Navvye/Olympiad-Problems-CSV huggingface dataset:
2008 Singapore MO Open
let n,m be positive integers st $m>n\geq 5$ with m depending on n.

consider the sequence $a_1,a_2,...a_m$ where

$a_i=i$ for $i=1,...,n$

$a_{n+j}=a_{3j}+a_{3j-1}+a_{3j-2}$ for $j=1,..,m-n$

with $m-3(m-n)=$1 or 2, ie $a_m=a_{m-k}+a_{m-k-1}+a_{m-k-2}$ where k=1 or 2

(Thus if $n=5$, the sequence is 1,2,3,4,5,6,15

and if $n=8$, the sequence is 1,2,3,4,5,6,7,8,6,15,21)

Find $S=a_1+...+a_m$ if (i) $n=2007$ (ii) $n=2008$
2008 Singapore MO Open
let $0<a,b<\pi/2$. Show that

$\frac{5}{cos^2(a)}+\frac{5}{sin^2(a)sin^2(b)cos^2(b)} \geq 27cos(a)+36sin(a) $
2008 Singapore MO Open
consider a $2008 \times 2008$ chess board. let $M$ be the smallest no of rectangles that can be drawn on the chess board so that sides of every cell of the board is contained in the sides of one of the rectangles. find the value of $M$. (eg for $2\times 3$ chessboard, the value of $M$ is 3.)
2007 Singapore MO Open
Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$.
2007 Singapore MO Open
Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial

$f(x) = (x -a_1)(x - a_2)\cdot  ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients

and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.
2007 Singapore MO Open
Let $A_1$, $B_1$ be two points on the base $AB$ of an isosceles triangle $ABC$, with $\angle C>60^{\circ}$, such that $\angle A_1CB_1=\angle ABC$. A circle externally tangent to the circumcircle of $\triangle A_1B_1C$ is tangent to the rays $CA$ and $CB$ at points $A_2$ and $B_2$, respectively. Prove that $A_2B_2=2AB$.
2007 Singapore MO Open
find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ st



$f(f(m)+f(n))=m+n \,\forall m,n\in\mathbb{N}$



related:
2007 Singapore MO Open
Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.
2006 Singapore MO Open
In the triangle $ABC,\angle A=\frac{\pi}{3},D,M$ are points on the line $AC$ and $E,N$ are points on the line $AB$ such that $DN$ and $EM$ are the perpendicular bisectors of $AC$ and $AB$ respectively. Let $L$ be the midpoint of $MN$. Prove that $\angle EDL=\angle ELD$
2006 Singapore MO Open
Show that any representation of 1 as the sum of distinct reciprocals of numbers drawn from the arithmetic progression $\{2,5,8,11,...\}$ such as given in the following example must have at least eight terms: $$1=\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{20}+\frac{1}{41}+\frac{1}{110}+\frac{1}{1640}$$
2006 Singapore MO Open
Consider the sequence $p_{1},p_{2},...$ of primes such that for each $i\geq2$, either $p_{i}=2p_{i-1}-1$ or $p_{i}=2p_{i-1}+1$. Show that any such sequence has a finite number of terms.
2006 Singapore MO Open
Let $n$ be positive integer. Let $S_1,S_2,\cdots,S_k$ be a collection of $2n$-element subsets of $\{1,2,3,4,...,4n-1,4n\}$ so that $S_{i}\cap S_{j}$ contains at most $n$ elements for all $1\leq i<j\leq k$. Show that $$k\leq 6^{(n+1)/2}$$
2006 Singapore MO Open
Let $a,b,n$ be positive integers. Prove that $n!$ divides $$b^{n-1}a(a+b)(a+2b)...(a+(n-1)b)$$
2005 Singapore MO Open
An integer is square-free if it is not divisible by $a^2$ for any integer $a>1$. Let $S$ be the set of positive square-free integers. Determine, with justification, the value of$$\sum_{k\epsilon S}\left[\sqrt{\frac{10^{10}}{k}}\right]$$where $[x]$ denote the greatest integer less than or equal to $x$
2005 Singapore MO Open
Let $G$ be the centroid of triangle $ABC$. Through $G$ draw a line parallel to $BC$ and intersecting the sides $AB$ and $AC$ at $P$ and $Q$ respectively. Let $BQ$ intersect $GC$ at $E$ and $CP$ intersect $GB$ at $F$. If $D$ is midpoint of $BC$, prove that triangles $ABC$ and $DEF$ are similar
2005 Singapore MO Open
Let $a,b,c$ be real numbers satisfying $a<b<c,a+b+c=6,ab+bc+ac=9$. Prove that $0<a<1<b<3<c<4$



