Navvye/Olympiad-Problems-CSV huggingface dataset:
1992 Dutch Mathematical Olympiad
Consider the configuration of six squares as shown on the picture. Prove that the sum of the area of the three outer squares ($ I,II$ and $ III$) equals three times the sum of the areas of the three inner squares ($ IV,V$ and $ VI$).
1992 Dutch Mathematical Olympiad
For every positive integer $ n$, we define $ n?$ as $ 1?=1$ and $ n?=\frac{n}{(n-1)?}$ for $ n \ge 2$.



Prove that $ \sqrt{1992}<1992?<\frac{4}{3} \sqrt{1992}.$
1992 Dutch Mathematical Olympiad
We consider regular $ n$-gons with a fixed circumference $ 4$. Let $ r_n$ and $ a_n$ respectively be the distances from the center of such an $ n$-gon to a vertex and to an edge.

$ (a)$ Determine $ a_4,r_4,a_8,r_8$.

$ (b)$ Give an appropriate interpretation for $ a_2$ and $ r_2$

$ (c)$ Prove that $ a_{2n}=\frac{1}{2} (a_n+r_n)$ and $ r_{2n}=\sqrt{a_2n r_n}.$

$ (d)$ Define $ u_0=0, u_1=1$ and $ u_n=\frac{1}{2}(u_{n-2}+u_{n-1})$ for $ n$ even or $ u_n=\sqrt{u_{n-2} u_{n-1}}$ for $ n$ odd. Determine $ \displaystyle\lim_{n\to\infty}u_n$.
1991 Dutch Mathematical Olympiad
Prove that for any three positive real numbers $ a,b,c, \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \ge \frac{9}{2} \cdot \frac{1}{a+b+c}$.
1991 Dutch Mathematical Olympiad
An angle with vertex $ A$ and measure $ \alpha$ and a point $ P_0$ on one of its rays are given so that $ AP_0=2$. Point $ P_1$ is chose on the other ray. The sequence of points $ P_1,P_2,P_3,...$ is defined so that $ P_n$ lies on the segment $ AP_{n-2}$ and the triangle $ P_n P_{n-1} P_{n-2}$ is isosceles with $ P_n P_{n-1}=P_n P_{n-2}$ for all $ n \ge 2$.

$ (a)$ Prove that for each value of $ \alpha$ there is a unique point $ P_1$ for which the sequence $ P_1,P_2,...,P_n,...$ does not terminate.

$ (b)$ Suppose that the sequence $ P_1,P_2,...$ does not terminate and that the length of the polygonal line $ P_0 P_1 P_2 ... P_k$ tends to $ 5$ when $ k \rightarrow \infty$. Compute the length of $ P_0 P_1$.
1991 Dutch Mathematical Olympiad
A real function $ f$ satisfies $ 4f(f(x))-2f(x)-3x=0$ for all real numbers $ x$. Prove that $ f(0)=0$.
1991 Dutch Mathematical Olympiad
Three real numbers $ a,b,c$ satisfy the equations $ a+b+c=3, a^2+b^2+c^2=9, a^3+b^3+c^3=24.$ Find $ a^4+b^4+c^4$.
1991 Dutch Mathematical Olympiad
Let $ H$ be the orthocenter, $ O$ the circumcenter, and $ R$ the circumradius of an acute-angled triangle $ ABC$. Consider the circles $ k_a,k_b,k_c,k_h,k$, all with radius $ R$, centered at $ A,B,C,H,M,$ respectively. Circles $ k_a$ and $ k_b$ meet at $ M$ and $ F$; $ k_a$ and $ k_c$ meet at $ M$ and $ E$; and $ k_b$ and $ k_c$ meet at $ M$ and $ D$.

$ (a)$ Prove that the points $ D,E,F$ lie on the circle $ k_h$.

$ (b)$ Prove that the set of the points inside $ k_h$ that are inside exactly one of the circles $ k_a,k_b,k_c$ has the area twice the area of $ \triangle ABC$.
1990 Dutch Mathematical Olympiad
Prove that for every integer $ n>1, 1 \cdot 3 \cdot 5 \cdot ... \cdot (2n-1)<n^n.$
1990 Dutch Mathematical Olympiad
Consider the sequence $ a_1=\frac{3}{2}, a_{n+1}=\frac{3a_n^2+4a_n-3}{4a_n^2}.$

$ (a)$ Prove that $ 1<a_n$ and $ a_{n+1}<a_n$ for all $ n$.

