Navvye/Olympiad-Problems-CSV huggingface dataset:
1998 Turkey MO (2nd round)
Some of the vertices of unit squares of an $n\times n$ chessboard are colored so that any $k\times k$ ( $1\le k\le n$) square consisting of these unit squares has a colored point on at least one of its sides. Let $l(n)$ denote the minimum number of colored points required to satisfy this condition. Prove that $\underset{n\to \infty }{\mathop \lim }\,\frac{l(n)}{{{n}^{2}}}=\frac{2}{7}$.
1997 Turkey MO (2nd round)
Find all pairs of integers $(x, y)$ such that $5x^{2}-6xy+7y^{2}=383$.
1997 Turkey MO (2nd round)
Let $F$ be a point inside a convex pentagon $ABCDE$, and let $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$ denote the distances from $F$ to the lines $AB$, $BC$, $CD$, $DE$, $EA$, respectively. The points $F_{1}$, $F_{2}$, $F_{3}$, $F_{4}$, $F_{5}$ are chosen on the inner bisectors of the angles $A$, $B$, $C$, $D$, $E$ of the pentagon respectively, so that $AF_{1} = AF$  , $BF_{2} = BF$ , $CF_{3} = CF$ , $DF_{4} = DF$ and $EF_{5} = EF$ . If the distances from $F_{1}$, $F_{2}$, $F_{3}$, $F_{4}$, $F_{5}$ to the lines $EA$, $AB$, $BC$, $CD$, $DE$ are $b_{1}$, $b_{2}$, $b_{3}$, $b_{4}$, $b_{5}$, respectively.

Prove that $a_{1} + a_{2} + a_{3} + a_{4} + a_{5} \leq b_{1} + b_{2} + b_{3} + b_{4} + b_{5}$
1997 Turkey MO (2nd round)
Let $n$ and $k$ be positive integers, where $n > 1$ is odd. Suppose $n$ voters are to elect one of the $k$ cadidates from a set $A$ according to the rule of "majoritarian compromise" described below. After each voter ranks the candidates in a column according to his/her preferences, these columns are concatenated to form a $k$ x $n$ voting matrix. We denote the number of  ccurences of $a \in A$ in the $i$-th row of the voting matrix by $a_{i}$ . Let $l_{a}$ stand for the minimum integer $l$ for which $\sum^{l}_{i=1}{a_{i}}> \frac{n}{2}$.

Setting $l'= min	\{l_{a} | a \in A\}$, we will regard the voting matrices which make the set $\{a \in A | l_{a} = l' \}$ as admissible. For each such matrix, the single candidate in this set will get elected according to majoritarian compromise. Moreover, if $w_{1} \geq w_{2} \geq ... \geq  w_{k} \geq 0$ are given, for each admissible voting matrix, $\sum^{k}_{i=1}{w_{i}a_{i}}$ is called the total weighted score of $a \in A$. We will say that the system $(w_{1},w_{2},  . . . , w_{k})$ of weights represents majoritarian compromise if the total score of the elected candidate is maximum among the scores of all candidates.

(a) Determine whether there is a system of weights representing majoritarian compromise if $k = 3$.

(b) Show that such a system of weights does not exist for $k > 3$.
1997 Turkey MO (2nd round)
Let $e > 0$ be a given real number. Find the least value of $f(e)$ (in terms of $e$ only) such that the inequality $a^{3}+ b^{3}+ c^{3}+ d^{3} \leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})$ holds for all real numbers $a, b, c, d$.
1997 Turkey MO (2nd round)
In a triangle $ABC$, the inner and outer bisectors of the $\angle A$ meet the line $BC$ at $D$ and $E$, respectively. Let $d$ be a common tangent of the circumcircle $(O)$ of $\triangle ABC$ and the circle with diameter $DE$ and center $F$. The projections of the tangency points onto $FO$ are denoted by $P$ and $Q$, and the length of their common chord is denoted by $m$. Prove that $PQ = m$
1997 Turkey MO (2nd round)
Let $D_{1}, D_{2}, . . . , D_{n}$ be rectangular parallelepipeds in space, with edges parallel to the $x, y, z$ axes. For each $D_{i}$, let $x_{i} , y_{i} , z_{i}$ be the lengths of its projections onto the $x, y, z$ axes, respectively. Suppose that for all pairs $D_{i}$ , $D_{j}$, if at least one of $x_{i} < x_{j}$ , $y_{i} < y_{j}$, $z_{i} < z_{j}$ holds, then $x_{i} \leq x_{j}$ , $y_{i} \leq y_{j}$, and $z_{i} < z_{j}$ . If the volume of the region $\bigcup^{n}_{i=1}{D_{i}}$ equals 1997, prove that there is a subset $\{D_{i_{1}}, D_{i_{2}}, . . . , D_{i_{m}}\}$ of the set $\{D_{1}, . . . , D_{n}\}$ such that