SolutionLet $abc=k$, then $a,b,c\ (a<b<c)$ are the roots of cubic equation $x^3-6x^2+9x-k=0\Longleftrightarrow x(x-3)^2=k$



that is to say, $a,b,c\ (a<b<c)$ are the $x$-coordinates of the interception of points between $y=x(x-3)^2$ and



$y=k$.



$y=x(x-3)^2$ have local maximuml value of $4$ at $x=1$ and  local minimum value of $0$ at $x=3$.



Since the $x$-coordinate of  the interception point between $y=x(x-3)^2$ and $y=4$ which is the tangent line at



local maximum point $(1,4)$ is a point $(4,4)$,Moving the line $y=k$ so that the two graphs $y=x(x-3)^2$ and



$y=k$ have the distinct three interception points,we can find that the range of $a,b,c$ are



$0<a<1,1<b<3,3<c<4

$,we are done.
2005 Singapore MO Open
Place 2005 points on the circumference of a circle. Two points $P,Q$ are said to form a pair of neighbours if the chord $PQ$ subtends an angle of at most 10 degrees at the centre. Find the smallest number of pairs of neighbours.
2004 Singapore MO Open
Let $m,n$ be integers so that $m \ge  n > 1$. Let $F_1,...,F_k$ be a collection of $n$-element subsets of $\{1,...,m\}$ so that $F_i\cap F_j$ contains at most $1$ element, $1 \le  i < j \le k$. Show that $k\le \frac{m(m-1)}{n(n-1)} $
2004 Singapore MO Open
Find the number of ordered pairs $(a, b)$ of integers, where $1 \le a, b \le 2004$, such that $x^2 + ax + b = 167 y$

has integer solutions in $x$ and $y$. Justify your answer.
2004 Singapore MO Open
Let $AD$ be the common chord of two circles $\Gamma_1$ and $\Gamma_2$. A line through $D$ intersects $\Gamma_1$ at $B$ and $\Gamma_2$ at $C$. Let $E$ be a point on the segment $AD$, different from $A$ and $D$. The line $CE$ intersect $\Gamma_1$ at $P$ and $Q$. The line $BE$ intersects $\Gamma_2$ at $M$ and $N$.

(i) Prove that $P,Q,M,N$ lie on the circumference of a circle $\Gamma_3$.

(ii) If the centre of $\Gamma_3$ is $O$, prove that $OD$ is perpendicular to $BC$.
2004 Singapore MO Open
If $0 <x_1,x_2,...,x_n\le 1$, where $n \ge 1$, show that

