Q$ be any points on the sides $AB$ and $AC$ respectively, such that $PQ$ is tangent to $S$. Show that $PB \cdot CQ = \left(\frac{BC}{2}\right)^2$
2000 Croatia National Olympiad
Let $n\ge3$ positive integers $a_1,\ldots,a_n$ be written on a circle so that each of them divides the sum of its two neighbors. Let us denote

$$S_n=\frac{a_n+a_2}{a_1}+\frac{a_1+a_3}{a_2}+\ldots+\frac{a_{n-2}+a_n}{a_{n-1}}+\ldots+\frac{a_{n-1}+a_1}{a_n}.$$Determine the minimum and maximum values of $S_n$.
2000 Croatia National Olympiad
Let $S$ be the set of all squarefree numbers and $n$ be a natural number. Prove that

$$\sum_{k\in S}\left\lfloor\sqrt{\frac nk}\right\rfloor=n.$$
2001 Croatia National Olympiad
Find all integers $x$ for which $2x^2-x-36$ is the square of a prime number.
2001 Croatia National Olympiad
Let $S$ be the center of a square $ABCD$ and $P$ be the midpoint of $AB$. The lines $AC$ and $PD$ meet at $M$, and the lines $BD$ and $PC$ meet at $N$. Prove that the radius of the incircle of the quadrilateral $PMSN$ equals $MP-MS$.
2001 Croatia National Olympiad
Let $a$ and $b$ be positive numbers. Prove the inequality

$$\sqrt[3]{\frac ab}+\sqrt[3]{\frac ba}\le\sqrt[3]{2(a+b)\left(\frac1a+\frac1b\right)}.$$
2001 Croatia National Olympiad
Find all possible values of $n$ for which a rectangular board $9\times n$ can be partitioned into tiles of the shape:
2001 Croatia National Olympiad
Let $z\ne0$ be a complex number such that $z^8=\overline z$. What are the possible values of $z^{2001}$?
2001 Croatia National Olympiad
The excircle of a triangle $ABC$ corresponding to $A$ touches the side $BC$ at $K$ and the rays $AB$ and $AC$ at $P$ and $Q$, respectively. The lines $OB$ and $OC$ intersect $PQ$ at $M$ and $N$, respectively. Prove that

$$\frac{QN}{AB}=\frac{NM}{BC}=\frac{MP}{CA}.$$
2001 Croatia National Olympiad
Let there be given triples of integers $(r_j,s_j,t_j),~j=1,2,\ldots,N$, such that for each $j$, $r_j,t_j,s_j$ are not all even. Show that one can find integers $a,b,c$ such that $ar_j+bs_j+ct_j$ is odd for at least $\frac{4N}7$ of the indices $j$.
2001 Croatia National Olympiad
On the coordinate plane is given a polygon $\mathcal P$ with area greater than $1$. Prove that there exist two different points $(x_1,y_1)$ and $(x_2,y_2)$ inside the polygon $\mathcal P$ such that $x_1-x_2$ and $y_1-y_2$ are both integers.
2001 Croatia National Olympiad
Let $O$ and $P$ be fixed points on a plane, and let $ABCD$ be any parallelogram with center $O$. Let $M$ and $N$ be the midpoints of $AP$ and $BP$ respectively. Lines $MC$ and $ND$ meet at $Q$. Prove that the point $Q$ lies on the lines $OP$, and show that it is independent of the choice of the parallelogram $ABCD$.
2001 Croatia National Olympiad
In a triangle $ABC$ with $AC\ne BC$, $M$ is the midpoint of $AB$ and $\angle A=\alpha$, $\angle B=\beta$, $\angle ACM=\varphi$ and $\angle BSM=\Psi$. Prove that

