Navvye/Olympiad-Problems-CSV huggingface dataset:
2003 China Western Mathematical Olympiad
$ 1650$ students are arranged in $ 22$ rows and $ 75$ columns.  It is known that in any two columns, the number of pairs of students in the same row and of the same sex is not greater than $ 11$.  Prove that the number of boys is not greater than $ 928$.
2002 China Western Mathematical Olympiad
Find all positive integers $ n$ such that $ n^4-4n^3+22n^2-36n+18$ is a perfect square.
2002 China Western Mathematical Olympiad
Let $ O$ be the circumcenter of acute triangle $ ABC$. Point $ P$ is in the interior of triangle $ AOB$. Let $ D,E,F$ be the projections of $ P$ on the sides $ BC,CA,AB$, respectively. Prove that the parallelogram consisting of $ FE$ and $ FD$ as its adjacent sides lies inside triangle $ ABC$.
2002 China Western Mathematical Olympiad
In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $ x^4 + px^3 + qx^2 + rx + s = 0$. Find the minimum of  square areas.
2002 China Western Mathematical Olympiad
Let $ n$ be a positive integer, let the sets $ A_{1},A_{2},\cdots,A_{n + 1}$ be non-empty subsets of the set $ \{1,2,\cdots,n\}.$ prove that there exist two disjoint non-empty subsets of the set $ \{1,2,\cdots,n + 1\}$: $ \{i_{1},i_{2},\cdots,i_{k}\}$ and $ \{j_{1},j_{2},\cdots,j_{m}\}$ such that $ A_{i_{1}}\cup A_{i_{2}}\cup\cdots\cup A_{i_{k}} = A_{j_{1}}\cup A_{j_{2}}\cup\cdots\cup A_{j_{m}}$.
2002 China Western Mathematical Olympiad
Given a trapezoid $ ABCD$ with $ AD\parallel BC, E$ is a moving point on the side $ AB,$ let $ O_{1},O_{2}$ be the circumcenters of triangles $ AED,BEC$, respectively. Prove that the length of $ O_{1}O_{2}$ is a constant value.
2002 China Western Mathematical Olympiad
Given a positive integer $ n$, find all integers $ (a_{1},a_{2},\cdots,a_{n})$ satisfying the following conditions:

$ (1): a_{1}+a_{2}+\cdots+a_{n}\ge n^2;$

$ (2): a_{1}^2+a_{2}^2+\cdots+a_{n}^2\le n^3+1.$
2002 China Western Mathematical Olympiad
Assume that $ \alpha$ and $ \beta$ are two roots of the equation: $ x^2-x-1=0$. Let $ a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$, $ n=1, 2, \cdots$.

(1) Prove that for any positive integer $ n$, we have $ a_{n+2}=a_{n+1}+a_n$.

(2) Find all positive integers $ a$ and $ b$, $ a<b$, satisfying $ b \mid a_n-2na^n$ for any positive integer $ n$.
2002 China Western Mathematical Olympiad
Assume that $ S=(a_1, a_2, \cdots, a_n)$ consists of $ 0$ and $ 1$ and is the longest sequence of number, which satisfies the following condition: Every two sections of successive $ 5$ terms in the sequence of numbers $ S$ are different, i.e., for arbitrary $ 1\le i<j\le n-4$, $ (a_i, a_{i+1}, a_{i+2}, a_{i+3}, a_{i+4})$ and $ (a_j, a_{j+1}, a_{j+2}, a_{j+3}, a_{j+4})$ are different. Prove that the first four terms and the last four terms in the sequence are the same.
2001 China Western Mathematical Olympiad
The sequence $ \{x_n\}$ satisfies $ x_1 = \frac {1}{2}, x_{n + 1} = x_n + \frac {x_n^2}{n^2}$.  Prove that $ x_{2001} < 1001$.
2001 China Western Mathematical Olympiad
$ ABCD$ is a rectangle of area 2.  $ P$ is a point on side $ CD$ and $ Q$ is the point where the incircle of $ \triangle PAB$ touches the side $ AB$.  The product $ PA \cdot PB$ varies as $ ABCD$ and $ P$ vary.  When $ PA \cdot PB$ attains its minimum value,