$$\frac{x_1}{1+(n-1)x_1}+\frac{x_2}{1+(n-1)x_2}+...+\frac{x_n}{1+(n-1)x_n}\le 1$$
2003 Singapore MO Open
A sequence $(a_1,a_2,...,a_{675})$ is given so that each term is an alphabet in the English language (no distinction is made between lower and upper case letters). It is known that in the sequence $a$ is never followed by $b$ and $c$ is never followed by $d$. Show that there are integers $m$ and $n$ with $1 \le m < n \le 674$ such that $a_m = a_n$ and $a_{m+1} = a_{n+1}$·
2003 Singapore MO Open
Find the maximum value of $\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}$ as $x, y$ and $z$ range over the set of all positive real numbers. Justify your answer.
2003 Singapore MO Open
For any given prime $p$, determine whether the equation $x^2 + y^2 + p^z = 2003$ always has integer solutions in $x, y, z$. Justify your answer
2003 Singapore MO Open
The pentagon $ABCDE$ which is inscribed in a circle with $AB < DE$ is the base of a pyramid with apex $S$. If the longest side from $S$ is $SA$, prove that $BS > CS$.
2002 Singapore MO Open
In the plane, $\Gamma$ is a circle with centre $O$ and radius $r, P$ and $Q$ are distinct points on $\Gamma , A$ is a point outside $\Gamma , M$ and $N$ are the midpoints of $PQ$ and $AO$ respectively. Suppose$ OA = 2a$ and $\angle PAQ$ is a right angle. Find the length of $MN$ in terms of $r$ and $a$. Express your answer in its simplest form, and justify your answer.
2002 Singapore MO Open
Let $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$ be real numbers between $1001$ and $2002$ inclusive. Suppose $ \sum_{i=1}^n a_i^2=  \sum_{i=1}^n b_i^2$.  Prove that  $$\sum_{i=1}^n\frac{a_i^3}{b_i} \le  \frac{17}{10} \sum_{i=1}^n a_i^2$$Determine when equality holds.
2002 Singapore MO Open
Let $n$ be a positive integer. Determine the smallest value of the sum $a_1b_1+a_2b_2+...+a_{2n+2}b_{2n+2}$

where $(a_1,a_2,...,a_{2n+2})$ and $(b_1,b_2,...,b_{2n+2})$ are rearrangements of the binomial coefficients $2n+1 \choose 0$, $2n+1 \choose 1$,...,$2n+1 \choose 2n+1$. Justify your answer
2002 Singapore MO Open
Find all real-valued functions $f : Q \to R$ defined on the set of all rational numbers $Q$ satisfying the conditions $f(x + y) = f(x) + f(y) + 2xy$ for all $x, y$ in $Q$ and $f(1) = 2002.$ Justify your answers.
2001 Singapore MO Open
In a parallelogram $ABCD$, the perpendiculars from $A$ to $BC$ and $CD$ meet the line segments $BC$ and $CD$ at the points $E$ and $F$ respectively. Suppose $AC = 37$ cm and $EF = 35$ cm. Let $H$ be the orthocentre of $\vartriangle AEF$. Find the length of $AH$ in cm. Show the steps in your calculations.
2001 Singapore MO Open
Let $n$ be a positive integer, and let $a_1,a_2,...,a_n$ be $n$ positive real numbers such that $a_1+a_2+...+a_n = 1$. Is it true that $\frac{a_1^4}{a_1^2+a_2^2}+\frac{a_2^4}{a_2^2+a_3^2}+\frac{a_3^4}{a_3^2+a_4^2}+...+\frac{a_{n-1}^4}{a_{n-1}^2+a_n^2}+\frac{a_n^4}{a_n^2+a_1^2}\ge \frac{1}{2n}$ ?

Justify your answer.
2001 Singapore MO Open
Suppose that there are $2001$ golf balls which are numbered from $1$ to $2001$ respectively, and some of these golf balls are placed inside a box. It is known that the difference between the two numbers of any two golf balls inside the box is neither $5$ nor $8$. How many such golf balls the box can contain at most? Justify your answer.
2001 Singapore MO Open
A positive integer $n$ is said to possess Property ($A$) if there exists a positive integer $N$ such that $N^2$ can be written as the sum of the squares of $n$ consecutive positive integers. Is it true that there are infinitely many positive integers which possess Property ($A$)? Justify your answer.