$$\frac{\sin\alpha\sin\beta}{\sin(\alpha-\beta)}=\frac{\sin\varphi\sin\Psi}{\sin(\varphi-\Psi)}.$$
2001 Croatia National Olympiad
Numbers $1,\frac12,\frac13,\ldots,\frac1{2001}$ are written on a blackboard. A student erases two numbers $x,y$ and writes down the number $x+y+xy$ instead. Determine the number that will be written on the board after $2000$ such operations.
2001 Croatia National Olympiad
Let $S$ be a set of $100$ positive integers less than $200$. Prove that there exists a nonempty subset $T$ of $S$ the product of whose elements is a perfect square.
2001 Croatia National Olympiad
On the unit circle $k$ with center $O$, points $A$ and $B$ with $AB=1$ are chosen and unit circles $k_1$ and $k_2$ with centers $A$ and $B$ are drawn. A sequence of circles $(l_n)$ is defined as follows: circle $l_1$ is tangent to $k$ internally at $D_1$ and to $k_1,k_2$ externally, and for $n>1$ circle $l_n$ is tangent to $k_1$ and $k_2$ and to $l_{n-1}$ at $D_n$. For each $n$, compute $d_n=OD_n$ and the radius $r_n$ of $l_n$.
2001 Croatia National Olympiad
A piece of paper in the shape of a square $FBHD$ with side $a$ is given. Points $G,A$ on $FB$ and $E,C$ on $BH$ are marked so that $FG=GA=AB$ and $BE=EC=CH$. The paper is folded along $DG,DA,DC$ and $AC$ so that $G$ overlaps with $B$, and $F$ and $H$ overlap with $E$. Compute the volume of the obtained tetrahedron $ABCD$.
2001 Croatia National Olympiad
Let $p_1,p_2,p_3,p_4$ be four distinct primes, and let $1=d_1<d_2<\ldots<d_{16}=n$ be the divisors of $n=p_1p_2p_3p_4$. Determine all $n<2001$ with the property that

$d_9-d_8=22$.
2001 Croatia National Olympiad
Suppose that zeros and ones are written in the cells of an $n\times n$ board, in such a way that the four cells in the intersection of any two rows and any two columns contain at least one zero. Prove that the number of ones does not exceed $\frac n2\left(1+\sqrt{4n-3}\right)$.
2002 Croatia National Olympiad
The length of the middle line of a trapezoid is $4$ and the angles at one of the bases are $40^\circ$ and $50^\circ$. Determine the lengths of the bases if the distance between their midpoints is $1$.
2002 Croatia National Olympiad
Prove that for any positive number $a,b,c$ and any nonnegative integer $p$

$$a^{p+2}+b^{p+2}+c^{p+2}\ge a^pbc+b^pca+c^pab.$$
2002 Croatia National Olympiad
Find all triples $(x,y,z)$ of natural numbers that verify the equation

$$2x^2y^2+2y^2z^2+2z^2x^2-x^4-y^4-z^4=576.$$
2002 Croatia National Olympiad
A disc is divided into $30$ segments which are labelled by $50,100,150,\ldots,1500$ in some order. Show that there always exist three successive segments, the sum of whose labels is at least $2350$.
2002 Croatia National Olympiad
Solve the equation

$$\left(x^2+3x-4\right)^3+\left(2x^2-5x+3\right)^3=\left(3x^2-2x-1\right)^3.$$
2002 Croatia National Olympiad
Let $a,b,c$ be real numbers greater than $1$. Prove the inequality

$$\log_a\left(\frac{b^2}{ac}-b+ac\right)\log_b\left(\frac{c^2}{ab}-c+ab\right)\log_c\left(\frac{a^2}{bc}-a+bc\right)\ge1.$$
2002 Croatia National Olympiad
If two triangles with side lengths $a,b,c$ and $a',b',c'$ and the corresponding angle $\alpha,\beta,\gamma$ and $\alpha',\beta',\gamma'$ satisfy $\alpha+\alpha'=\pi$ and $\beta=\beta'$, prove that $aa'=bb'+cc'$.
2002 Croatia National Olympiad
Find all natural numbers $n$ for which the equation $\frac1x+\frac1y=\frac1n$ has exactly five solutions $(x,y)$ in the set of natural numbers.
2002 Croatia National Olympiad
In triangle $ABC$, the angles $\alpha=\angle A$ and $\beta=\angle B$ are acute. The isosceles triangle $ACD$ and $BCD$ with the bases $AC$ and $BC$ and $\angle ADC=\beta$, $\angle BEC=\alpha$ are constructed in the exterior of the triangle $ABC$. Let $O$ be the circumcenter of $\triangle ABC$. Prove that $DO+EO$ equals the perimeter of triangle $ABC$ if and only if $\angle ACB$ is right.
2002 Croatia National Olympiad
Prove that a natural number can be written as a sum of two or more consecutive positive integers if and only if that number is not a power of two.
2002 Croatia National Olympiad
Points $E$ and $F$ are taken on the diagonals $AB_1$ and $CA_1$ of the lateral faces $ABB_1A_1$ and $CAA_1C_1$ of a triangular prism $ABCA_1B_1C_1$ so that $EF\parallel BC_1$. Find the ratio of the lengths of $EF$ and $BC_1$.
2002 Croatia National Olympiad
Among the $n$ inhabitants of an island, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that the necklaces of two friends have at least one marble of the same type and that the necklaces of two enemies differ at all marbles. (A necklace may also be marbleless). Show that the chief’s order can be achieved by using $\left\lfloor\frac{n^2}4\right\rfloor$ different types of stones, but not necessarily by using fewer types.
2002 Croatia National Olympiad
For each $x$ with $|x|<1$, compute the sum of the series