$ (b)$ From $ (a)$ it follows that $ \displaystyle\lim_{n\to\infty}a_n$ exists. Find this limit.

$ (c)$ Determine $ \displaystyle\lim_{n\to\infty}a_1a_2a_3...a_n$.
1990 Dutch Mathematical Olympiad
A polynomial $ f(x)=ax^4+bx^3+cx^2+dx$ with $ a,b,c,d>0$ is such that $ f(x)$ is an integer for $ x \in \{ -2,-1,0,1,2 \}$ and $ f(1)=1$ and $ f(5)=70$.

$ (a)$ Show that $ a=\frac{1}{24}, b=\frac{1}{4},c=\frac{11}{24},d=\frac{1}{4}$.

$ (b)$ Prove that $ f(x)$ is an integer for all $ x \in \mathbb{Z}$.
1990 Dutch Mathematical Olympiad
If $ ABCDEFG$ is a regular $ 7$-gon with side $ 1$, show that: $ \frac{1}{AC}+\frac{1}{AD}=1$.
1985 Dutch Mathematical Olympiad
For some $ p$, the equation $ x^3 + px^2 + 3x - 10 = 0$ has three real solutions $ a,b,c$ such that $ c - b = b - a > 0$. Determine $ a,b,c,$ and $ p$.
1985 Dutch Mathematical Olympiad
Among the numbers $ 11n + 10^{10}$, where $ 1 \le n \le 10^{10}$ is an integer, how many are squares?
1985 Dutch Mathematical Olympiad
In a factory, square tables of $ 40 \times 40$ are tiled with four tiles of size $ 20 \times 20$. All tiles are the same and decorated in the same way with an asymmetric pattern such as the letter $ J$. How many different types of tables can be produced in this way?
1985 Dutch Mathematical Olympiad
A convex hexagon $ ABCDEF$ is such that each of the diagonals $ AD,BE,CF$ divides the hexagon into two parts of equal area. Prove that these three diagonals are concurrent.
1983 Dutch Mathematical Olympiad
A triangle $ ABC$ can be divided into two isosceles triangles by a line through $ A$. Given that one of the angles of the triangles is $ 30^{\circ}$, find all possible values of the other two angles.
1983 Dutch Mathematical Olympiad
Prove that if $ n$ is an odd positive integer, then the last two digits of $ 2^{2n}(2^{2n+1}-1)$ in base $ 10$ are $ 28$.
1983 Dutch Mathematical Olympiad
Suppose that $ a,b,c,p$ are real numbers with $ a,b,c$ not all equal, such that: $ a+\frac{1}{b}=b+\frac{1}{c}=c+\frac{1}{a}=p.$ Determine all possible values of $ p$ and prove that $ abc+p=0$.
1983 Dutch Mathematical Olympiad
Within an equilateral triangle of side $ 15$ are $ 111$ points. Prove that it is always possible to cover three of these points by a round coin of diameter $ \sqrt{3}$, part of which may lie outside the triangle.
1982 Dutch Mathematical Olympiad
Which is greater: $ 17091982!^2$ or $ 17091982^{17091982}$?
1982 Dutch Mathematical Olympiad
In a triangle $ ABC$, $ M$ is the midpoint of $ AB$ and $ P$ an arbitrary point on side $ AC$. Using only a straight edge, construct point $ Q$ on $ BC$ such that $ P$ and $ Q$ are at equal distance from $ CM$.
1982 Dutch Mathematical Olympiad
Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?
1982 Dutch Mathematical Olympiad
Determine $ \gcd (n^2+2,n^3+1)$ for $ n=9^{753}$.
1983 Federal Competition For Advanced Students, P2
For every natural number $ x$, let $ Q(x)$ be the sum and $ P(x)$ the product of the (decimal) digits of $ x$. Show that for each $ n \in \mathbb{N}$ there exist infinitely many values of $ x$ such that:



$ Q(Q(x))+P(Q(x))+Q(P(x))+P(P(x))=n$.
1983 Federal Competition For Advanced Students, P2
Let $ x_1,x_2,x_3$ be the roots of: $ x^3-6x^2+ax+a=0$. Find all real numbers $ a$ for which $ (x_1-1)^3+(x_2-1)^3+(x_3-1)^3=0$. Also, for each such $ a$, determine the corresponding values of $ x_1,x_2,$ and $ x_3$.
1983 Federal Competition For Advanced Students, P2
Let $ P$ be a point in the plane of a triangle $ ABC$. Lines $ AP,BP,CP$ respectively meet lines $ BC,CA,AB$ at points $ A',B',C'$. Points $ A'',B'',C''$ are symmetric to $ A,B,C$ with respect to $ A',B',C',$ respectively. Show that: $ S_{A''B''C''}=3S_{ABC}+4S_{A'B'C'}$.
1983 Federal Competition For Advanced Students, P2
The sequence $ (x_n)_{n \in \mathbb{N}}$ is defined by $ x_1=2, x_2=3,$ and



$ x_{2m+1}=x_{2m}+x_{2m-1}$ for $ m \ge 1;$

$ x_{2m}=x_{2m-1}+2x_{2m-2}$ for $ m \ge 2.$



Determine $ x_n$ as a function of $ n$.
1983 Federal Competition For Advanced Students, P2
Given positive integers $ a,b,$ find all positive integers $ x,y$ satisfying the equation: $ x^{a+b}+y=x^a y^b$.
1983 Federal Competition For Advanced Students, P2
Planes $ \pi _1$ and $ \pi _2$ in Euclidean space $ \mathbb{R} ^3$ partition $ S=\mathbb{R} ^3 \setminus (\pi _1 \cup \pi _2)$ into several components. Show that for any cube in $ \mathbb{R} ^3$, at least one of the components of $ S$ meets at least three faces of the cube.
1985 Federal Competition For Advanced Students, P2
Determine all quadruples $ (a,b,c,d)$ of nonnegative integers satisfying:



$ a^2+b^2+c^2+d^2=a^2 b^2 c^2$.
1985 Federal Competition For Advanced Students, P2
For $ n \in \mathbb{N}$, let $ f(n)=1^n+2^{n-1}+3^{n-2}+...+n^1$. Determine the minimum value of: $ \frac{f(n+1)}{f(n)}.$
1985 Federal Competition For Advanced Students, P2
A line meets the lines containing sides $ BC,CA,AB$ of a triangle $ ABC$ at $ A_1,B_1,C_1,$ respectively. Points $ A_2,B_2,C_2$ are symmetric to $ A_1,B_1,C_1$ with respect to the midpoints of $ BC,CA,AB,$ respectively. Prove that $ A_2,B_2,$ and $ C_2$ are collinear.
1985 Federal Competition For Advanced Students, P2
Find all natural numbers $ n$ such that the equation:



$ a_{n+1} x^2-2x \sqrt{a_1^2+a_2^2+...+a_{n+1}^2}+a_1+a_2+...+a_n=0$



has real solutions for all real numbers $ a_1,a_2,...,a_{n+1}$.
1985 Federal Competition For Advanced Students, P2
A sequence $ (a_n)$ of positive integers satisfies: $ a_n=\sqrt{\frac{a_{n-1}^2+a_{n+1}^2}{2}}$ for all $ n \ge 1$. Prove that this sequence is constant.
1985 Federal Competition For Advanced Students, P2
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying: $ x^2 f(x)+f(1-x)=2x-x^4$ for all $ x \in \mathbb{R}$.
1986 Federal Competition For Advanced Students, P2
Show that a square can be inscribed in any regular polygon.
1986 Federal Competition For Advanced Students, P2
For $ s,t \in \mathbb{N}$, consider the set $ M=\{ (x,y) \in \mathbb{N} ^2 | 1 \le x \le s, 1 \le y \le t \}$. Find the number of rhombi with the vertices in $ M$ and the diagonals parallel to the coordinate axes.
1986 Federal Competition For Advanced Students, P2
Find all possible values of $ x_0$ and $ x_1$ such that the sequence defined by:



$ x_{n+1}=\frac{x_{n-1} x_n}{3x_{n-1}-2x_n}$ for $ n \ge 1$



contains infinitely many natural numbers.
1986 Federal Competition For Advanced Students, P2
Find the largest $ n$ for which there is a natural number $ N$ with $ n$ decimal digits which are all different such that $ n!$ divides $ N$. Furthermore, for this largest $ n$ find all possible numbers $ N$.
1986 Federal Competition For Advanced Students, P2
Show that for every convex $ n$-gon $ ( n \ge 4)$, the arithmetic mean of the lengths of its sides is less than the arithmetic mean of the lengths of all its diagonals.
1986 Federal Competition For Advanced Students, P2
Given a positive integer $ n$, find all functions $ F: \mathbb{N} \rightarrow \mathbb{R}$ such that $ F(x+y)=F(xy-n)$ whenever $ x,y \in \mathbb{N}$ satisfy $ xy>n$.
1987 Federal Competition For Advanced Students, P2
The sides $ a,b$ and the bisector of the included angle $ \gamma$ of a triangle are given. Determine necessary and sufficient conditions for such triangles to be constructible and show how to reconstruct the triangle.
1987 Federal Competition For Advanced Students, P2
Find the number of all sequences $ (x_1,...,x_n)$ of letters $ a,b,c$ that satisfy $ x_1=x_n=a$ and $ x_i \not= x_{i+1}$ for $ 1 \le i \le n-1.$
1987 Federal Competition For Advanced Students, P2
Let $ x_1,...,x_n$ be positive real numbers. Prove that:



$ \displaystyle\sum_{k=1}^{n}x_k+\sqrt{\displaystyle\sum_{k=1}^{n}x_k^2} \le \frac{n+\sqrt{n}}{n^2} \left( \displaystyle\sum_{k=1}^{n} \frac{1}{x_k} \right) \left( \displaystyle\sum_{k=1}^{n} x_k^2 \right).$
1987 Federal Competition For Advanced Students, P2
Find all triples $ (x,y,z)$ of natural numbers satisfying $ 2xz=y^2$ and $ x+z=1987$.
1987 Federal Competition For Advanced Students, P2
Let $ P$ be a point in the interior of a convex $ n$-gon $ A_1 A_2 ... A_n$ $ (n \ge 3)$. Show that among the angles $ \beta _{ij}=\angle A_i P A_j,1 \le i \le n$, there are at least $ n-1$ angles satisfying $ 90^{\circ} \le \beta_{ij} \le 180^{\circ}$.
1987 Federal Competition For Advanced Students, P2
Determine all polynomials $ P_n(x)=x^n+a_1 x^{n-1}+...+a_{n-1} x+a_n$ with integer coefficients whose $ n$ zeros are precisely the numbers $ a_1,...,a_n$ (counted with their respective multiplicities).
1988 Federal Competition For Advanced Students, P2
If $ a_1,...,a_{1988}$ are positive numbers whose arithmetic mean is $ 1988$, show that:



$ \sqrt[1988]{\displaystyle\prod_{i,j=1}^{1988} \left( 1+\frac{a_i}{a_j} \right)} \ge 2^{1988}$



and determine when equality holds.
1988 Federal Competition For Advanced Students, P2
An equilateral triangle $ A_1 A_2 A_3$ is divided into four smaller equilateral triangles by joining the midpoints $ A_4,A_5,A_6$ of its sides. Let $ A_7,...,A_{15}$ be the midpoints of the sides of these smaller triangles. The $ 15$ points $ A_1,...,A_{15}$ are colored either green or blue. Show that with any such colouring there are always three mutually equidistant points $ A_i,A_j,A_k$ having the same color.
1988 Federal Competition For Advanced Students, P2
Show that there is precisely one sequence $ a_1,a_2,...$ of integers which satisfies $ a_1=1, a_2>1,$ and $ a_{n+1}^3+1=a_n a_{n+2}$ for $ n \ge 1$.
1988 Federal Competition For Advanced Students, P2
Let $ a_{ij}$ be nonnegative integers such that $ a_{ij}=0$ if and only if $ i>j$ and that $ \displaystyle\sum_{j=1}^{1988}a_{ij}=1988$ holds for all $ i=1,...,1988$. Find all real solutions of the system of equations:



$ \displaystyle\sum_{j=1}^{1988} (1+a_{ij})x_j=i+1, 1 \le i \le 1988$.
1988 Federal Competition For Advanced Students, P2
The bisectors of angles $ B$ and $ C$ of triangle $ ABC$ intersect the opposite sides in points $ B'$ and $ C'$ respectively. Show that the line $ B'C'$ intersects the incircle of the triangle.
1988 Federal Competition For Advanced Students, P2
Determine all monic polynomials $ p(x)$ of fifth degree having real coefficients and the following property: Whenever $ a$ is a (real or complex) root of $ p(x)$, then so are $ \frac{1}{a}$ and $ 1-a$.
1989 Federal Competition For Advanced Students, P2
Consider the set $ S_n$ of all the $ 2^n$ numbers of the type $ 2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},$ where number $ 2$ appears $ n+1$ times.

