Navvye/Olympiad-Problems-CSV huggingface dataset:
2007 Czech and Slovak Olympiad III A
In an acute-angled triangle $ABC$ ($AC\ne BC$), let $D$ and $E$ be points on $BC$ and $AC$, respectively, such that the points $A,B,D,E$ are concyclic and $AD$ intersects $BE$ at $P$. Knowing that $CP\bot AB$, prove that $P$ is the orthocenter of triangle $ABC$.
2007 Czech and Slovak Olympiad III A
Find all pariwise distinct real numbers $x,y,z$ such that $\left\{\frac{x-y}{y-z},\frac{y-z}{z-x},\frac{z-x}{x-y} \right\} = \{x,y,z\}$. (It means, those three fractions make a permutation of $x, y$, and $z$.)
2006 Czech and Slovak Olympiad III A
Define a sequence of positive integers $\{a_n\}$ through the recursive formula:

$a_{n+1}=a_n+b_n(n\ge 1)$,where $b_n$ is obtained by rearranging the digits of $a_n$ (in decimal representation) in reverse order (for example,if $a_1=250$,then $b_1=52,a_2=302$,and so on). Can $a_7$ be a prime?
2006 Czech and Slovak Olympiad III A
Let $m,n$ be positive integers such that the equation (in respect of $x$)

$$(x+m)(x+n)=x+m+n$$

has at least one integer root. Prove that $\frac{1}{2}n<m<2n$.
2006 Czech and Slovak Olympiad III A
In a scalene triangle $ABC$,the bisectors of angle $A,B$ intersect their corresponding sides at $K,L$ respectively.$I,O,H$ denote respectively the incenter,circumcenter and orthocenter of triangle $ABC$. Prove that $A,B,K,L,O$ are concyclic iff $KL$ is the common tangent line of the circumcircles of the three triangles $ALI,BHI$ and $BKI$.
2006 Czech and Slovak Olympiad III A
Given a segment $AB$ in the plane. Let $C$ be another point in the same plane,$H,I,G$ denote the orthocenter,incenter and centroid of triangle $ABC$. Find the locus of $M$ for which $A,B,H,I$ are concyclic.
2006 Czech and Slovak Olympiad III A
Find all triples $(p,q,r)$ of pairwise distinct primes such that

$$p\mid q+r, q\mid r+2p, r\mid p+3q.$$
2006 Czech and Slovak Olympiad III A
Find all real solutions $(x,y,z)$ of the system of equations:

$$

\begin{cases}

 \tan ^2x+2\cot^22y=1 \\ 

 \tan^2y+2\cot^22z=1  \\
  \tan^2z+2\cot^22x=1  \\
\end{cases}

$$
2005 Czech And Slovak Olympiad III A
Consider all arithmetical sequences of real numbers $(x_i)^{\infty}=1$ and $(y_i)^{\infty} =1$ with the common first term, such that for some $k > 1, x_{k-1}y_{k-1} = 42, x_ky_k = 30$, and $x_{k+1}y_{k+1} = 16$. Find all such pairs of sequences with the maximum possible $k$.
2005 Czech And Slovak Olympiad III A
Determine for which $m$ there exist exactly $2^{15}$ subsets $X$ of $\{1,2,...,47\}$ with the following property: $m$ is the smallest element of $X$, and for every $x \in  X$, either $x+m \in X$ or $x+m > 47$.
2005 Czech And Slovak Olympiad III A
In a trapezoid $ABCD$ with $AB // CD, E$ is the midpoint of $BC$. Prove that if the quadrilaterals $ABED$ and $AECD$ are tangent, then the sides $a = AB, b = BC, c =CD, d = DA$ of the trapezoid satisfy the equalities $a+c = \frac{b}{3} +d$ and $\frac1a +\frac1c = \frac3b$ .
2005 Czech And Slovak Olympiad III A
An acute-angled triangle $AKL$ is given on a plane. Consider all rectangles $ABCD$ circumscribed to triangle $AKL$ such that point $K$ lies on side $BC$ and point $L$ lieson side $CD$. Find the locus of the intersection $S$ of the diagonals $AC$ and $BD$.
2005 Czech And Slovak Olympiad III A
Let $p,q, r, s$ be real numbers with $q \ne -1$ and $s \ne -1$. Prove that the quadratic equations $x^2 + px+q = 0$ and $x^2 +rx+s = 0$ have a common root, while their other roots are inverse of each other, if and only if $pr = (q+1)(s+1)$ and $p(q+1)s = r(s+1)q$.