(As an example, the number $n = 2$ possesses Property ($A$) since $5^2 = 3^2 + 4^2$).
2000 Singapore MO Open
Triangle $ABC$ is inscribed in a circle with center $O$. Let $D$ and $E$ be points on the respective sides $AB$ and $AC$ so that $DE$ is perpendicular to $AO$. Show that the four points $B,D,E$ and $C$ lie on a circle.
2000 Singapore MO Open
Show that $240$ divides all numbers of the form $p^4 - q^4$, where p and q are prime numbers strictly greater than $5$. Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$, with $p$ and $q$ prime numbers strictly greater than $5$.
2000 Singapore MO Open
Is there a positive integer with at most four digits whose value is increased by exactly $60\%$ when the first digit is moved to the end of the number? For example, when the first digit of $1234$ is moved to the end of the number, the result is the integer $2341$.
2000 Singapore MO Open
In a party of $1000$ people, the number of people who have shaken hands with at most $962$ people is less than or equal to $37$. Show that one can find $27$ people in the party who have all shaken hands with each other.
1999 Singapore MO Open
Let $n$ be a positive integer. A square $ABCD$ is divided into $n^2$ identical small squares by drawing $(n-1)$ equally spaced lines parallel to the side $AB$ and another $(n- 1)$ equally spaced lines parallel to $BC$, thus giving rise to $(n+1)^2$ intersection points. The points $A, C$ are coloured red and the points $B, D$ are coloured blue. The rest of the intersection points are coloured either red or blue. Prove that the number of small squares having exactly $3$ vertices of the same colour is even.
1999 Singapore MO Open
Call a natural number $n$ a magic  number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer
1999 Singapore MO Open
For each positive integer $n$, let $f(n)$ be a positive integer. Show that if $f(n + 1) > f(f(n))$ for every positive integer n, then $f(x) = x$ for all positive integers $x$.
1999 Singapore MO Open
Let $ABCD$ be a quadrilateral with each interior angle less than $180^o$. Show that if $A, B, C, D$ do not lie on a circle, then $AB \cdot  CD + AD\cdot  BC > AC \cdot  BD$
1998 Singapore MO Open
In Fig. , $PA$ and $QB$ are tangents to the circle at $A$ and $B$ respectively. The line $AB$ is extended to meet $PQ$ at $S$. Suppose that $PA = QB$. Prove that $QS = SP$.
1998 Singapore MO Open
Let $N$ be the set of natural numbers, and let $f: N \to N$ be a function satisfying $f(x) + f(x + 2) < 2 f(x + 1)$ for any $x \in N$. Prove that there exists a straight line in the $xy$-plane which contains infinitely many points with coordinates $(n,f(n))$.
1998 Singapore MO Open
Do there exist integers $x$ and $y$ such that $19^{19} = x^3 +y^4$ ? Justify your answer.
1998 Singapore MO Open
Let $n$ be a fixed positive integer. Find all the positive integers $m$ such that

$$\frac{m^2+4m}{a_1}+\frac{m^2+8m}{a_1+a_2}+\frac{m^2+12m}{a_1+a_2+a_3}+...+\frac{m^2+4nm}{a_1+a_2+...+a_n}<2500 \left(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)$$for any positive numbers $a_1,a_2,...,a_n$. Justify your answer.
1997 Singapore MO Open
$\vartriangle ABC$ is an equilateral triangle. $L, M$ and $N$ are points on $BC, CA$ and $AB$ respectively. Prove that $MA  \cdot AN + NB  \cdot  BL + LC \cdot CM < BC^2$.
1997 Singapore MO Open
Observe that the number $4$ is such that $4 \choose k$ $= \frac{4!}{k!(4-k)!}$ divisible by $k + 1$ for $k = 0,1,2,3$. Find all the natural numbers $n$ between $50$ and $90$ such that  $n \choose k$ is divisible by $k + 1$ for $k = 0,1,2,..., n - 1$. Justify your answers.
1997 Singapore MO Open
Find all the natural numbers $N$ which satisfy the following properties:

(i) $N$ has exactly $6$ distinct factors $1, d_1, d_2, d_3, d_4, N$ and

(ii) $1 + N = 5(d_1 + d_2+d_3 + d_4)$.