$(i)$ if $k 	\not= l $  then $D_{i_{k}} \cap D_{i_{l}} = 	\emptyset $, and

$(ii)$ the volume of $\bigcup^{m}_{k=1}{D_{i_{k}}}$  is at least 73.
1996 Turkey MO (2nd round)
Let $({{A}_{n}})_{n=1}^{\infty }$ and $({{a}_{n}})_{n=1}^{\infty }$ be sequences of positive integers. Assume that for each positive integer $x$, there is a unique positive integer $N$ and a unique $N-tuple$ $({{x}_{1}},...,{{x}_{N}})$ such that

$0\le {{x}_{k}}\le {{a}_{k}}$ for $k=1,2,...N$, ${{x}_{N}}\ne 0$, and $x=\sum\limits_{k=1}^{N}{{{A}_{k}}{{x}_{k}}}$.



(a) Prove that ${{A}_{k}}=1$ for some $k$;

(b) Prove that ${{A}_{k}}={{A}_{j}}\Leftrightarrow k=j$;

(c) Prove that if ${{A}_{k}}\le {{A}_{j}}$, then $\left. {{A}_{k}} \right|{{A}_{j}}$.
1996 Turkey MO (2nd round)
Let $ABCD$ be a square of side length 2, and let $M$ and $N$ be points on the sides $AB$ and $CD$ respectively. The lines $CM$ and $BN$ meet at $P$, while the lines $AN$ and $DM$ meet at $Q$. Prove that $\left| PQ \right|\ge 1$.
1996 Turkey MO (2nd round)
Let $n$ integers on the real axis be colored. Determine for which positive integers $k$ there exists a family $K$ of closed intervals with the following properties:

i)  The union of the intervals in $K$ contains all of the colored points;

ii) Any two distinct intervals in $K$ are disjoint;

iii) For each interval $I$ at $K$ we have ${{a}_{I}}=k.{{b}_{I}}$, where ${{a}_{I}}$ denotes the number of integers in $I$,  and ${{b}_{I}}$ the number of colored integers in $I$.
1996 Turkey MO (2nd round)
A circle is tangent to sides $AD,\text{ }DC,\text{ }CB$ of a convex quadrilateral $ABCD$ at $\text{K},\text{ L},\text{ M}$ respectively. A line $l$, passing through $L$ and parallel to $AD$, meets $KM$ at $N$ and $KC$ at $P$. Prove that $PL=PN$.
1996 Turkey MO (2nd round)
Prove that $\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})}$ is divisible by $n!$ for all positive integers $n$.
1996 Turkey MO (2nd round)
Show that there is no function $f:{{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}$ such that $f(x+y)>f(x)(1+yf(x))$

for all $x,y\in {{\mathbb{R}}^{+}}$.
1995 Turkey MO (2nd round)
Let $m_{1},m_{2},\ldots,m_{k}$ be integers with $2\leq m_{1}$ and $2m_{1}\leq m_{i+1}$ for all $i$. Show that for any integers $a_{1},a_{2},\ldots,a_{k}$ there are infinitely many integers $x$ which do not satisfy any of the congruences $$x\equiv a_{i}\ (\bmod \ m_{i}),\ i=1,2,\ldots k.$$
1995 Turkey MO (2nd round)
Let $ABC$ be an acute triangle and let $k_{1},k_{2},k_{3}$ be the circles with diameters $BC,CA,AB$, respectively. Let $K$ be the radical center of these circles. Segments $AK,CK,BK$ meet $k_{1},k_{2},k_{3}$ again at $D,E,F$, respectively. If the areas of triangles $ABC,DBC,ECA,FAB$ are $u,x,y,z$, respectively, prove that $$u^{2}=x^{2}+y^{2}+z^{2}.$$
1995 Turkey MO (2nd round)
Let $A$ be a real number and $(a_{n})$ be a sequence of real numbers such that $a_{1}=1$ and $$1<\frac{a_{n+1}}{a_{n}}\leq A \mbox{ for all }n\in\mathbb{N}.$$