(A double root is counted twice.)
2005 Czech And Slovak Olympiad III A
Decide whether for every arrangement of the numbers $1,2,3, . . . ,15$ in a sequence one can color these numbers with at most four different colors in such a way that the numbers of each color form a monotone subsequence.
2004 Czech and Slovak Olympiad III A
Find all triples $(x,y,z)$ of real numbers such that

$$x^2+y^2+z^2\le 6+\min (x^2-\frac{8}{x^4},y^2-\frac{8}{y^4},z^2-\frac{8}{z^4}).$$
2004 Czech and Slovak Olympiad III A
Consider all words containing only letters $A$ and $B$. For any positive integer $n$, $p(n)$ denotes the number of all $n$-letter words without four consecutive $A$'s or three consecutive $B$'s. Find the value of the expression

$$\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.$$
2004 Czech and Slovak Olympiad III A
Given a circle $S$ and its $121$ chords $P_i (i=1,2,\ldots,121)$, each with a point $A_i(i=1,2,\ldots,121)$ on it. Prove that there exists a point $X$ on the circumference of $S$ such that: there exist $29$ distinct indices $1\le k_1\le k_2\le\ldots\le k_{29}\le 121$, such that the angle formed by ${A_{k_j}}X$ and ${P_{k_j}}$ is smaller than $21$ degrees for every $j=1,2,\ldots,29$.
2004 Czech and Slovak Olympiad III A
Find all positive integers $n$ such that $\sum_{k=1}^{n}\frac{n}{k!}$ is an integer.
2004 Czech and Slovak Olympiad III A
Let $L$ be an arbitrary point on the minor arc $CD$ of the circumcircle of square $ABCD$. Let $K,M,N$ be the intersection points of $AL,CD$; $CL,AD$; and $MK,BC$ respectively. Prove that $B,M,L,N$ are concyclic.
2004 Czech and Slovak Olympiad III A
Find all functions $f:\mathbb R^+ \rightarrow \mathbb R^+$ such that for all positive real numbers $x,y$,

$$x^2[f(x)+f(y)]=(x+y)f(yf(x)).$$
2003 Czech And Slovak Olympiad III A
Solve the following system in the set of real numbers:

$x^2 -xy+y^2 = 7$,

$x^2y+xy^2 = -2$.
2003 Czech And Slovak Olympiad III A
On sides $BC,CA,AB$ of a triangle $ABC$ points $D,E,F$ respectively are chosen so that $AD,BE,CF$ have a common point, say $G$. Suppose that one can inscribe circles in the quadrilaterals $AFGE,BDGF,CEGD$ so that each two of them have a common point. Prove that triangle $ABC$ is equilateral.
2003 Czech And Slovak Olympiad III A
A sequence $(x_n)_{n= 1}^{\infty}$ satisfies $x_1 = 1$ and for each $n > 1, x_n = \pm (n-1)x_{n-1} \pm  (n-2)x_{n-2} \pm  ... \pm 2x_2  \pm x_1$. Prove that the signs ” $\pm$” can be chosen so that $x_n \ne 12$ holds only for finitely many $n$.
2003 Czech And Slovak Olympiad III A
Let be given an obtuse angle $AKS$ in the plane. Construct a triangle $ABC$ such that $S$ is the midpoint of $BC$ and $K$ is the intersection point of $BC$ with the bisector of  $\angle BAC$.
2003 Czech And Slovak Olympiad III A
Show that, for each integer $z \ge 3$, there exist two two-digit numbers $A$ and $B$ in base $z$, one equal to the other one read in reverse order, such that the equation $x^2 -Ax+B$ has one double root. Prove that this pair is unique for a given $z$. For instance, in base $10$ these numbers are $A = 18, B = 81$.
2003 Czech And Slovak Olympiad III A
a，b，c>0，abc=1，prove  that(a/b)+(b/c)+(c/a)≥a+b+c.
2002 Czech and Slovak Olympiad III A
Solve the system

