Navvye/Olympiad-Problems-CSV huggingface dataset:
2005 Hungary-Israel Binational
Let $F_{n}$ be the $n-$ th Fibonacci number (where $F_{1}= F_{2}= 1$). Consider the functions

$f_{n}(x)=\parallel . . . \parallel |x|-F_{n}|-F_{n-1}|-...-F_{2}|-F_{1}|, g_{n}(x)=| . . . \parallel x-1|-1|-...-1|$ ($F_{1}+...+F_{n}$ one’s).

Show that $f_{n}(x) = g_{n}(x)$ for every real number $x.$
2005 Hungary-Israel Binational
There are seven rods erected at the vertices of a regular heptagonal area. The top of each rod is connected to the top of its second neighbor by a straight piece of wire so that, looking from above, one sees each wire crossing exactly two others. Is it possible to set the respective heights of the rods in such a way that no four tops of the rods are coplanar and each wire passes one of the crossings from above and the other one from below?
2003 Hungary-Israel Binational
If $x_{1}, x_{2}, . . . , x_{n}$ are positive numbers, prove the inequality $\frac{x_{1}^{3}}{x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}}+\frac{x_{2}^{3}}{x_{2}^{2}+x_{2}x_{3}+x_{3}^{2}}+...+\frac{x_{n}^{3}}{x_{n}^{2}+x_{n}x_{1}+x_{1}^{2}}\geq\frac{x_{1}+x_{2}+...+x_{n}}{3}$.
2003 Hungary-Israel Binational
Let $ABC$ be an acute-angled triangle. The tangents to its circumcircle at

$A, B, C$ form a triangle $PQR$ with $C \in PQ$ and $B \in PR$. Let $C_{1}$ be the foot of the altitude from $C$ in $\Delta ABC$ . Prove that $CC_{1}$ bisects $\widehat{QC_{1}P}$ .
2003 Hungary-Israel Binational
Let $d > 0$ be an arbitrary real number. Consider the set $S_{n}(d)=\{s=\frac{1}{x_{1}}+\frac{1}{x_{2}}+...+\frac{1}{x_{n}}|x_{i}\in\mathbb{N},s<d\}$. Prove that $S_{n}(d)$ has a maximum element.
2003 Hungary-Israel Binational
Two players play the following game. They alternately write divisors of

$100!$ on the blackboard, not repeating any of the numbers written before. The player after whose move the greatest common divisor of the written numbers equals $1,$ loses the game. Which player has a winning strategy?
2003 Hungary-Israel Binational
Let $M$ be a point inside a triangle $ABC$ . The lines $AM , BM , CM$ intersect $BC, CA, AB$ at $A_{1}, B_{1}, C_{1}$, respectively. Assume that

$S_{MAC_{1}}+S_{MBA_{1}}+S_{MCB_{1}}= S_{MA_{1}C}+S_{MB_{1}A}+S_{MC_{1}B}$ . Prove that one of the lines $AA_{1}, BB_{1}, CC_{1}$ is a median of the triangle $ABC.$
2003 Hungary-Israel Binational
Let $n$ be a positive integer. Show that there exist three distinct integers

between $n^{2}$ and $n^{2}+n+3\sqrt{n}$, such that one of them divides the product of the other two.
2002 Hungary-Israel Binational
Find the greatest exponent $k$ for which $2001^{k}$ divides $2000^{2001^{2002}}+2002^{2001^{2000}}$.
2002 Hungary-Israel Binational
Points $A_{1}, B_{1}, C_{1}$ are given inside an equilateral triangle $ABC$ such that $\widehat{B_{1}AB}= \widehat{A1BA}= 15^{0}, \widehat{C_{1}BC}= \widehat{B_{1}CB}= 20^{0}, \widehat{A_{1}CA}= \widehat{C_{1}AC}= 25^{0}$.