a) Prove that $ AB \geq 2BC$,

b) Find the value of $ AQ \cdot BQ$.
2001 China Western Mathematical Olympiad
Let $ n, m$ be positive integers of different parity, and $ n > m$.  Find all integers $ x$ such that $ \frac {x^{2^n} - 1}{x^{2^m} - 1}$ is a perfect square.
2001 China Western Mathematical Olympiad
Let $ x, y, z$ be real numbers such that $ x + y + z \geq xyz$.  Find the smallest possible value of $ \frac {x^2 + y^2 + z^2}{xyz}$.
2001 China Western Mathematical Olympiad
Find all real numbers $ x$ such that $ \lfloor x^3 \rfloor = 4x + 3$.
2001 China Western Mathematical Olympiad
$ P$ is a point on the exterior of a circle centered at $ O$.  The tangents to the circle from $ P$ touch the circle at $ A$ and $ B$.  Let $ Q$ be the point of intersection of $ PO$ and $ AB$.  Let $ CD$ be any chord of the circle passing through $ Q$.  Prove that $ \triangle PAB$ and $ \triangle PCD$ have the same incentre.
2001 China Western Mathematical Olympiad
Find, with proof, all real numbers $ x \in \lbrack 0, \frac {\pi}{2} \rbrack$, such that $ (2 - \sin 2x)\sin (x + \frac {\pi}{4}) = 1$.
2001 China Western Mathematical Olympiad
We call $ A_1, A_2, \ldots, A_n$ an $ n$-division of $ A$ if



(i) $ A_1 \cap A_2 \cap \cdots \cap A_n = A$,

(ii) $ A_i \cap A_j \neq \emptyset$.



Find the smallest positive integer $ m$ such that for any $ 14$-division $ A_1, A_2, \ldots, A_{14}$ of $ A = \{1, 2, \ldots, m\}$, there exists a set $ A_i$ ($ 1 \leq i \leq 14$) such that there are two elements $ a, b$ of $ A_i$ such that $ b < a \leq \frac {4}{3}b$.
2010 Costa Rica - Final Round
Consider points $D,E$ and $F$ on sides $BC,AC$ and $AB$, respectively, of a triangle $ABC$, such that $AD, BE$ and $CF$ concurr at a point $G$. The parallel through $G$ to $BC$ cuts $DF$ and $DE$ at $H$ and $I$, respectively. Show that triangles $AHG$ and $AIG$ have the same areas.
2010 Costa Rica - Final Round
Consider the sequence $x_n>0$ defined with the following recurrence relation:



$$x_1 = 0$$



and for $n>1$ $$(n+1)^2x_{n+1}^2 + (2^n+4)(n+1)x_{n+1}+ 2^{n+1}+2^{2n-2} = 9n^2x_n^2+36nx_n+32.$$



Show that if $n$ is a prime number larger or equal to $5$, then $x_n$ is an integer.
2010 Costa Rica - Final Round
Christian Reiher and Reid Barton want to open a security box, they already managed to discover the algorithm to generate the key codes and they obtained the following information:



$i)$ In the screen of the box will appear a sequence of $n+1$ numbers, $C_0 = (a_{0,1},a_{0,2},...,a_{0,n+1})$



$ii)$ If the code $K = (k_1,k_2,...,k_n)$ opens the security box then the following must happen:



a) A sequence $C_i = (a_{i,1},a_{i,2},...,a_{i,n+1})$ will be asigned to each $k_i$ defined as follows:



$a_{i,1} = 1$ and $a_{i,j} = a_{i-1,j}-k_ia_{i,j-1}$, for $i,j \ge 1$



b) The sequence $(C_n)$ asigned to $k_n$ satisfies that $S_n = \sum_{i=1}^{n+1}|a_i|$ has its least possible value, considering all possible sequences $K$.