$$(4x)_5+7y=14 \\ (2y)_5 -(3x)_7=74$$

in the domain of integers, where $(n)_k$ stands for the multiple of the number $k$ closest to the number $n$.
2002 Czech and Slovak Olympiad III A
Consider an arbitrary equilateral triangle $KLM$, whose vertices $K, L$ and $M$ lie on the sides $AB, BC$ and $CD$, respectively, of a given square $ABCD$. Find the locus of the midpoints of the sides $KL$ of all such triangles $KLM$.
2002 Czech and Slovak Olympiad III A
Show that a given natural number $A$ is the square of a natural number if and only if for any natural number $n$, at least one of the differences

$$(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A, \cdots , (A + n)^2 - A$$

is divisible by $n$.
2002 Czech and Slovak Olympiad III A
Find all pairs of real numbers $a, b$ for which the equation in the domain of the real numbers

$$\frac{ax^2-24x+b}{x^2-1}=x$$

has two solutions and the sum of them equals $12$.
2002 Czech and Slovak Olympiad III A
A triangle $KLM$ is given in the plane together with a point $A$ lying on the half-line opposite to $KL$. Construct a rectangle $ABCD$ whose vertices $B, C$ and $D$ lie on the lines $KM, KL$ and $LM$, respectively. (We allow the rectangle to be a square.)
2002 Czech and Slovak Olympiad III A
Let $\mathbb{R}^{+}$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^{+} \to \mathbb{R}^{+}$ satisfying for all $x, y \in \mathbb{R}^{+}$ the equality

$$f(xf(y))=f(xy)+x$$
2001 Czech And Slovak Olympiad IIIA
Determine all polynomials $P$ such that for every real number $x$, $P(x)^2 +P(-x) = P(x^2)+P(x)$
2001 Czech And Slovak Olympiad IIIA
Given a triangle $PQX$ in the plane, with $PQ = 3, PX = 2.6$ and $QX = 3.8$. Construct a right-angled triangle $ABC$ such that the incircle of $\vartriangle ABC$ touches $AB$ at $P$ and $BC$ at $Q$, and point $X$ lies on the line $AC$.
2001 Czech And Slovak Olympiad IIIA
Find all triples of real numbers $(a,b,c)$ for which the set of solutions $x$ of $\sqrt{2x^2 +ax+b} > x-c$ is the set $(-\infty,0]\cup(1,\infty)$.
2001 Czech And Slovak Olympiad IIIA
In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.
2001 Czech And Slovak Olympiad IIIA
A sheet of paper has the shape of an isosceles trapezoid $C_1AB_2C_2$ with the shorter base $B_2C_2$. The foot of the perpendicular from the midpoint $D$ of $C_1C_2$ to $AC_1$ is denoted by $B_1$. Suppose that upon folding the paper along $DB_1, AD$ and $AC_1$ points $C_1,C_2$ become a single point $C$ and points $B_1,B_2$ become a point $B$. The area of the tetrahedron $ABCD$ is $64$ cm$^2$ . Find the sides of the initial trapezoid.
2001 Czech And Slovak Olympiad IIIA
Let be given natural numbers $a_1,a_2,...,a_n$ and a function $f : Z \to R$ such that $f(x) = 1$ for all integers $x < 0$ and $f(x) = 1- f(x-a_1)f(x-a_2)... f(x-a_n)$ for all integers $x \ge 0$. Prove that there exist natural numbers $s$ and $t$ such that for all integers $x > s$ it holds that $f(x+t) = f(x)$.
2000 Czech And Slovak Olympiad IIIA
Let $n$ be a natural number. Prove that the number $4 \cdot 3^{2^n}+ 3 \cdot4^{2^n}$ is divisible by $13$ if and only if $n$ is even.
2000 Czech And Slovak Olympiad IIIA
Let be given an isosceles triangle $ABC$ with the base $AB$. A point $P$ is chosen on the altitude $CD$ so that the incircles of $ABP$ and $PECF$ are congruent, where E and F are the intersections of $AP$ and $BP$ with the opposite sides of the triangle, respectively. Prove that the incircles of triangles $ADP$ and $BCP$ are also congruent.
2000 Czech And Slovak Olympiad IIIA
In the plane are given $2000$ congruent triangles of area $1$, which are all images of one triangle under translations. Each of these triangles contains the centroid of every other triangle. Prove that the union of these triangles has area less than $22/9$.
2000 Czech And Slovak Olympiad IIIA
For which quadratic polynomials $f(x)$ does there exist a quadratic polynomial $g(x)$ such that the equations $g(f(x)) = 0$ and $f(x)g(x) = 0$ have the same roots, which are mutually distinct and form an arithmetic progression?
2000 Czech And Slovak Olympiad IIIA
Monika made a paper model of a tetrahedron whose base is a right-angled triangle. When she cut the model along the legs of the base and the median of a lateral face corresponding to one of the legs, she obtained a square of side a. Compute the volume of the tetrahedron.
2000 Czech And Slovak Olympiad IIIA
Find all four-digit numbers $\overline{abcd}$ (in decimal system) such that $\overline{abcd}= (\overline{ac}+1).(\overline{bd} +1)$
1999 Czech And Slovak Olympiad IIIA
We are allowed to put several brackets in the expression