$(a)$ Show that there is a unique non-decreasing surjective function $f: \mathbb{N}\rightarrow \mathbb{N}$ such that $1<A^{k(n)}/a_{n}\leq A$ for all $n\in \mathbb{N}$.



$(b)$ If $k$ takes every value at most $m$ times, show that there is a real number $C>1$ such that $Aa_{n}\geq C^{n}$ for all $n\in \mathbb{N}$.
1995 Turkey MO (2nd round)
In a triangle $ABC$ with $AB\neq AC$, the internal and external bisectors of angle $A$ meet the line $BC$ at $D$ and $E$ respectively. If the feet of the perpendiculars from a point $F$ on the circle with diameter $DE$ to $BC,CA,AB$ are $K,L,M$, respectively, show that $KL=KM$.
1995 Turkey MO (2nd round)
Let $t(A)$ denote the sum of elements of a nonempty set $A$ of integers, and define $t(\emptyset)=0$. Find a set $X$ of positive integers such that for every integers $k$ there is a unique ordered pair of disjoint subsets $(A_{k},B_{k})$ of $X$ such that $t(A_{k})-t(B_{k}) = k$.
1995 Turkey MO (2nd round)
Find all surjective functions $f: \mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in \mathbb{N}$ $$f(m)\mid f(n) \mbox{ if and only if }m\mid n.$$
1994 Turkey MO (2nd round)
For $n\in\mathbb{N}$, let $a_{n}$ denote the closest integer to $\sqrt{n}$. Evaluate $$\sum_{n=1}^\infty{\frac{1}{a_{n}^{3}}}.$$
1994 Turkey MO (2nd round)
Let $ABCD$ be a cyclic quadrilateral $\angle{BAD}< 90^\circ$ and $\angle BCA = \angle DCA$. Point $E$ is taken on segment $DA$ such that $BD=2DE$. The line through $E$ parallel to $CD$ intersects the diagonal $AC$ at $F$. Prove that $$ \frac{AC\cdot BD}{AB\cdot FC}=2.$$
1994 Turkey MO (2nd round)
Let $n$ blue lines, no two of which are parallel and no three concurrent, be drawn on a plane. An intersection of two blue lines is called a blue point. Through any two blue points that have not already been joined by a blue line, a red line is drawn. An intersection of two red lines is called a red point, and an intersection of red line and a blue line is called a purple point. What is the maximum possible number of purple points?
1994 Turkey MO (2nd round)
Let $f: \mathbb{R}^{+}\rightarrow \mathbb{R}+$ be an increasing function. For each $u\in\mathbb{R}^{+}$, we denote $g(u)=\inf\{ f(t)+u/t \mid t>0\}$. Prove that:



$(a)$ If $x\leq g(xy)$, then $x\leq 2f(2y)$;



$(b)$ If $x\leq f(y)$, then $x\leq 2g(xy)$.
1994 Turkey MO (2nd round)
Find the set of all ordered pairs $(s,t)$ of positive integers such that $$t^{2}+1=s(s+1).$$
1994 Turkey MO (2nd round)
The incircle of triangle $ABC$ touches $BC$ at $D$ and $AC$ at $E$. Let $K$ be the point on $CB$ with $CK=BD$, and $L$ be the point on $CA$ with $AE=CL$. Lines $AK$ and $BL$ meet at $P$. If $Q$ is the midpoint of $BC$, $I$ the incenter, and $G$ the centroid of $\triangle ABC$, show that:



$(a)$ $IQ$ and $AK$ are parallel,



$(b)$ the triangles $AIG$ and $QPG$ have equal area.
1993 Turkey MO (2nd round)
Prove that there is a number such that its decimal represantation ends with 1994 and it can be written as $1994\cdot 1993^{n}$ ($n\in{Z^{+}}$)
1993 Turkey MO (2nd round)
I centered incircle of triangle $ABC$ $(m(\hat{B})=90^\circ)$ touches $\left[AB\right], \left[BC\right], \left[AC\right]$ respectively at $F, D, E$. $\left[CI\right]\cap\left[EF\right]={L}$ and $\left[DL\right]\cap\left[AB\right]=N$. Prove that $\left[AI\right]=\left[ND\right]$.
1993 Turkey MO (2nd round)
$n\in{Z^{+}}$ and $A={1,\ldots ,n}$. $f: N\rightarrow N$ and $\sigma: N\rightarrow N$ are two permutations, if there is one $k\in A$ such that $(f\circ\sigma)(1),\ldots ,(f\circ\sigma)(k)$ is increasing and $(f\circ\sigma)(k),\ldots ,(f\circ\sigma)(n)$ is decreasing sequences we say that $f$ is good for $\sigma$. $S_\sigma$ shows the set of good functions for $\sigma$.

a) Prove that, $S_\sigma$ has got $2^{n-1}$ elements for every $\sigma$ permutation.

b)$n\geq 4$, prove that there are permutations $\sigma$ and $\tau$ such that, $S_{\sigma}\cap S_{\tau}=\phi$

.
1993 Turkey MO (2nd round)
$a_{n}$ is a sequence of positive integers such that, for every $n\geq 1$, $0<a_{n+1}-a_{n}<\sqrt{a_{n}}$. Prove that for every $x,y\in{R}$ such that $0<x<y<1$ $x< \frac{a_{k}}{a_{m}}<y$ we can find such $k,m\in{Z^{+}}$.
1993 Turkey MO (2nd round)
Prove that we can draw a line (by a ruler and a compass) from a vertice of a convex quadrilateral such that, the line divides the quadrilateral to two equal areas.
1993 Turkey MO (2nd round)
$n_{1},\ldots ,n_{k}, a$ are integers that satisfies the above conditions A)For every $i\neq j$, $(n_{i}, n_{j})=1$ B)For every $i, a^{n_{i}}\equiv 1 (mod n_{i})$ C)For every $i, X^{a-1}\equiv 0(mod n_{i})$.

Prove that $a^{x}\equiv 1(mod x)$ congruence has at least $2^{k+1}-2$ solutions. ($x>1$)
2015 Turkmenistan National Math Olympiad
Solve : $y(x+y)^2=9 $ ; $y(x^3-y^3)=7$
2015 Turkmenistan National Math Olympiad
Find $ \lim_{n\to\infty}(\sum_{i=0}^{n}\frac{1}{n+i})$
2015 Turkmenistan National Math Olympiad
Find the sum : $C^{n}_{1}$ - $\frac{1}{3} \cdot C^{n}_{3}$ + $\frac{1}{9} \cdot C^{n}_{5}$ - $\frac{1}{27} \cdot C^{n}_{9}$ + ...
2015 Turkmenistan National Math Olympiad
Find the max and minimum without using dervivate:

$\sqrt{x} +4 \cdot \sqrt{\frac{1}{2} - x}$
1997 Ukraine National Mathematical Olympiad
Which number is greater: $ 4^{4^{4^{4^4}}}$ or $ 5^{5^{5^5}}$?
1997 Ukraine National Mathematical Olympiad
There are $ n$ candidates on a table. Petrik and Mikola alternately take candies from the table according to the following rule. Petrik starts by taking one candy; then Mikola takes $ i$ candies, where $ i$ divides $ 2$, then Petrik takes $ j$ candies, where $ j$ divides $ 3$, and so on. The player who takes the last candy wins the game. Which player has a winning strategy?
1997 Ukraine National Mathematical Olympiad
Consider acute-angled triangles $ ABC$ and $ APQ$, where $ P$ and $ Q$ lie on the side $ BC.$ Prove that the circumcenter of $ \triangle ABC$ is closer to line $ BC$ than the circumcenter of $ \triangle APQ.$
1997 Ukraine National Mathematical Olympiad
Two players alternately write the numbers $ 1,2,...,7$ in the cells of the figure on the picture. After the figure is filled up, the sum of numbers along each line on the picture is calculated. If three of these sums are equal, the first player wins; otherwise the second player wins. Which of the players has a winning strategy?
1997 Ukraine National Mathematical Olympiad
Construct the bisector of a given angle using a ruler and a compass, but without marking any auxiliary points inside the angle.
1997 Ukraine National Mathematical Olympiad
Find all natural numbers $ n$ for which $ 2^n-n^2$ is divisible by $ 7$.
1997 Ukraine National Mathematical Olympiad
There are four apples on a plate, of weights $ 600,400,300,$ and $ 250$ grammes. Two pupils take an apple each (one by one) and start eating simultaneously. A pupil can take another apple only after he has eaten the previous apple. It is assumed that they equaly fast (measured in grammes per second). How should the first pupil do in order to eat as large amount as possible?
1997 Ukraine National Mathematical Olympiad
Solve in real numbers the equation $ 9^x+4^x+1=6^x+3^x+2^x.$
1997 Ukraine National Mathematical Olympiad
The incircle of a triangle $ ABC$ is tangent to its sides $ AB,BC,CA$ at $ M,N,K,$ respectively. A line $ l$ through the midpoint $ D$ of $ AC$ is parallel to $ MN$ and intersects the lines $ BC$ and $ BD$ at $ T$ and $ S$, respectively. Prove that $ TC=KD=AS.$
1997 Ukraine National Mathematical Olympiad
Does there exist a positvie integer $ a$ such that all the numbers $ a,2a,3a,...,1997a$ are perfect powers (i.e. numbers of the form $ m^k$, where $ m,k \in \mathbb{N}$, $ k \ge 2$)?
1997 Ukraine National Mathematical Olympiad
Find the smallest natural number which can be represented in the form $ 19m+97n$, where $ m,n$ are positive integers, in at least two different ways.
1997 Ukraine National Mathematical Olympiad
The cells of a rectangular board are colored in the chess manner. Every cell is filled with an integer. Assume that the sum of numbers in any row and in any column is even. Prove that the sum of numbers in the black cells is even.
1997 Ukraine National Mathematical Olympiad
Triangles $ ABC$ and $ A_1 B_1 C_1$ are non-congruent, but $ AC=A_1 C_1=b,$ $ BC=B_1 C_1=a$, and $ BH=B_1 H_1$, where $ BH$ and $ B_1 H_1$ are the altitudes. Prove the inequality:



$ a \cdot AB+b \cdot A_1 B_1 \le \sqrt{2}(a^2+b^2).$
1997 Ukraine National Mathematical Olympiad
For every natural number $ n$ prove the inequality:



$ \sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt{2}}}}+\sqrt{6+\sqrt{6+\sqrt{6+...+\sqrt{6}}}}<5$



where there are $ n$ twos and $ n$ sixs on the left hand side.
1997 Ukraine National Mathematical Olympiad
Find the maximum value of the function:



$ f(x)=\frac{x}{x^2+9}+\frac{1}{x^2-6x+21}+\cos {2 \pi x}, x>0.$
1997 Ukraine National Mathematical Olympiad
Each side of an equilateral triangle is divided into $ 1997$ equal parts, and each vertex of the triangle is joined by a segment to each of the division points on the opposite side. All the intersection points of these segments inside the triangle are marked. How many marked points are there?
1997 Ukraine National Mathematical Olympiad
A sequence is defined by $ a_1=a_2=a_3=1$ and $ a_{n+3}=-a_n-a_{n+1}$ for all $ n \in \mathbb{N}$. Prove that this sequence is not bounded , i.e. that for every $ M \in \mathbb{R}$ there is an $ n$ for which $ |a_n|>M.$
1997 Ukraine National Mathematical Olympiad
Two regular pentagons $ ABCDE$ and $ AEKPL$ are placed in space so that $ \angle DAK=60^{\circ}$. Prove that the planes $ ACK$ and $ BAL$ are perpendicular.
1997 Ukraine National Mathematical Olympiad
Solve in the real numbers the system:



$ x_1+x_2+...+x_{1997}=1997,$



$ x_1^4+x_2^4+...+x_{1997}^4=x_1^3+x_2^3+...+x_{1997}^3.$
1997 Ukraine National Mathematical Olympiad
In a parallelogram $ ABCD$, $ M$ is the midpoint of $ BC$ and $ N$ an arbitrary point on the side $ AD$. Let $ P$ be the intersection of $ MN$ and $ AC$, and $ Q$ the intersection of $ AM$ and $ BN$. Prove that the triangles $ BDQ$ and $ DMP$ have equal areas.
1997 Ukraine National Mathematical Olympiad
Let $ d(n)$ denote the largest odd divisor of a positive integer $ n$. The function $ f: \mathbb{N} \rightarrow \mathbb{N}$ is defined by $ f(2n-1)=2^n$ and $ f(2n)=n+\frac{2n}{d(n)}$ for all $ n \in \mathbb{N}$. Find all natural numbers $ k$ such that: $ f(f(...f(1)...))=1997.$ (where the paranthesis appear $ k$ times)
1997 Ukraine National Mathematical Olympiad
Let $ ABCD$ be a parallelogram with $ AB=1$. Suppose that $ K$ is a point on the side $ AD$ such that $ KD=1, \angle ABK=90^{\circ}$ and $ \angle DBK=30^{\circ}$. Determine $ AD$.
1997 Ukraine National Mathematical Olympiad
Prove that among any four distinct numbers from the interval $ (0,\frac{\pi}{2})$ there are two, say $ x,y,$ such that:



$ 8 \cos x \cos y \cos (x-y)+1>4(\cos ^2 x+\cos ^2 y).$
1997 Ukraine National Mathematical Olympiad
In space are given $ 1997$ points. Let $ M$ and $ m$ be the largest and smallest distance between two of the given points. Prove that $ \frac{M}{m}>9$.
1997 Ukraine National Mathematical Olympiad
The equation $ ax^3+bx^2+cx+d=0$ has three distinct solutions. How many distinct solutions does the following equation have:



$ 4(ax^3+bx^2+cx+d)(3ax+b)=(3ax^2+2bx+c)^2?$
1997 Ukraine National Mathematical Olympiad
Find all real solutions to the equation:



$ (2+\sqrt{3})^x+1=\left( 2\sqrt{2+\sqrt{3}} \right)^x$.
1997 Ukraine National Mathematical Olympiad
Find all functions $ f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ which satisfy the conditions:



$ (i)$ $ f(x+1)=f(x)+1$ for all $ x \in \mathbb{Q}^{+}$



$ (ii)$ $ f(x^2)=f(x)^2$ for all $ x \in \mathbb{Q}^{+}$.
1997 Ukraine National Mathematical Olympiad
Determine the smallest natural number $ n$ such that among any $ n$ integers one can choose $ 18$ integers whose sum is divisible by $ 18$.
1997 Ukraine National Mathematical Olympiad
On the edges $ AB,BC,CD,DA$ of a parallelepiped $ ABCDA_1 B_1 C_1 D_1$ points $ K,L,M,N$ are selected, respectively. Prove that the circumcenters of the tetrahedra $ A_1 AKN, B_1 BKL, C_1 CLM, D_1 DMN$ are vertices of a parallelogram.
1998 Ukraine National Math Olympiad
In a family of five members (father, mother and three children), the product of the (integral) ages of all members equals $1998$. Knowing that the father is $10$ years older than the mother, determine the ages of all members.
1998 Ukraine National Math Olympiad
The plane is partitioned into congruent regular hexagons. Of these hexagons, some $1998$ are marked. Show that one can select $666$ of the marked hexagons in such a way that no two of them share a vertex.
1998 Ukraine National Math Olympiad
On the lower-left corner of an $n\times n$ board there is a piece. In a move a player can move the piece to the adjacent cell up, right, or up-right. Two players alternate moving the piece, and the winner is the one who places the piece on the upper-right corner. Who has a winning strategy?
1998 Ukraine National Math Olympiad
Determine the last four decimal digits of the number $1997\cdot5^{1998}$.
1998 Ukraine National Math Olympiad
A line $l$ and points $A,B$ on the same side of $l$ are given in the plane. Construct (with a ruler and a compass) a point $C$ such that the line $l$ intersects $AC$ at $M$ and $BC$ at $N$, where $BM$ is the altitude and $AN$ the median.
1998 Ukraine National Math Olympiad
Show that it is possible to write a number from the set $\{1,2,3,4,5\}$ in each square of a $5\times120$ board ($5$ columns and $120$ rows) so that the following conditions are satisfied:



(i) all numbers in the same row are distinct;

(ii) all rows are distinct;

(iii) the board can be partitioned into $24$ $5\times5$ boards which can be reassembled (without rotating and turning over) into a $120\times5$ board whose $120$ columns are distinct.
1998 Ukraine National Math Olympiad
Is there a triangle in the coordinate plane whose vertices, centroid, orthocenter, incenter and circumcenter all have integral coordinates?
1998 Ukraine National Math Olympiad
A quadrilateral $ABCD$ is inscribed in a circle with diameter $AD$. Using a ruler and a compass, construct a triangle inscribed in the same circle and having the same area as $ABCD$.
1998 Ukraine National Math Olympiad




sqing wrote:

Let $ \displaystyle{a,b,c}$ be positive real numbers such that $\displaystyle{abc=1.}$ Prove that $$ \displaystyle{\frac{1+ab}{a+b}+\frac{1+bc}{b+c}+\frac{1+ca}{c+a}> 2.}$$



Let $ \displaystyle{a,b,c}$ be positive real numbers such that $\displaystyle{abc=1.}$ Prove that $$ \displaystyle{\frac{1+ab}{1+a}+\frac{1+bc}{1+b}+\frac{1+ca}{1+c}\geq 3.}$$here
1998 Ukraine National Math Olympiad
Real numbers $x$ and $y$ not less than $1$ have the property that

$$\left\lfloor\frac xy\right\rfloor=\frac{\lfloor nx\rfloor}{\lfloor ny\rfloor}\enspace\text{for any }n\in\mathbb N.$$Prove that either $x=y$ or $x$ and $y$ are integers, one dividing the other.
1998 Ukraine National Math Olympiad
Prove that the equation $x^3-x=y^2-19y+98$ has no solutions in integers.
1998 Ukraine National Math Olympiad
Prove that the sum of squared lengths of the medians of a triangle does not exceed the square of its semiperimeter.
1998 Ukraine National Math Olympiad
Two players alternately write numbers in the cells of an $n\times n$ square board. Hereby, in the intersection of the $i$-th row and the $j$-th column the first player may write the greatest common divisor of numbers $i$ and $j$, whereas the second player may write their least common multiple. When the board is filled up, the numbers of the first column are divided by $1$, those of the second column by $2$, etc., those of the last column by $n$. Then the product of the obtained numbers in the board is computed. If the result is smaller than $1$, the first player wins, otherwise the second player wins. Which player has a winning strategy?
1998 Ukraine National Math Olympiad
A convex $2000$-gon is given in the plane. Show that one can select $1998$ points in the plane such that every triangle with the vertices at vertices of the $2000$-gon contains exactly one of the selected points in its interior (excluding the boundary).
1998 Ukraine National Math Olympiad
Find all pairs of real numbers $(x,y)$ satisfying the equation

$$\sin^2x+\sin^2y=\frac{|x|}x+\frac{|y|}y.$$
1998 Ukraine National Math Olympiad
Point $M$ is arbitrarily taken on side $AC$ of a triangle $ABC$. Let $O$ be the intersection of the perpendiculars from the midpoints of segments $AM$ and $MC$ to lines $BC$ and $AB$, respectively. For which position of $M$ is the distance $OM$ minimal?
1998 Ukraine National Math Olympiad
A finite set of segments on a line has the following property: In any subset of $1998$ segments there are two having a common point. Show that there exist $1997$ points on the line such that each segment contains at least one of these points.
1998 Ukraine National Math Olympiad
A function $f$ defined on the interval $[1,+\infty)$ satisfies

$$\frac{f(2x)}{\sqrt2}\le f(x)\le x\enspace\text{for }x\ge1.$$Prove that $f(x)<\sqrt{2x}$ for all $x\ge1$.
1998 Ukraine National Math Olympiad
Let $m$ be the smallest of the four numbers $1,x^9,y^9,z^7$, where $x,y,z$ are nonnegative numbers. Prove that $m\le xy^2z^3$.