The sequence $C_0$ that appears in the screen is the following:



$a_{0,1} = 1$ and $a_0,i$ is the sum of the products of the elements of each of the subsets with $i-1$ elements of the set $A =$ {$1,2,3,...,n$}, $i\ge 2$, such that $a_{0, n+1} = n!$



Find a sequence $K = (k_1,k_2,...,k_n)$ that satisfies the conditions of the problem and show that there exists at least $n!$ of them.
2010 Costa Rica - Final Round
Find all integer solutions $(a,b)$ of the equation $$ (a+b+3)^2 + 2ab = 3ab(a+2)(b+2)$$
2010 Costa Rica - Final Round
Let $C_1$ be a circle with center $O$ and let $B$ and $C$ be points in $C_1$ such that $BOC$ is an equilateral triangle. Let $D$ be the midpoint of the minor arc $BC$ of $C_1$. Let $C_2$ be the circle with center $C$ that passes through $B$ and $O$. Let $E$ be the second intersection of $C_1$ and $C_2$. The parallel to $DE$ through $B$ intersects $C_1$ for second time in $A$. Let $C_3$ be the circumcircle of triangle $AOC$. The second intersection of $C_2$ and $C_3$ is $F$. Show that $BE$ and $BF$ trisect the angle $\angle ABC$.
2010 Costa Rica - Final Round
Let $F$ be the family of all sets of positive integers with $2010$ elements that satisfy the following condition:

The difference between any two of its elements is never the same as the difference of any other two of its elements. Let $f$ be a function defined from $F$ to the positive integers such that $f(K)$ is the biggest element of $K \in F$. Determine the least value of $f(K)$.
2009 Costa Rica - Final Round
Let $ x$ and $ y$ positive real numbers such that $ (1+x)(1+y)=2$. Show that  $ xy+\frac{1}{xy}\geq\ 6$
2009 Costa Rica - Final Round
Prove that for that for every positive integer $ n$, the smallest integer that is greater than $ (\sqrt {3} + 1)^{2n}$ is divisible by $ 2^{n + 1}$.
2009 Costa Rica - Final Round
Let triangle $ ABC$ acutangle, with $ m \angle ACB\leq\ m \angle ABC$. $ M$ the midpoint of side $ BC$ and $ P$ a point over the side $ MC$. Let $ C_{1}$ the circunference with center $ C$. Let $ C_{2}$ the circunference with center $ B$. $ P$ is a point of $ C_{1}$ and $ C_{2}$. Let $ X$ a point on the opposite semiplane than $ B$ respecting with the straight line $ AP$; Let $ Y$ the intersection of side $ XB$ with $ C_{2}$ and $ Z$ the intersection of side $ XC$ with $ C_{1}$. Let $ m\angle PAX = \alpha$ and $ m\angle ABC = \beta$. Find the geometric place of $ X$ if it satisfies the following conditions:

$ (a) \frac {XY}{XZ} = \frac {XC + CP}{XB + BP}$

$ (b) \cos(\alpha) = AB\cdot \frac {\sin(\beta )}{AP}$
2009 Costa Rica - Final Round
Show that the number $ 3^{{4}^{5}} + 4^{{5}^{6}}$ can be expresed as the product of two integers greater than $ 10^{2009}$
2009 Costa Rica - Final Round
Suppose the polynomial $ x^{n} + a_{n - 1}x^{n - 1} + ... + a_{1} + a_{0}$ can be factorized as  $ (x + r_{1})(x + r_{2})...(x + r_{n})$, with $ r_{1}, r_{2}, ..., r_{n}$ real numbers.

Show that $ (n - 1)a_{n - 1}^{2}\geq\ 2na_{n - 2}$
2009 Costa Rica - Final Round
Let $ \Delta ABC$ with incircle $ \Gamma$, let $ D, E$ and $ F$ the tangency points of $ \Gamma$ with sides $ BC, AC$ and $ AB$, respectively and let $ P$ the intersection point of $ AD$ with $ \Gamma$.

$ a)$ Prove that $ BC, EF$ and the straight line tangent to $ \Gamma$ for $ P$ concur at a point $ A'$.