$$\frac{29 : 28 : 27 : 26 :... : 17 : 16}{15 : 14 : 13 : 12 : ... : 3 : 2}$$(a) Find the smallest possible integer value we can obtain in that way.

(b) Find all possible integer values that can be obtained.
1999 Czech And Slovak Olympiad IIIA
In a tetrahedron $ABCD, E$ and $F$ are the midpoints of the medians from $A$ and $D$. Find the ratio of the volumes of tetrahedra $BCEF$ and $ABCD$.
1999 Czech And Slovak Olympiad IIIA
Show that there exists a triangle $ABC$ such that $a \ne b$ and $a+t_a = b+t_b$, where $t_a,t_b$ are the medians corresponding to $a,b$, respectively. Also prove that there exists a number $k$ such that every such triangle satisfies $a+t_a = b+t_b = k(a+b)$. Finally, find all possible ratios $a : b$ in such triangles.
1999 Czech And Slovak Olympiad IIIA
In a certain language there are only two letters, $A$ and $B$. We know that

(i) There are no words of length $1$, and the only words of length $2$ are $AB$ and $BB$.

(ii) A segment of length $n > 2$ is a word if and only if it can be obtained from a word of length less than $n$ by replacing each letter $B$ by some (not necessarily the same) word.

Prove that the number of words of length $n$ is equal to $\frac{2^n +2\cdot (-1)^n}{3}$
1999 Czech And Slovak Olympiad IIIA
Given an acute angle $APX$ in the plane, construct a square $ABCD$ such that $P$ lies on the side $BC$ and ray $PX$ meets $CD$ in a point $Q$ such that $AP$ bisects the angle $BAQ$.
1999 Czech And Slovak Olympiad IIIA
Find all pairs of real numbers $a,b$ for which the system of equations $\frac{x+y}{x^2 +y^2} = a , \frac{x^3 +y^3}{x^2 +y^2} = b$ has a real solution.
1998 Czech And Slovak Olympiad IIIA
Solve the equation $x\cdot  [x\cdot [x \cdot [x]]] = 88$ in the set of real numbers.
1998 Czech And Slovak Olympiad IIIA
Given any set of $14$ (different) natural numbers, prove that for some $k$ ($1 \le k \le 7$) there exist two disjoint $k$-element subsets $\{a_1,...,a_k\}$ and $\{b_1,...,b_k\}$ such that $A =\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}$ and $B =\frac{1}{b_1}+\frac{1}{b_2}+...+\frac{1}{b_k}$ differ by less than $0.001$, i.e. $|A-B| < 0.001$
1998 Czech And Slovak Olympiad IIIA
A sphere is inscribed in a tetrahedron $ABCD$. The tangent planes to the sphere parallel to the faces of the tetrahedron cut off four smaller tetrahedra. Prove that sum of all the $24$ edges of the smaller tetrahedra equals twice the sum of edges of the tetrahedron $ABCD$.
1998 Czech And Slovak Olympiad IIIA
For each date of year $1998$, we calculate day$^{month}$ −year and determine the greatest power of $3$ that divides it. For example, for April $21$ we get $21^4 - 1998 =192483 = 3^3 \cdot 7129$, which is divisible by $3^3$ and not by $3^4$ . Find all dates for which this power of $3$ is the greatest.
1998 Czech And Slovak Olympiad IIIA
A circle $k$ and a point $A$ outside it are given in the plane. Prove that all trapezoids, whose non-parallel sides meet at $A$, have the same intersection of diagonals.
1998 Czech And Slovak Olympiad IIIA
Let $a,b,c$ be positive numbers. Prove that a triangle with sides $a,b,c$ exists if and only if the system of equations