$$1+4x+9x^2+\ldots+n^2x^{n-1}+\ldots.$$
2002 Croatia National Olympiad
Consider the cube with the vertices $A(1,1,1)$, $B(-1,1,1)$, $C(-1,-1,1)$, $D(1,-1,1)$ and $A',B',C',D'$ symmetric to $A,B,C,D$ respectively with respect to the origin $O$. Let $T$ be a point not on the circumsphere of the cube and let $OT=d$. Denote $\alpha=\angle ATA'$, $\beta=\angle BTB'$, $\gamma=\angle CTC'$, $\delta=\angle DTD'$. Prove that

$$\tan^2\alpha+\tan^2\beta+\tan^2\gamma+\tan^2\delta=\frac{32d^2}{\left(d^2-3\right)^2}.$$
2002 Croatia National Olympiad
Let $f(x)=x^{2002}-x^{2001}+1$. Prove that for every positive integer $m$, the numbers $m,f(m),f(f(m)),\ldots$ are pairwise coprime.
2002 Croatia National Olympiad
Let $(a_n)_{n\in\mathbb N}$ be an increasing sequence of positive integers. A term $a_k$ in the sequence is said to be good if it a sum of some other terms (not necessarily distinct). Prove that all terms of the sequence, apart from finitely many of them, are good.
2003 Croatia National Olympiad
Show that a triangle whose side lengths are prime numbers cannot have integer area.
2003 Croatia National Olympiad
The product of the positive real numbers $x, y, z$ is 1. Show that if $$ \frac{1}{x}+\frac{1}{y} + \frac{1}{z} \geq x+y+z $$then $$ \frac{1}{x^{k}}+\frac{1}{y^{k}} + \frac{1}{z^{k}} \geq x^{k}+y^{k}+z^{k} $$

for all positive integers $k$.
2003 Croatia National Olympiad
In an isosceles triangle with base $a$, lateral side $b$, and height to the base $v$, it holds that $\frac a2+v\ge b\sqrt2$. Find the angles of the triangle. Compute its area if $b=8\sqrt2$.
2003 Croatia National Olympiad
How many divisors of $30^{2003}$ are there which do not divide $20^{2000}$?
2003 Croatia National Olympiad
Find all pairs of real numbers $(x,y)$ satisfying

$$(2x+1)^2+y^2+(y-2x)^2=\frac13.$$
2003 Croatia National Olympiad
Let $M$ be a point inside square $ABCD$ and $A_1,B_1,C_1,D_1$ be the second intersection points of $AM$, $BM$, $CM$, $DM$ with the circumcircle of the square. Prove that $A_1B_1\cdot C_1D_1=A_1D_1\cdot B_1C_1$.
2003 Croatia National Olympiad
For positive numbers $a_1,a_2,\ldots,a_n$ ($n\ge2$) denote $s=a_1+\ldots+a_n$. Prove that

$$\frac{a_1}{s-a_1}+\ldots+\frac{a_n}{s-a_n}\ge\frac n{n-1}.$$
2003 Croatia National Olympiad
Find the least possible cardinality of a set $A$ of natural numbers, the smallest and greatest of which are $1$ and $100$, and having the property that every element of $A$ except for $1$ equals the sum of two elements of $A$.
2003 Croatia National Olympiad
Let $a,b,c$ be the sides of triangle $ABC$ and let $\alpha,\beta,\gamma$ be the corresponding angles.



(a) If $\alpha=3\beta$, prove that $\left(a^2-b^2\right)(a-b)=bc^2$.