Find the angles of triangle $A_{1}B_{1}C_{1}$.
2002 Hungary-Israel Binational
Let $p \geq 5$ be a prime number. Prove that there exists a positive integer $a < p-1$ such that neither of $a^{p-1}-1$ and $(a+1)^{p-1}-1$ is divisible by $p^{2}$ .
2002 Hungary-Israel Binational
Suppose that positive numbers $x$ and $y$ satisfy $x^{3}+y^{4}\leq x^{2}+y^{3}$. Prove that $x^{3}+y^{3}\leq 2.$
2002 Hungary-Israel Binational
Let $A', B' , C'$ be the projections of a point $M$ inside a triangle $ABC$ onto the sides $BC, CA, AB$, respectively. Deﬁne $p(M ) = \frac{MA'\cdot MB'\cdot MC'}{MA \cdot MB \cdot MC}$ . Find the position of point $M$ that maximizes $p(M )$.
2002 Hungary-Israel Binational
Let $p(x)$ be a polynomial with rational coefficients, of degree at least $2$. Suppose that a sequence $(r_{n})$ of rational numbers satisﬁes $r_{n}= p(r_{n+1})$ for every $n\geq 1$. Prove that the sequence $(r_{n})$ is periodic.
2001 Hungary-Israel Binational
Find positive integers $x, y, z$ such that $x > z > 1999 \cdot 2000 \cdot 2001 > y$ and $2000x^{2}+y^{2}= 2001z^{2}.$
2001 Hungary-Israel Binational
Points $A, B, C, D$ lie on a line $l$, in that order. Find the locus of points $P$ in the plane for which $\angle{APB}=\angle{CPD}$.
2001 Hungary-Israel Binational
Find all continuous functions $f : \mathbb{R}\to\mathbb{R}$ such that for all $x \in\mathbb{ R}$,

$$f (f (x)) = f (x)+x.$$
2001 Hungary-Israel Binational
Let $P (x) = x^{3}-3x+1.$ Find the polynomial $Q$ whose roots are the ﬁfth powers of the roots of $P$.
2001 Hungary-Israel Binational
In a triangle $ABC$ , $B_{1}$ and $C_{1}$ are the midpoints of $AC$ and $AB$ respectively, and $I$ is the incenter. The lines $B_{1}I$ and $C_{1}I$ meet $AB$ and $AC$ respectively at $C_{2}$ and $B_{2}$ . If the areas of $\Delta ABC$ and $\Delta AB_{2}C_{2}$ are equal, ﬁnd $\angle{BAC}$ .
2001 Hungary-Israel Binational
Let be given $32$ positive integers with the sum $120$, none of which is greater than $60.$ Prove that these integers can be divided into two disjoint subsets with the same sum of elements.
2001 Hungary-Israel Binational
Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$.



The edges of $K_{n}(n \geq 3)$ are colored with $n$ colors, and every color is used.

Show that there is a triangle whose sides have different colors.
2001 Hungary-Israel Binational
Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$.



If $n \geq 5$ and $e(G_{n}) \geq \frac{n^{2}}{4}+2$, prove that $G_{n}$ contains two triangles that share exactly one vertex.
2001 Hungary-Israel Binational
Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$.



If $e(G_{n}) \geq\frac{n\sqrt{n}}{2}+\frac{n}{4}$ ,prove that $G_{n}$ contains $C_{4}$ .
2001 Hungary-Israel Binational
Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$.



(a) If $G_{n}$ does not contain $K_{2,3}$ , prove that $e(G_{n}) \leq\frac{n\sqrt{n}}{\sqrt{2}}+n$.

(b) Given $n \geq 16$ distinct points $P_{1}, . . . , P_{n}$ in the plane, prove that at most $n\sqrt{n}$ of the segments $P_{i}P_{j}$ have unit length.
2001 Hungary-Israel Binational
Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$.



(a) Let $p$ be a prime. Consider the graph whose vertices are the ordered pairs $(x, y)$ with $x, y \in\{0, 1, . . . , p-1\}$ and whose edges join vertices $(x, y)$ and $(x' , y')$ if and only if $xx'+yy'\equiv 1 \pmod{p}$ . Prove that this graph does not contain $C_{4}$ .