$\frac{y}{z}+\frac{z}{y}=\frac{a}{x},\frac{z}{x}+\frac{x}{z}=\frac{b}{y},\frac{x}{y}+\frac{y}{x}=\frac{c}{z}$ has a real solution.
1997 Abels Math Contest (Norwegian MO)
We call a positive integer $n$ happy  if there exist integers $a,b$ such that $a^2+b^2 = n$. If $t$ is happy, show that

(a) $2t$ is happy,

(b) $3t$ is not happy
1997 Abels Math Contest (Norwegian MO)
Let $P$ be an interior point of an equilateral triangle $ABC$, and let $Q,R,S$ be the feet of perpendiculars from $P$ to $AB,BC,CA$, respectively. Show that the sum $PQ+PR+PS$ is independent of the choice of $P$.
1997 Abels Math Contest (Norwegian MO)
Let $A,B,C$ be different points on a circle such that $AB = AC$.

Point $E$ lies on the segment $BC$, and $D \ne A$ is the point of intersection of the circle and line $AE$.

Show that the product $AE \cdot AD$ is independent of the choice of $E$.
1997 Abels Math Contest (Norwegian MO)
Each subset of $97$ out of $1997$ given real numbers has positive sum.

Show that the sum of all the $1997$ numbers is positive.
1997 Abels Math Contest (Norwegian MO)
Ninety-one students in a school are distributed in three classes.  Each student took part in a competition. It is known that among any six students of the same sex some two got the same number of points. Show that here are four students of the same sex who are in the same class and who got the same number of points.
1997 Abels Math Contest (Norwegian MO)
Let $p(x)$ be a polynomial with integer coefficients. Suppose that there exist different integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$. Show that the equation $f(x) = x$ has at most one integer solution.
1997 Czech And Slovak Olympiad IIIA
Let $ABC$ be a triangle with sides $a,b,c$ and corresponding angles $\alpha,\beta\gamma$ . Prove that if $\alpha = 3\beta$  then $(a^2 -b^2)(a-b) = bc^2$ . Is the converse true?
1997 Czech And Slovak Olympiad IIIA
Each side and diagonal of a regular $n$-gon ($n \ge 3$) is colored red or blue. One may choose a vertex and change the color of all segments emanating from that vertex. Prove that, no matter how the edges were colored initially, one can achieve that the number of blue segments at each vertex is even. Show also that the resulting coloring depends only on the initial coloring.
1997 Czech And Slovak Olympiad IIIA
A tetrahedron $ABCD$ is divided into five polyhedra so that each face of the tetrahedron is a face of (exactly) one polyhedron, and that the intersection of any two of the polyhedra is either a common vertex, a common edge, or a common face. What is the smallest possible sum of the numbers of faces of the five polyhedra?
1997 Czech And Slovak Olympiad IIIA
Show that there exists an increasing sequence $a_1,a_2,a_3,...$ of natural numbers such that, for any integer $k \ge 2$, the sequence $k+a_n$ ($n \in N$) contains only finitely many primes.
1997 Czech And Slovak Olympiad IIIA
For a given integer $n \ge  2$, find the maximum possible value of $V_n = \sin x_1 \cos x_2 +\sin x_2 \cos x_3 +...+\sin x_n \cos x_1$, where $x_1,x_2,...,x_n$ are real numbers.
1997 Czech And Slovak Olympiad IIIA
In a parallelogram $ABCD$, triangle $ABD$ is acute-angled and $\angle BAD = \pi /4$. Consider all possible choices of points $K,L,M,N$ on sides $AB,BC, CD,DA$ respectively, such that $KLMN$ is a cyclic quadrilateral whose circumradius equals those of triangles $ANK$ and $CLM$. Find the locus of the intersection of the diagonals of $KLMN$
1996 Czech And Slovak Olympiad IIIA
A sequence $(G_n)_{n=0}^{\infty}$ satisfies $G(0) = 0$ and $G(n) = n-G(G(n-1))$ for each $n \in N$. Show that