$ (a)$ Show that all members of $ S_n$ are real.

$ (b)$ Find the product $ P_n$ of the elements of $ S_n$.
1989 Federal Competition For Advanced Students, P2
Find all triples $ (a,b,c)$ of integers with $ abc=1989$ and $ a+b-c=89$.
1989 Federal Competition For Advanced Students, P2
Show that it is possible to situate eight parallel planes at equal distances such that each plane contains precisely one vertex of a given cube. How many such configurations of planes are there?
1989 Federal Competition For Advanced Students, P2
We are given a circle $ k$ and nonparallel tangents $ t_1,t_2$ at points $ P_1,P_2$ on $ k$, respectively. Lines $ t_1$ and $ t_2$ meet at $ A_0$. For a point $ A_3$ on the smaller arc $ P_1 P_2,$ the tangent $ t_3$ to $ k$ at $ P_3$ meets $ t_1$ at $ A_1$ and $ t_2$ at $ A_2$. How must $ P_3$ be chosen so that the triangle $ A_0 A_1 A_2$ has maximum area?
1989 Federal Competition For Advanced Students, P2
Find all real solutions of the system:



$ x^2+2yz=x,$

$ y^2+2zx=y,$

$ z^2+2xy=z.$
1989 Federal Competition For Advanced Students, P2
Determine all functions $ f: \mathbb{N}_0 \rightarrow \mathbb{N}_0$ such that $ f(f(n))+f(n)=2n+6$ for all $ n \in \mathbb{N}_0$.
1990 Federal Competition For Advanced Students, P2
Determine the number of integers $ n$ with $ 1 \le n \le N=1990^{1990}$ such that $ n^2-1$ and $ N$ are coprime.
1990 Federal Competition For Advanced Students, P2
Show that for all integers $ n \ge 2$, $ \sqrt { 2\sqrt[3]{3 \sqrt[4]{4...\sqrt[n]{n}}}}<2$
1990 Federal Competition For Advanced Students, P2
In a convex quadrilateral $ ABCD$, let $ E$ be the intersection point of the diagonals, and let $ F_1,F_2,$ and $ F$ be the areas of $ ABE,CDE,$ and $ ABCD,$ respectively. Prove that:



$ \sqrt {F_1}+\sqrt {F_2} \le \sqrt {F}.$
1990 Federal Competition For Advanced Students, P2
For each nonzero integer $ n$ find all functions $ f: \mathbb{R} - \{-3,0 \} \rightarrow \mathbb{R}$ satisfying:



$ f(x+3)+f \left( -\frac{9}{x} \right)=\frac{(1-n)(x^2+3x-9)}{9n(x+3)}+\frac{2}{n}$ for all $ x \not= 0,-3.$



Furthermore, for each fixed $ n$ find all integers $ x$ for which $ f(x)$ is an integer.
1990 Federal Competition For Advanced Students, P2
Determine all rational numbers $ r$ such that all solutions of the equation:



$ rx^2+(r+1)x+(r-1)=0$ are integers.
1990 Federal Competition For Advanced Students, P2
A convex pentagon $ ABCDE$ is inscribed in a circle. The distances of $ A$ from the lines $ BC,CD,DE$ are $ a,b,c,$ respectively. Compute the distance of $ A$ from the line $ BE$.
1991 Federal Competition For Advanced Students, P2
Consider a convex solid $ K$ in space and two parallel planes $ \epsilon _1$ and $ \epsilon _2$ on the distance $ 1$ tangent to $ K$. A plane $ \epsilon$ between $ \epsilon _1$ and $ \epsilon _2$ is on the distance $ d_1$ from $ \epsilon _1$. Find all $ d_1$ such that the part of $ K$ between $ \epsilon _1$ and $ \epsilon$ always has a volume not exceeding half the volume of $ K$.
1991 Federal Competition For Advanced Students, P2
Find all functions $ f: \mathbb{Z} - \{ 0 \} \rightarrow \mathbb{Q}$ satisfying:



$ f \left( \frac{x+y}{3} \right)=\frac {f(x)+f(y)}{2},$ whenever $ x,y,\frac{x+y}{3} \in \mathbb{Z} - \{ 0 \}.$
1991 Federal Competition For Advanced Students, P2
$ (a)$ Prove that $ 91$ divides $ n^{37}-n$ for all integers $ n$.