Justify your answers.
1997 Singapore MO Open
Let $n \ge  2$ be a positive integer. Suppose that $a_1,a_2,...,a_n$ and  $b_1,b_2,...,b_n$ are 2n numbers such that $\sum_{i=1}^n a_i =\sum_{i=1}^n n_i= 1$ and $a_i\ge 0, 0 \le b_i\le \frac{n-1}{n}, i = 1, 2,..., n$. Show that

$$b_1a_2a_3...a_n+a_1b_2a_3...a_n+...+a_1a_2...a_{k-1}b_ka_{k+1}...a_n+ ...+ a_1a_2...a_{n-1}b_n \le \frac{1}{n(n-1)^{n-2}}$$
1996 Singapore MO Open
Three numbers are selected at random from the interval $[0,1]$. What is the probability that they form the lengths of the sides of a triangle?
1996 Singapore MO Open
In the following figure, $ABCD$ is a square of unit length and $P, Q$ are points on $AD$ and $AB$ respectively. Find $\angle PCQ$ if $|AP| + |AQ| + |PQ| = 2$.
1996 Singapore MO Open
Let $n$ be a positive integer. Prove that there is no positive integer solution to thxe equation $(x + 2)^n - x^n = 1 + 7^n$.
1996 Singapore MO Open
Determine all the solutions of the equation $x^3 + y^3 + z^3 = wx^2y^2z^2$ in natural numbers $x, y, z, w$. Justify your answer
1995 Singapore MO Open
Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer
1995 Singapore MO Open
Let $A_1A_2A_3$ be a triangle and $M$ an interior point. The straight lines $MA_1, MA_2, MA_3$ intersect the opposite sides at the points $B_1, B_2, B_3$ respectively (see Fig.). Show that if the areas of triangles $A_2B_1M, A_3B_2M$ and $A_1B_3M$ are equal, then $M$ coincides with the centroid of triangle $A_1A_2A_3$.
1995 Singapore MO Open
Let $P$ be a point inside $\vartriangle ABC$. Let $D, E, F$ be the feet of the perpendiculars from $P$ to the lines $BC, CA$ and $AB$, respectively (see Fig. ). Show that

(i) $EF = AP \sin A$,

(ii) $PA+ PB + PC \ge  2(PE+ PD+ PF)$
1995 Singapore MO Open
Let $a, b$ and $c$ be positive integers such that $1 < a < b < c$. Suppose that $(ab-l)(bc-1 )(ca-1)$ is divisible by $abc$. Find the values of $a, b$ and $c$. Justify your answer.
1995 Singapore MO Open
Let $a, b, c, d$ be four positive real numbers. Prove that

$$a^{10} + b^{10}+c^{10} + d^{10} \ge (0.1a + 0.2b + 0.3c + 0.4d)^{10} + (0.4a + 0.3b + 0.2c + 0.ld)^{10} + (0.2a + 0.4b + 0.1c + 0.3d)^{10} + (0.3a + 0.1b + 0.4c + 0.2d)^{10}$$
1997 Slovenia National Olympiad
Let $k$ be a positive integer. Prove that:

(a) If $k=m+2mn+n$ for some positive integers $m,n$, then $2k+1$ is composite.