(a) $G(k) \ge G(k -1)$ for every $k \in N$;

(b) there is no integer $k$ for which $G(k -1) = G(k) = G(k +1)$.
1996 Czech And Slovak Olympiad IIIA
Let $AP,BQ$ and $CR$ be altitudes of an acute-angled triangle $ABC$. Show that for any point $X$ inside the triangle $PQR$ there exists a tetrahedron $ABCD$ such that $X$ is the point on the face $ABC$ at the greatest distance from $D$ (measured along the surface of the tetrahedron).
1996 Czech And Slovak Olympiad IIIA
Given six three-element subsets of a finite set $X$, show that it is possible to color the elements of $X$ in two colors so that none of the given subsets is in one color
1996 Czech And Slovak Olympiad IIIA
Points $A$ and $B$ on the rays $CX$ and $CY$ respectively of an acute angle $XCY$ are given so that $CX < CA = CB < CY$. Construct a line meeting the ray $CX$ and the segments $AB,BC$ at $K,L,M$, respectively, such that $KA \cdot YB = XA \cdot MB = LA\cdot LB \ne 0$.
1996 Czech And Slovak Olympiad IIIA
For which integers $k$ does there exist a function $f : N \to Z$ such that

$f(1995) =1996$ and $f(xy) = f(x)+ f(y)+k f(gcd(x,y))$ for all $x,y \in N$?
1996 Czech And Slovak Olympiad IIIA
Let $K,L,M$ be points on sides $AB,BC,CA$, respectively, of a triangle $ABC$ such that $AK/AB = BL/BC = CM/CA = 1/3$. Show that if the circumcircles of the triangles $AKM, BLK, CML$ are equal, then so are the incircles of these triangles.
1995 Czech And Slovak Olympiad IIIA
Suppose that tetrahedron $ABCD$ satisfies $\angle BAC+\angle CAD+\angle DAB = \angle ABC+\angle CBD+\angle DBA = 180^o$. Prove that $CD \ge AB$.
1995 Czech And Slovak Olympiad IIIA
Find the positive real numbers $x,y$ for which $\frac{x+y}{2},\sqrt{xy},\frac{2xy}{x+y},\sqrt{\frac{x^2 +y^2}{2}}$ are integers whose sum is $66$.
1995 Czech And Slovak Olympiad IIIA
Five distinct points and five distinct lines are given in the plane. Prove that one can select two of the points and two of the lines so that none of the selected lines contains any of the selected points.
1995 Czech And Slovak Olympiad IIIA
Do there exist $10000$ ten-digit numbers divisible by $7$, all of which can be obtained from one another by a reordering of their digits?
1995 Czech And Slovak Olympiad IIIA
Let $A,B$ be points on a circle $k$ with center $S$ such that $\angle ASB = 90^o$ . Circles $k_1$ and $k_2$ are tangent to each other at $Z$ and touch $k$ at $A$ and $B$ respectively. Circle $k_3$ inside $\angle ASB$ is internally tangent to $k$ at $C$ and externally tangent to $k_1$ and $k_2$ at $X$ and $Y$, respectively. Prove that $\angle XCY = 45^o$
1995 Czech And Slovak Olympiad IIIA
Find all real parameters $p$ for which the equation $x^3 -2p(p+1)x^2+(p^4 +4p^3 -1)x-3p^3 = 0$

has three distinct real roots which are sides of a right triangle.
1994 Czech And Slovak Olympiad IIIA
Let $f : N \to  N$ be a function which satisfies $f(x)+ f(x+2) \le 2 f(x+1)$ for any $x \in N$.