$ b)$ Define $ B'$ and $ C'$ in an anologous way than $ A'$. Prove that $ A'-B'-C'$
2008 Costa Rica - Final Round
We want to colour all the squares of an $ nxn$ board of red or black. The colorations should be such that any subsquare of $ 2x2$ of the board have exactly two squares of each color. If $ n\geq 2$  how many such colorations are possible?
2008 Costa Rica - Final Round
Let $ ABC$ be a triangle and let $ P$ be a point on the angle bisector $ AD$, with $ D$ on $ BC$. Let $ E$, $ F$ and $ G$ be the intersections of $ AP$, $ BP$ and $ CP$ with the circumcircle of the triangle, respectively. Let $ H$ be the intersection of $ EF$ and $ AC$, and let $ I$ be the intersection of $ EG$ and $ AB$. Determine the geometric place of the intersection of $ BH$ and $ CI$ when $ P$ varies.
2008 Costa Rica - Final Round
Find all polinomials $ P(x)$ with real coefficients, such that



$ P(\sqrt {3}(a - b)) + P(\sqrt {3}(b - c)) + P(\sqrt {3}(c - a)) = P(2a - b - c) + P( - a + 2b - c) + P( - a - b + 2c)$

for any $ a$,$ b$ and $ c$ real numbers
2008 Costa Rica - Final Round
Let $ x$, $ y$ and $ z$ be non negative reals, such that there are not two simultaneously equal to $ 0$. Show that



$ \frac {x + y}{y + z} + \frac {y + z}{x + y} + \frac {y + z}{z + x} + \frac {z + x}{y + z} + \frac {z + x}{x + y} + \frac {x + y}{z + x}\geq\ 5 + \frac {x^{2} + y^{2} + z^{2}}{xy + yz + zx}$

and determine the equality cases.
2008 Costa Rica - Final Round
Let $ p$ be a prime number such that $ p-1$ is a perfect square. Prove that the equation

$ a^{2}+(p-1)b^{2}=pc^{2}$

has infinite many integer solutions $ a$, $ b$ and $ c$ with $ (a,b,c)=1$
2008 Costa Rica - Final Round
Let $ O$ be the circumcircle of a $ \Delta ABC$ and let $ I$ be its incenter, for a point $ P$ of the plane let $ f(P)$ be the point obtained by reflecting $ P'$ by the midpoint of $ OI$, with $ P'$ the homothety  of $ P$ with center $ O$ and ratio $ \frac{R}{r}$  with $ r$ the inradii and $ R$ the circumradii,(understand it by $ \frac{OP}{OP'}=\frac{R}{r}$). Let $ A_1$, $ B_1$ and $ C_1$ the midpoints of $ BC$, $ AC$ and $ AB$, respectively. Show that the rays $ A_1f(A)$, $ B_1f(B)$ and $ C_1f(C)$ concur on the incircle.
2006 Costa Rica - Final Round
Consider the set $S=\{1,2,...,n\}$. For every $k\in S$, define $S_{k}=\{X \subseteq S, \ k \notin X, X\neq \emptyset\}$. Determine the value of the sum $$S_{k}^{*}=\sum_{\{i_{1},i_{2},...,i_{r}\}\in S_{k}}\frac{1}{i_{1}\cdot i_{2}\cdot...\cdot i_{r}}$$ Click to reveal hidden textin fact, this problem was taken from an austrian-polish
2006 Costa Rica - Final Round
If $ a$, $ b$, $ c$ are the sidelengths of a triangle, then prove that

$ \frac {3\left(a^4 + b^4 + c^4\right)}{\left(a^2 + b^2 + c^2\right)^2} + \frac {bc + ca + ab}{a^2 + b^2 + c^2}\geq 2$.
2006 Costa Rica - Final Round
Let $ABC$ be a triangle. Let $P, Q, R$ be the midpoints of $BC, CA, AB$ respectively. Let $U, V, W$ be the midpoints of $QR, RP, PQ$ respectively. Let $x=AU, y=BV, z=CW$.

Prove that there exist a triangle with sides $x, y, z$.
2006 Costa Rica - Final Round
Let $f$ be a function that satisfies :

$$ \displaystyle f(x)+2f\left(\frac{x+\frac{2001}2}{x-1}\right) = 4014-x. $$

Find $f(2004)$.
2006 Costa Rica - Final Round
Let $n$ be a positive integer, and let $p$ be a prime, such that $n>p$.

Prove that :

$$ \displaystyle \binom np \equiv \left\lfloor\frac{n}{p}\right\rfloor \ \pmod p. $$
2006 Costa Rica - Final Round
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.