(b) If $2k+1$ is composite, then there exist positive integers $m,n$ such that $k=m+2mn+n$.
1997 Slovenia National Olympiad
Let $a$ be an integer and $p$ a prime number that divides both $5a-1$ and $a-10$. Show that $p$ also divides $a-3$.
1997 Slovenia National Olympiad
Let $MN$ be a chord of a circle with diameter $AB$, and let $A'$ and $B'$ be the orthogonal projections of $A$ and $B$ onto $MN$. Prove that $MA'=B'N$.
1997 Slovenia National Olympiad
Janez wants to make an $m\times n$ grid (consisting of unit squares) using equal elements of the form $\llcorner$, where each leg of an element has the unit length. No two elements can overlap. For which values of $m$ and $n$ can Janez do the task?
1997 Slovenia National Olympiad
Prove that if real numbers $a,b,c,d$ satisfy $a^2+b^2+(a+b)^2=c^2+d^2+(c+d)^2$, then they also satisfy $a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4$.
1997 Slovenia National Olympiad
Points $M,N,P,Q$ are taken on the sides $AB,BC,CD,DA$ respectively of a square $ABCD$ such that $AM=BN=CP=DQ=\frac1nAB$. Find the ratio of the area of the square determined by the lines $MN,NP,PQ,QM$ to the ratio of $ABCD$.
1997 Slovenia National Olympiad
Let $C$ and $D$ be different points on the semicircle with diameter $AB$. The lines $AC$ and $BD$ intersect at $E$, and the lines $AD$ and $BC$ intersect at $F$. Prove that the midpoints $X,Y,Z$ of the segments $AB,CD,EF$ respectively are collinear.
1997 Slovenia National Olympiad
Prove that among any $1001$ numbers taken from the numbers $1,2,\ldots,1997$ there exist two with the difference $4$.
1997 Slovenia National Olympiad
Suppose that $m,n$ are integers greater than $1$ such that $m+n-1$ divides $m^2+n^2-1$. Prove that $m+n-1$ cannot be a prime number.
1997 Slovenia National Olympiad
Determine all positive integers $n$ for which there exists a polynomial $p(x)$ of degree $n$ with integer coefficients such that it takes the value $n$ in $n$ distinct integer points and takes the value $0$ at point $0$.
1997 Slovenia National Olympiad
In a convex quadrilateral $ABCD$ we have $\angle ADB=\angle ACD$ and $AC=CD=DB$. If the diagonals $AC$ and $BD$ intersect at $X$, prove that $\frac{CX}{BX}-\frac{AX}{DX}=1$.
1997 Slovenia National Olympiad
In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims:

(i) Less than $12$ employees have a more difficult work;

(ii) At least $30$ employees take a higher salary.

Assuming that an employee either always lies or always tells the truth, find how many employees are there in the enterprise.
1997 Slovenia National Olympiad
Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.
1997 Slovenia National Olympiad
The Fibonacci sequence $f_n$ is defined by $f_1=f_2=1$ and $f_{n+2}=f_{n+1}+f_n$ for $n\in\mathbb N$.



(a) Show that $f_{1005}$ is divisible by $10$.