Prove that there exists a line in the coordinate plane containing infinitely many points of the form $(n, f(n)), n \in N$.
1994 Czech And Slovak Olympiad IIIA
A cuboid of volume $V$ contains a convex polyhedron $M$. The orthogonal projection of $M$ onto each face of the cuboid covers the entire face. What is the smallest possible volume of polyhedron $M$?
1994 Czech And Slovak Olympiad IIIA
A convex $1994$-gon $M$ is given in the plane. A closed polygonal line consists of $997$ of its diagonals. Every vertex is adjacent to exactly one diagonal. Each diagonal divides $M$ into two sides, and the smaller of the numbers of edges on the two sides of $M$ is defined to be the length of the diagonal. Is it posible to have

(a) $991$ diagonals of length $3$ and $6$ of length $2$?

(b) $985$ diagonals of length $6, 4$ of length $8$, and $8$ of length $3$?
1994 Czech And Slovak Olympiad IIIA
Let $a_1,a_2,...$ be a sequence of natural numbers such that for each $n$, the product $(a_n - 1)(a_n- 2)...(a_n - n^2)$ is a positive integral multiple of $n^{n^2-1}$. Prove that for any finite set $P$ of prime numbers the following inequality holds:

$$\sum_{p\in P}\frac{1}{\log_p a_p}< 1$$
1994 Czech And Slovak Olympiad IIIA
In an acute-angled triangle $ABC$, the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$. If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?
1994 Czech And Slovak Olympiad IIIA
Show that from any four distinct numbers lying in the interval $(0,1)$ one can choose two distinct numbers $a$ and $b$ such that

$$\sqrt{(1-a^2)(1-b^2)} > \frac{a}{2b}+\frac{b}{2a}-ab-\frac{1}{8ab} $$
1993 Czech And Slovak Olympiad IIIA
Find all natural numbers $n$ for which $7n -1$ is divisible by $6n -1$
1993 Czech And Slovak Olympiad IIIA
In fields of a $19 \times 19$ table are written integers so that any two lying on neighboring fields differ at most by $2$ (two fields are neighboring if they share a side). Find the greatest possible number of mutually different integers in such a table.
1993 Czech And Slovak Olympiad IIIA
Let $AKL$ be a triangle such that $\angle ALK > 90^o +\angle LAK$. Construct an equilateral trapezoid $ABCD$ (i.e. with three sides equal) with $AB \perp CD$ such that $K$ lies on the side $BC, L$ on the diagonal $AC$ and the lines $AK$ and $BL$ intersect at the circumcenter of the trapezoid.
1993 Czech And Slovak Olympiad IIIA
The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_n+1$ equals the sum of tenth powers of the digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?
1993 Czech And Slovak Olympiad IIIA
Find all functions $f : Z \to Z$ such that $f(-1) = f(1)$ and $f(x)+ f(y) = f(x+2xy)+ f(y-2xy)$ for all $x,y \in Z$
1993 Czech And Slovak Olympiad IIIA
Show that there exists a tetrahedron which can be partitioned into eight congruent tetrahedra, each of which is similar to the original one.
1992 Czech And Slovak Olympiad IIIA
For a permutation $p(a_1,a_2,...,a_{17})$ of $1,2,...,17$, let $k_p$ denote the largest $k$ for which $a_1 +...+a_k < a_{k+1} +...+a_{17}$. Find the maximum and minimum values of $k_p$ and find the sum $\sum_{p} k_p$ over all permutations$ p$.