Proposed by Dimitris Kontogiannis, Greece
2003 Costa Rica - Final Round
Two players $A$ and $B$ participate in the following game. Initially we have a pile of 2003 stones. $A$ plays first, and he picks a divisor of 2003 and removes that number of stones from the pile. Then $B$ picks a divisor of the number of remaining stones, and removes that number of stones from the pile, and so forth. The players who removes the last stone loses. Prove that one of the players has a winning strategy and describe it.
2003 Costa Rica - Final Round
Let $AB$ be a diameter of circle $\omega$. $\ell$ is the tangent line to $\omega$ at $B$. Take two points $C$, $D$ on $\ell$ such that $B$ is between $C$ and $D$. $E$, $F$ are the intersections of $\omega$ and $AC$, $AD$, respectively, and $G$, $H$ are the intersections of $\omega$ and $CF$, $DE$, respectively. Prove that $AH=AG$.
2003 Costa Rica - Final Round
If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds.
2003 Costa Rica - Final Round
$S_{1}$ and $S_{2}$ are two circles that intersect at distinct points $P$ and $Q$. $\ell_{1}$ and $\ell_{2}$ are two parallel lines through $P$ and $Q$. $\ell_{1}$ intersects $S_{1}$ and $S_{2}$ at points $A_{1}$ and $A_{2}$, different from $P$, respectively. $\ell_{2}$ intersects $S_{1}$ and $S_{2}$ at points $B_{1}$ and $B_{2}$, different from $Q$, respectively. Show that the perimeters of the triangles $A_{1}QA_{2}$ and $B_{1}PB_{2}$ are equal.
2003 Costa Rica - Final Round
Each of the squares of an $8 \times 8$ board can be colored white or black. Find the number of colorings of the board such that every $2 \times 2$ square contains exactly 2 black squares and 2 white squares.
2003 Costa Rica - Final Round
Say a number is tico if the sum of it's digits is a multiple of $2003$.



$\text{(i)}$ Show that there exists a positive integer $N$ such that the first $2003$ multiples, $N,2N,3N,\ldots 2003N$ are all tico.



$\text{(ii)}$ Does there exist a positive integer $N$ such that all it's multiples are tico?
1997 Croatia National Olympiad
Let $n$ be a natural number. Solve the equation

$$||\cdots|||x-1|-2|-3|-\ldots-(n-1)|-n|=0.$$
1997 Croatia National Olympiad
Given are real numbers $a<b<c<d$. Determine all permutations $p,q,r,s$ of the numbers $a,b,c,d$ for which the value of the sum

$$(p-q)^2+(q-r)^2+(r-s)^2+(s-p)^2$$is minimal.
1997 Croatia National Olympiad
A chord divides the interior of a circle $k$ into two parts. Variable circles $k_1$ and $k_2$ are inscribed in these two parts, touching the chord at the same point. Show that the ratio of the radii of circles $k_1$ and $k_2$ is constant, i.e. independent of the tangency point with the chord.
1997 Croatia National Olympiad
An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.)
1997 Croatia National Olympiad
In a regular hexagon $ABCDEF$ with center $O$, points $M$ and $N$ are the midpoints of the sides $CD$ and $DE$, and $L$ the intersection point of $AM$ and $BN$. Prove that:



(a) $ABL$ and $DMLN$ have equal areas;

(b) $\angle ALD=\angle OLN=60^\circ$;

(c) $\angle OLD=90^\circ$.
1997 Croatia National Olympiad
For any different positive numbers $a,b,c$ prove the inequality

$$a^ab^bc^c>a^bb^cc^a.$$
1997 Croatia National Olympiad
The number $2^{1997}$ has $m$ decimal digits, while the number $5^{1997}$ has $n$ digits. Evaluate $m+n$.
1997 Croatia National Olympiad
In the plane are given $1997$ points. Show that among the pairwise distances between these points, there are at least $32$ different values.
1997 Croatia National Olympiad
Integers $x,y,z$ and $a,b,c$ satisfy

$$x^2+y^2=a^2,\enspace y^2+z^2=b^2\enspace z^2+x^2=c^2.$$Prove that the product $xyz$ is divisible by (a) $5$, and (b) $55$.
1997 Croatia National Olympiad
Prove that for every real number $x$ and positive integer $n$