$ (b)$ Find the largest $ k$ that divides $ n^{37}-n$ for all integers $ n$.
1991 Federal Competition For Advanced Students, P2
The sequence $ (a_n)$ is given by $ a_1=1,a_2=0$ and:



$ a_{2k+1}=a_k+a_{k+1}, a_{2k+2}=2a_{k+1}$ for $ k \in \mathbb{N}.$



Find $ a_m$ for $ m=2^{19}+91.$
1991 Federal Competition For Advanced Students, P2
For all positive integers $ n$ prove the inequality:



$ \left( \frac{1+(n+1)^{n+1}}{n+2} \right)^{n-1}>\left( \frac{1+n^n}{n+1} \right)^n.$
1991 Federal Competition For Advanced Students, P2
Find the number of ten-digit natural numbers (which do not start with zero) containing no block $ 1991$.
1997 Federal Competition For Advanced Students, P2
Let $ a$ be a fixed integer. Find all integer solutions $ x,y,z$ of the system:



$ 5x+(a+2)y+(a+2)z=a,$

$ (2a+4)x+(a^2+3)y+(2a+2)z=3a-1,$

$ (2a+4)x+(2a+2)y+(a^2+3)z=a+1.$
1997 Federal Competition For Advanced Students, P2
A positive integer $ K$ is given. Define the sequence $ (a_n)$ by $ a_1=1$ and $ a_n$ is the $ n$-th natural number greater than $ a_{n-1}$ which is congruent to $ n$ modulo $ K$.

$ (a)$ Find an explicit formula for $ a_n$.

$ (b)$ What is the result if $ K=2?$
1997 Federal Competition For Advanced Students, P2
Let be given a triangle $ ABC$. Points $ P$ on side $ AC$ and $ Y$ on the production of $ CB$ beyond $ B$ are chosen so that $ Y$ subtends equal angles with $ AP$ and $ PC$. Similarly, $ Q$ on side $ BC$ and $ X$ on the production of $ AC$ beyond $ C$ are such that $ X$ subtends equal angles with $ BQ$ and $ QC$. Lines $ YP$ and $ XB$ meet at $ R$, $ XQ$ and $ YA$ meet at $ S$, and $ XB$ and $ YA$ meet at $ D$. Prove that $ PQRS$ is a parallelogram if and only if $ ACBD$ is a cyclic quadrilateral.
1997 Federal Competition For Advanced Students, P2
Determine all quadruples $ (a,b,c,d)$ of real numbers satisfying the equation:



$ 256a^3 b^3 c^3 d^3=(a^6+b^2+c^2+d^2)(a^2+b^6+c^2+d^2)(a^2+b^2+c^6+d^2)(a^2+b^2+c^2+d^6).$
1997 Federal Competition For Advanced Students, P2
We define the following operation which will be applied to a row of bars being situated side-by-side on positions $ 1,2,...,N$. Each bar situated at an odd numbered position is left as is, while each bar at an even numbered position is replaced by two bars. After that, all bars will be put side-by-side in such a way that all bars form a new row and are situated on positions $ 1,...,M.$

From an initial number $ a_0>0$ of bars there originates a sequence $ (a_n)_{n \ge 0},$ where $ a_n$ is the number of bars after having applied the operation $ n$ times.

$ (a)$ Prove that for no $ n>0$ can we have $ a_n=1997.$

$ (b)$ Determine all natural numbers that can only occur as $ a_0$ or $ a_1$.
1997 Federal Competition For Advanced Students, P2
For every natural number $ n$, find all polynomials $ x^2+ax+b$, where $ a^2 \ge 4b$, that divide $ x^{2n}+ax^n+b$.
1997 Federal Competition For Advanced Students, Part 2
Let $a$ be a fixed integer. Find all integer solutions $x, y, z$ of the system

$$5x + (a + 2)y + (a + 2)z = a,$$$$(2a + 4)x + (a^2 + 3)y + (2a + 2)z = 3a - 1,$$$$(2a + 4)x + (2a + 2)y + (a^2 + 3)z = a + 1.$$
1997 Federal Competition For Advanced Students, Part 2
A positive integer $K$ is given. Define the sequence $(a_n)$ by $a_1 = 1$ and $a_n$ is the $n$-th positive integer greater than $a_{n-1}$ which is congruent to $n$ modulo $K$.



(a) Find an explicit formula for $a_n$.