(b) Is the converse true?
2003 Croatia National Olympiad
For every integer $n>2$, prove the equality

$$\left\lfloor\frac{n(n+1)}{4n-2}\right\rfloor=\left\lfloor\frac{n+1}4\right\rfloor.$$
2003 Croatia National Olympiad
In a tetrahedron $ABCD$, all angles at vertex $D$ are equal to $\alpha$ and all dihedral angles between faces having $D$ as a vertex are equal to $\phi$. Prove that there exists a unique $\alpha$ for which $\phi=2\alpha$.
2003 Croatia National Olympiad
Given $8$ unit cubes, $24$ of their faces are painted in blue and the remaining $24$ faces in red. Show that it is always possible to assemble these cubes into a cube of edge $2$ on whose surface there are equally many blue and red unit squares.
2003 Croatia National Olympiad
Let $I$ be a point on the bisector of angle $BAC$ of a triangle $ABC$. Points $M,N$ are taken on the respective sides $AB$ and $AC$ so that $\angle ABI=\angle NIC$ and $\angle ACI=\angle MIB$. Show that $I$ is the incenter of triangle $ABC$ if and only if points $M,N$ and $I$ are collinear.
2003 Croatia National Olympiad
A sequence $(a_n)_{n\ge0}$ satisfies $a_{m+n}+a_{m-n}=\frac12\left(a_{2m}+a_{2n}\right)$ for all integers $m,n$ with $m\ge n\ge0$. Given that $a_1=1$, find $a_{2003}$.
2003 Croatia National Olympiad
The natural numbers $1$ through $2003$ are arranged in a sequence. We repeatedly perform the following operation: If the first number in the sequence is $k$, the order of the first $k$ terms is reversed. Prove that after several operations number $1$ will occur on the first place.
2003 Croatia National Olympiad
Prove that the number $\binom np-\left\lfloor\frac np\right\rfloor$ is divisible by $p$ for every prime number and integer $n\ge p$.
2004 Croatia National Olympiad
Find all real solutions of the system of equations

$$x^2-y^2=2(xz+yz+x+y),$$$$y^2-z^2=2(yx+zx+y+z),$$$$z^2-x^2=2(zy+xy+z+x).$$
2004 Croatia National Olympiad
Prove that the medians from the vertices $A$ and $B$ of a triangle $ABC$ are orthogonal if and only if $BC^2+AC^2=5AB^2$.
2004 Croatia National Olympiad
Prove that for any three real numbers $x,y,z$ the following inequality holds:

$$|x|+|y|+|z|-|x+y|-|y+z|-|z+x|+|x+y+z|\ge0.$$
2004 Croatia National Olympiad
The sequence $1,2,3,4,0,9,6,9,4,8,7,\ldots$ is formed so that each term, starting from the fifth, is the units digit of the sum of the previous four.

(a) Do the digits $2,0,0,4$ occur in the sequence in this order?

(b) Will the initial digits $1,2,3,4$ ever occur again in this order?
2004 Croatia National Olympiad
Parts of a pentagon have areas $x,y,z$ as shown in the picture. Given the area $x$, find the areas $y$ and $z$ and the area of the entire pentagon.
2004 Croatia National Olympiad
If $a,b,c$ are positive numbers, prove the inequality

$$\frac{a^2}{(a+b)(a+c)}+\frac{b^2}{(b+c)(b+a)}+\frac{c^2}{(c+a)(c+b)}\ge\frac34.$$
2004 Croatia National Olympiad
The sequence $(p_n)_{n\in\mathbb N}$ is defined by $p_1=2$ and, for $n\ge2$, $p_n$ is the largest prime factor of $p_1p_2\cdots p_{n-1}+1$. Show that $p_n\ne5$ for all $n$.
2004 Croatia National Olympiad
A frog jumps on the coordinate lattice, starting from the point $(1,1)$, according to the following rules:



(i) From point $(a,b)$ the frog can jump to either $(2a,b)$ or $(a,2b)$;

(ii) If $a>b$, the frog can also jump from $(a,b)$ to $(a-b,b)$, while for $a<b$ it can jump from $(a,b)$ to $(a,b-a)$.

Can the frog get to the point: (a) $(24,40)$; (b) $(40,60)$; (c) $(24,60)$; (d) $(200,4)$?
2004 Croatia National Olympiad
Let $ABCD$ be a square and $P$ be a point on the shorter arc $AB$ of the circumcircle of the square. Which values can the expression $\frac{AP+BP}{CP+DP}$ take?
2004 Croatia National Olympiad
If $a,b,c$ are the sides and $\alpha,\beta,\gamma$ the corresponding angles of a triangle, prove the inequality

$$\frac{\cos\alpha}{a^3}+\frac{\cos\beta}{b^3}+\frac{\cos\gamma}{c^3}\ge\frac3{2abc}.$$
2004 Croatia National Olympiad
The altitudes of a tetrahedron meet at a single point. Prove that this point, the centroid of one face of the tetrahedron, the foot of the altitude on that face, and the three points dividing the other three altitudes in ratio $2:1$ (closer to the feet) all lie on a sphere.
2004 Croatia National Olympiad
Finitely many cells of an infinite square board are colored black. Prove that one can choose finitely many squares in the plane of the board so that the following conditions are satisfied:



(i) The interiors of any two different squares are disjoint;

(ii) Each black cell lies in one of these squares;

(iii) In each of these squares, the black cells cover at least $\frac15$ and at most $\frac45$ of the area of that square.
2004 Croatia National Olympiad
Let $z_1,\ldots,z_n$ and $w_1,\ldots,w_n$ $(n\in\mathbb N)$ be complex numbers such that

$$|\epsilon_1z_1+\ldots+\epsilon_nz_n|\le|\epsilon_1w_1+\ldots+\epsilon_nw_n|$$holds for every choice of $\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}$. Prove that

$$|z_1|^2+\ldots+|z_n|^2\le|w_1|^2+\ldots+|w_n|^2.$$
2004 Croatia National Olympiad
Points $P$ and $Q$ inside a triangle $ABC$ with sides $a,b,c$ and the corresponding angle $\alpha,\beta,\gamma$ satisfy $\angle BPC=\angle CPA=\angle APB=120^\circ$ and $\angle BQC=60^\circ+\alpha$, $\angle CQA=60^\circ+\beta$, $\angle AQB=60^\circ+\gamma$. Prove the equality

$$(AP+BP+CP)^3\cdot AQ\cdot BQ\cdot CQ=(abc)^2.$$
2004 Croatia National Olympiad
The sequences $(x_n),(y_n),(z_n),n\in\mathbb N$, are defined by the relations

$$x_{n+1}=\frac{2x_n}{x_n^2-1},\qquad y_{n+1}=\frac{2y_n}{y_n^2-1},\qquad z_{n+1}=\frac{2z_n}{z_n^2-1},$$where $x_1=2$, $y_1=4$, and $x_1y_1z_1=x_1+y_1+z_1$.

(a) Show that $x_n^2\ne1$, $y_n^2\ne1$, $z_n^2\ne1$ for all $n$;

(b) Does there exist a $k\in\mathbb N$ for which $x_k+y_k+z_k=0$?
2004 Croatia National Olympiad
Determine all real numbers $\alpha$ with the property that all numbers in the sequence $\cos\alpha,\cos2\alpha,\cos2^2\alpha,\ldots,\cos2^n\alpha,\ldots$ are negative.
2005 Croatia National Olympiad
Find all possible digits $x, y, z$ such that the number $\overline{13xy45z}$ is divisible

by $792.$
2005 Croatia National Olympiad
The lines joining the incenter of a triangle to the vertices divide the triangle into three triangles. If one of these triangles is similar to the initial one,determine the angles of the triangle.
2005 Croatia National Olympiad
If $k, l, m$ are positive integers with $\frac{1}{k}+\frac{1}{l}+\frac{1}{m}<1$, ﬁnd the maximum possible value of $\frac{1}{k}+\frac{1}{l}+\frac{1}{m}$.
2005 Croatia National Olympiad
The circumradius $R$ of a triangle with side lengths $a, b, c$ satisﬁes $R =\frac{a\sqrt{bc}}{b+c}$. Find the angles of the triangle.
2005 Croatia National Olympiad
Let $a \not = 0, b, c$ be real numbers. If $x_{1}$ is a root of the equation $ax^{2}+bx+c = 0$ and $x_{2}$ a root of $-ax^{2}+bx+c = 0$, show that there is a root $x_{3}$ of $\frac{a}{2}\cdot x^{2}+bx+c = 0$ between $x_{1}$ and $x_{2}$.
2005 Croatia National Olympiad
Let $U$ be the incenter of a triangle $ABC$ and $O_{1}, O_{2}, O_{3}$ be the circumcenters of the triangles $BCU, CAU, ABU$ , respectively. Prove that the circumcircles of the triangles $ABC$ and $O_{1}O_{2}O_{3}$ have the same center.
2005 Croatia National Olympiad
If $a, b, c$ are real numbers greater than $1$, prove that for any real number $r$



$$(\log_{a}bc)^{r}+(\log_{b}ca)^{r}+(\log_{c}ab)^{r}\geq 3 \cdot 2^{r}. $$
2005 Croatia National Olympiad
Show that in any set of eleven integers there are six whose sum is divisible by $6$.
2005 Croatia National Olympiad
Find all positive integer solutions of the equation $k!l! = k!+l!+m!.$