$$|\cos x|+|\cos2x|+|\cos2^2x|+\ldots+|\cos2^nx|\ge\frac n{2\sqrt2}.$$
1997 Croatia National Olympiad
The areas of the faces $ABD,ACD,BCD,BCA$ of a tetrahedron $ABCD$ are $S_1,S_2,Q_1,Q_2$, respectively. The angle between the faces $ABD$ and $ACD$ equals $\alpha$, and the angle between $BCD$ and $BCA$ is $\beta$. Prove that

$$S_1^2+S_2^2-2S_1S_2\cos\alpha=Q_1^2+Q_2^2-2Q_1Q_2\cos\beta.$$
1997 Croatia National Olympiad
On the sides of a triangle $ABC$ are constructed similar triangles $ABD,BCE,CAF$ with $k=AD/DB=BE/EC=CF/FA$ and $\alpha=\angle ADB=\angle BEC=\angle CFA$. Prove that the midpoints of the segments $AC,BC,CD$ and $EF$ form a parallelogram with an angle $\alpha$ and two sides whose ratio is $k$.
1997 Croatia National Olympiad
Find the last four digits of each of the numbers $3^{1000}$ and $3^{1997}$.
1997 Croatia National Olympiad
Consider a circle $k$ and a point $K$ in the plane. For any two distinct points $P$ and $Q$ on $k$, denote by $k'$ the circle through $P,Q$ and $K$. The tangent to $k'$ at $K$ meets the line $PQ$ at point $M$. Describe the locus of the points $M$ when $P$ and $Q$ assume all possible positions.
1997 Croatia National Olympiad
Function $f$ is defined on the positive integers by $f(1)=1$, $f(2)=2$ and

$$f(n+2)=f(n+2-f(n+1))+f(n+1-f(n))\enspace\text{for }n\ge1.$$(a) Prove that $f(n+1)-f(n)\in\{0,1\}$ for each $n\ge1$.

(b) Show that if $f(n)$ is odd then $f(n+1)=f(n)+1$.

(c) For each positive integer $k$ find all $n$ for which $f(n)=2^{k-1}+1$.
1997 Croatia National Olympiad
Let $k$ be a natural number. Determine the number of non-congruent triangles with the vertices at vertices of a given regular $6k$-gon.
1998 Croatia National Olympiad
Which number is greater:

$$A=\frac{2.00\ldots04}{1.00\ldots04^2+2.00\ldots04},\text{ or }B=\frac{2.00\ldots02}{1.00\ldots02^2+2.00\ldots02},$$where each of the numbers above contains $1998$ zeros?
1998 Croatia National Olympiad
Find all positive integer solutions of the equation $10(m+n)=mn$.
1998 Croatia National Olympiad
Ivan and Krešo started to travel from Crikvenica to Kraljevica, whose distance is $15$ km, and at the same time Marko started from Kraljevica to Crikvenica. Each of them can go either walking at a speed of $5$ km/h, or by bicycle with the speed of $15$ km/h. Ivan started walking, and Krešo was driving a bicycle until meeting Marko. Then Krešo gave the bicycle to Marko and continued walking to Kraljevica, while Marko continued to Crikvenica by bicycle. When Marko met Ivan, he gave him the bicycle and continued on foot, so Ivan arrived at Kraljevica by bicycle. Find, for each of them, the time he spent in travel as well as the time spent in walking.
1998 Croatia National Olympiad
Let there be given a regular hexagon of side length $1$. Six circles with the sides of the hexagon as diameters are drawn. Find the area of the part of the hexagon lying outside all the circles.
1998 Croatia National Olympiad
Solve the equation $2z^3-(5+6i)z^2+9iz+1-3i=0$ over $\mathbb C$ given that one of the solutions is real.
1998 Croatia National Olympiad
If $a,b$ are nonnegative real numbers, prove the inequality

$$\frac{a+\sqrt[3]{a^2b}+\sqrt[3]{ab^2}+b}4\le\frac{\sqrt{a+\sqrt{ab}+b}}3.$$
1998 Croatia National Olympiad
Points $E$ and $F$ are chosen on the sides $AB$ and $BC$ respectively of a square $ABCD$ such that $BE=BF$. Let $BN$ be an altitude of the triangle $BCE$. Prove that the triangle $DNF$ is right-angled.
1998 Croatia National Olympiad
For natural numbers $m,n$, set $a=(n+1)^m-n$ and $b=(n+1)^{m+3}-n$.