(b) What is the result if $K = 2$?
1997 Federal Competition For Advanced Students, Part 2
Let be given a triangle $ABC$. Points $P$ on side $AC$ and $Y$ on the production of $CB$ beyond $B$ are chosen so that $Y$ subtends equal angles with $AP$ and $PC$. Similarly, $Q$ on side $BC$ and $X$ on the production of $AC$ beyond $C$ are such that $X$ subtends equal angles with $BQ$ and $QC$. Lines $YP$ and $XB$ meet at $R$, $XQ$ and $YA$ meet at $S$, and $XB$ and $YA$ meet at $D$. Prove that $PQRS$ is a parallelogram if and only if $ACBD$ is a cyclic quadrilateral.
1997 Federal Competition For Advanced Students, Part 2
Determine all quadruples $(a, b, c, d)$ of real numbers satisfying the equation

$$256a^3b^3c^3d^3 = (a^6+b^2+c^2+d^2)(a^2+b^6+c^2+d^2)(a^2+b^2+c^6+d^2)(a^2+b^2+c^2+d^6).$$
1997 Federal Competition For Advanced Students, Part 2
We define the following operation which will be applied to a row of bars being situated side-by-side on positions $1, 2, \ldots ,N$. Each bar situated at an odd numbered position is left as is, while each bar at an even numbered position is replaced by two bars. After that, all bars will be put side-by- side in such a way that all bars form a new row and are situated on positions $1, \ldots,M$. From an initial number $a_0 > 0$ of bars there originates a sequence $(a_n)_{n \geq 0}$, where an is the number of bars after having applied the operation $n$ times.



(a) Prove that for no $n > 0$ can we have $a_n = 1997$.



(b) Determine all natural numbers that can only occur as $a_0$ or $a_1$.
1997 Federal Competition For Advanced Students, Part 2
For every natural number $n$, find all polynomials $x^2+ax+b$, where $a^2 \geq 4b$, that divide $x^{2n} + ax^n + b$.
1998 Federal Competition For Advanced Students, Part 2
Let $a \geq 0$ be a natural number. Determine all rational $x$, so that

$$\sqrt{1+(a-1)\sqrt[3]x}=\sqrt{1+(a-1)\sqrt x}$$

All occurring square roots, are not negative.



Note. It seems the set of natural numbers = $\mathbb N = \{0,1,2,\ldots\}$ in this problem.
1998 Federal Competition For Advanced Students, Part 2
Let $Q_n$ be the product of the squares of even numbers less than or equal to $n$ and $K_n$ equal to the product of cubes of odd numbers less than or equal to $n$. What is the highest power of $98$, that a)$Q_n$, b) $K_n$ or c) $Q_nK_n$ divides? If one divides $Q_{98}K_{98}$ by the highest power of $98$, then one get a number $N$. By which power-of-two number is $N$ still divisible?
1998 Federal Competition For Advanced Students, Part 2
Let $a_n$ be a sequence recursively defined by $a_0 = 0, a_1 = 1$ and $a_{n+2} = a_{n+1} + a_n$. Calculate the sum of $a_n\left( \frac 25\right)^n$ for all positive integers $n$. For what value of the base $b$ we get the sum $1$?
1998 Federal Competition For Advanced Students, Part 2
Let $M$ be the set of the vertices of a regular hexagon, our Olympiad symbol. How many chains $\emptyset \subset A \subset B \subset C \subset D \subset M$ of six different set, beginning with the empty set and ending with the $M$, are there?
1998 Federal Competition For Advanced Students, Part 2
Let $P(x) = x^3 - px^2 + qx - r$ be a cubic polynomial with integer roots $a, b, c$.



(a) Show that the greatest common divisor of $p, q, r$ is equal to $1$ if the greatest common divisor of $a, b, c$ is equal to $1$.



(b) What are the roots of polynomial $Q(x) = x^3-98x^2+98sx-98t$ with $s, t$ positive integers.
1998 Federal Competition For Advanced Students, Part 2
In a parallelogram $ABCD$ with the side ratio $AB : BC = 2 : \sqrt 3$ the normal through $D$ to $AC$ and the normal through $C$ to $AB$ intersects in the point $E$ on the line $AB$. What is the relationship between the lengths of the diagonals $AC$ and $BD$?
1999 Federal Competition For Advanced Students, Part 2
Prove that for each positive integer $n$, the sum of the numbers of digits of $4^n$ and of $25^n$ (in the decimal system) is odd.