(b) Show that $f_{1005}$ is not divisible by $100$.
1997 Slovenia National Olympiad
Two disjoint circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively lie on the same side of a line $p$ and touch the line at $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects $k_1$ at $B_1$ and $k_2$ at $B_2$. Prove that $A_1B_1\perp A_2B_2$.
1997 Slovenia National Olympiad
The expression $*3^5*3^4*3^3*3^2*3*1$ is given. Ana and Branka alternately change the signs $*$ to $+$ or $-$ (one time each turn). Can Branka, who plays second, do this so as to obtain an expression whose value is divisible by $7$?
1998 Slovenia National Olympiad
Find all integers $x,y$ which satisfy the equation $xy=20-3x+y$.
1998 Slovenia National Olympiad
A four-digit number has the property that the units digit equals the tens digit increased by $1$, the hundreds digit equals twice the tens digit, and the thousands digit is at least twice the units. Determine this four-digit number, knowing that it is twice a prime number.
1998 Slovenia National Olympiad
A point $E$ on side $CD$ of a rectangle $ABCD$ is such that $\triangle DBE$ is isosceles and $\triangle ABE$ is right-angled. Find the ratio between the side lengths of the rectangle.
1998 Slovenia National Olympiad
In the lower-left $3\times3$ square of an $8\times8$ chessboard there are nine pawns. Every pawn can jump horizontally or vertically over a neighboring pawn to the cell across it if that cell is free. Is it possible to arrange the nine pawns in the upperleft $3\times3$ square of the chessboard using finitely many such moves?
1998 Slovenia National Olympiad
Find all positive integers $n$ that are equal to the sum of digits of $n^2$.
1998 Slovenia National Olympiad
Find all pairs $(p,q)$ of real numbers such that $p+q=1998$ and the solutions of the equation $x^2+px+q=0$ are integers.
1998 Slovenia National Olympiad
A point $A$ is outside a circle $\mathcal K$ with center $O$. Line $AO$ intersects the circle at $B$ and $C$, and a tangent through $A$ touches the circle in $D$. Let $E$ be an arbitrary point on the line $BD$ such that $D$ lies between $B$ and $E$. The circumcircle of the triangle $DCE$ meets line $AO$ at $C$ and $F$ and line $AD$ at $D$ and $G$. Prove that the lines $BD$ and $FG$ are parallel.
1998 Slovenia National Olympiad
Two players play the following game starting with one pile of at least two stones. A player in turn chooses one of the piles and divides it into two or three nonempty piles. The player who cannot make a legal move loses the game. Which player has a winning strategy?
1998 Slovenia National Olympiad
Show that for any integter $a$, the number $\frac{a^5}5+\frac{a^3}3+\frac{7a}{15}$ is an integer.
1998 Slovenia National Olympiad
Find all polynomials $p$ with real coefficients such that for all real $x$

$$(x-8)p(2x)=8(x-1)p(x).$$
1998 Slovenia National Olympiad
A rectangle $ABCD$ with $AB>AD$ is given. The circle with center $B$ and radius $AB$ intersects the line $CD$ at $E$ and $F$.



(a) Prove that the circumcircle of triangle $EBF$ is tangent to the circle with diameter $AD$. Denote the tangency point by $G$.

(b) Prove that the points $D,G,$ and $B$ are collinear.
1998 Slovenia National Olympiad
Alf was attending an eight-year elementary school on Melmac. At the end of each school year, he showed the certificate to his father. If he was promoted, his father gave him the number of cats equal to Alf’s age times the number of the grade he passed. During elementary education, Alf failed one grade and had to repeat it. When he finished elementary education he found out that the total number of cats he had received was divisible by $1998$. Which grade did Alf fail?
1998 Slovenia National Olympiad
Let $n$ be a positive integer. If the number $1998$ is written in base $n$, a three-digit number with the sum of digits equal to $24$ is obtained. Find all possible values of $n$.
1998 Slovenia National Olympiad
find all functions $f(x)$ satisfying:

$(\forall x\in R) f(x)+xf(1-x)=x^2+1$
1998 Slovenia National Olympiad
In a right-angled triangle $ABC$ with the hypotenuse $BC$, $D$ is the foot of the altitude from $A$. The line through the incenters of the triangles $ABD$ and $ADC$ intersects the legs of $\triangle ABC$ at $E$ and $F$. Prove that $A$ is the circumcenter of triangle $DEF$.
1998 Slovenia National Olympiad
On every square of a chessboard, there are as many grains as shown on the picture. Starting from an arbitrary square, a knight starts a journey over the chessboard. After every move it eats up all the grains from the square it arrived to, but when it leaves, the same number of grains is put back on the square. After some time the knight returns to its initial square. Prove that the total number of grains the knight has eaten up during the journey is divisible by $3$.
1999 Slovenia National Olympiad
Two three-digit numbers are given. The hundreds digit of each of them is equal to the units digit of the other. Find these numbers if their difference is $297$ and the sum of digits of the smaller number is $23$.
1999 Slovenia National Olympiad
Find all integers $x,y$ such that $2x+3y=185$ and $xy>x+y$.