(a) Prove that $a$ and $b$ are coprime if $m$ is not divisible by $3$.

(b) Find all numbers $m,n$ for which $a$ and $b$ are not coprime.
1998 Croatia National Olympiad
Let $a,b,c$ be the sides and $\alpha,\beta,\gamma$ be the corresponding angles of a triangle. Prove the equality

$$\left(\frac bc+\frac cb\right)\cos\alpha+\left(\frac ca+\frac ac\right)\cos\beta+\left(\frac ab+\frac ba\right)\cos\gamma=3.$$
1998 Croatia National Olympiad
A hemisphere is inscribed in a cone so that its base lies on the base of the cone. The ratio of the area of the entire surface of the cone to the area of the hemisphere (without the base) is $\frac{18}5$. Compute the angle at the vertex of the cone.
1998 Croatia National Olympiad
Let $AA_1,BB_1,CC_1$ be the altitudes of a triangle $ABC$. If $\overrightarrow{AA_1}+\overrightarrow{BB_1}+\overrightarrow{CC_1}=0$ prove that the triangle $ABC$ is equilateral.
1998 Croatia National Olympiad
Among any $79$ consecutive natural numbers there exists one whose sum of digits is divisible by $13$. Find a sequence of $78$ consecutive natural numbers for which the above statement fails.
1998 Croatia National Olympiad
Let there be a given parabola $y^2=4ax$ in the coordinate plane. Consider all chords of the parabola that are visible at a right angle from the origin of the coordinate system. Prove that all these chords pass through a fixed point.
1998 Croatia National Olympiad
Let $a$ and $m$ be positive integers and $p$ be an odd prime number such that $p^m\mid a-1$ and $p^{m+1}\nmid a-1$. Prove that



(a) $p^{m+n}\mid a^{p^n}-1$ for all $n\in\mathbb N$, and

(a) $p^{m+n+1}\nmid a^{p^n}-1$ for all $n\in\mathbb N$.
1998 Croatia National Olympiad
Let $A=\{1,2,\ldots,2n\}$ and let the function $g:A\to A$ be defined by $g(k)=2n-k+1$. Does there exist a function $f:A\to A$ such that $f(k)\ne g(k)$ and $f(f(f(k)))=g(k)$ for all $k\in A$, if (a) $n=999$; (b) $n=1000$?
1998 Croatia National Olympiad
Eight bulbs are arranged on a circle. In every step we perform the following operation: We simultaneously switch off all those bulbs whose two neighboring bulbs are in different states, and switch on the other bulbs. Prove that after at most four steps all the bulbs will be switched on.
1999 Croatia National Olympiad
Circles $k_1$ and $k_2$ with radii $r_1=6$ and $r_2=3$ are externally tangent and touch a circle $k$ with radius $r=9$ from inside. A common external tangent of $k_1$ and $k_2$ intersects $k$ at $P$ and $Q$. Determine the length of $PQ$.
1999 Croatia National Olympiad
How do I prove that, for every $a, b, c$ positive real numbers such that $a+b+c = 1$ the following inequality holds: $\frac{a^3}{a^2+b^2} +\frac{b^3}{b^2+c^2} +\frac {c^3}{c^2+a^2} \geq  \frac{1}{2}$?
1999 Croatia National Olympiad
For each $a$, $1<a<2$, the graphs of functions $y=1-|x-1|$ and $y=|2x-a|$ determine a figure. Prove that the area of this figure is less than $\frac13$.
1999 Croatia National Olympiad
A triple of numbers $(a_1,a_2,a_3)=(3,4,12)$ is given. The following operation is performed a finite number of times: choose two numbers $a,b$ from the triple and replace them by $0.6x-0.8y$ and $0.8x+0.6y$. Is it possible to obtain the (unordered) triple $(2,8,10)$?