(b) Prove that for inﬁnitely many values $n$ there is a graph $G_{n}$ with $e(G_{n}) \geq \frac{n\sqrt{n}}{2}-n$ that does not contain $C_{4}$.
2000 Hungary-Israel Binational
Let $S$ be the set of all partitions of $2000$ (in a sum of positive integers). For every such partition $p$, we deﬁne $f (p)$ to be the sum of the number of summands in $p$ and the maximal summand in $p$. Compute the minimum of $f (p)$ when $p \in S .$
2000 Hungary-Israel Binational
Prove or disprove: For any positive integer $k$ there exists an integer $n > 1$ such that the binomial coeffcient $\binom{n}{i}$  is divisible by $k$ for any $1 \leq i \leq n-1.$
2000 Hungary-Israel Binational
Let ${ABC}$ be a non-equilateral triangle. The incircle is tangent to the sides ${BC,CA,AB}$ at ${A_1,B_1,C_1}$, respectively, and M is the orthocenter of triangle ${A_1B_1C_1}$. Prove that ${M}$ lies on the line through the incenter and circumcenter of ${\vartriangle ABC}$.
2000 Hungary-Israel Binational
Let $A$ and $B$ be two subsets of $S = \{1, 2, . . . , 2000\}$ with $|A| \cdot |B| \geq 3999$. For a set $X$ , let $X-X$ denotes the set $\{s-t | s, t \in X, s \not = t\}$. Prove that $(A-A) \cap (B-B)$ is nonempty.
2000 Hungary-Israel Binational
For a given integer $d$, let us deﬁne $S = \{m^{2}+dn^{2}| m, n \in\mathbb{Z}\}$. Suppose that $p, q$ are two elements of $S$ , where $p$ is prime and $p | q$. Prove that $r = q/p$ also belongs to $S$ .
2000 Hungary-Israel Binational
Let $k$ and $l$ be two given positive integers and $a_{ij}(1 \leq i \leq k, 1 \leq j \leq l)$ be $kl$ positive integers. Show that if $q \geq p > 0$, then $$(\sum_{j=1}^{l}(\sum_{i=1}^{k}a_{ij}^{p})^{q/p})^{1/q}\leq (\sum_{i=1}^{k}(\sum_{j=1}^{l}a_{ij}^{q})^{p/q})^{1/p}.$$
1999 Hungary-Israel Binational
$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)=f(x), g_{n+1}(x)=f(g_n(x))$, for all $ n \in N$. Let $ r_n$ be the average of $ g_n(x)$'s roots. If $ r_{19}=99$, find $ r_{99}$.
1999 Hungary-Israel Binational
$ 2n+1$ lines are drawn in the plane, in such a way that every 3 lines define a triangle with no right angles. What is the maximal possible number of acute triangles that can be made in this way?
1999 Hungary-Israel Binational
Find all functions $ f:\mathbb{Q}\to\mathbb{R}$ that satisfy $ f(x+y)=f(x)f(y)-f(xy)+1$ for every $x,y\in\mathbb{Q}$.
1999 Hungary-Israel Binational
$ c$ is a positive integer. Consider the following recursive sequence: $ a_1=c, a_{n+1}=ca_{n}+\sqrt{(c^2-1)(a_n^2-1)}$, for all $ n \in N$.

Prove that all the terms of the sequence are positive integers.
1999 Hungary-Israel Binational
The function $ f(x,y,z)=\frac{x^2+y^2+z^2}{x+y+z}$ is defined for every $ x,y,z \in R$ whose sum is not 0. Find a point $ (x_0,y_0,z_0)$ such that $ 0 < x_0^2+y_0^2+z_0^2 < \frac{1}{1999}$ and $ 1.999 < f(x_0,y_0,z_0) < 2$.
1999 Hungary-Israel Binational
In a multiple-choice test, there are 4 problems, each having 3 possible answers.

In some group of examinees, it turned out that for every 3 of them, there was a question that each of them gave a different answer to. What is the maximal number of examinees in this group?
1998 Hungary-Israel Binational
A player is playing the following game. In each turn he ﬂips a coin and guesses the outcome. If his guess is correct, he gains $ 1$ point; otherwise he loses all his points. Initially the player has no points, and plays the game

until he has $ 2$ points.

(a) Find the probability $ p_{n}$ that the game ends after exactly $ n$ ﬂips.

(b) What is the expected number of ﬂips needed to ﬁnish the game?
1998 Hungary-Israel Binational
A triangle ABC is inscribed in a circle with center $ O$ and radius $ R$. If the inradii of the triangles $ OBC, OCA, OAB$ are $ r_{1}, r_{2}, r_{3}$ , respectively, prove that $ \frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}\geq\frac{4\sqrt{3}+6}{R}.$
1998 Hungary-Israel Binational
Let $ a, b, c, m, n$ be positive integers. Consider the trinomial $ f (x) = ax^{2}+bx+c$. Show that there exist $ n$ consecutive natural numbers $ a_{1}, a_{2}, . . . , a_{n}$ such that each of the numbers $ f (a_{1}), f (a_{2}), . . . , f (a_{n})$ has at least $ m$ different prime factors.
1998 Hungary-Israel Binational
Find all positive integers $ x$ and $ y$ such that $ 5^{x}-3^{y}= 16$.
1998 Hungary-Israel Binational
On the sides of a convex hexagon $ ABCDEF$ , equilateral triangles are constructd in its exterior. Prove that the third vertices of these six triangles are vertices of a regular hexagon if and only if the initial hexagon is affine regular. (A hexagon is called affine regular if it is centrally symmetric and any two opposite sides are parallel to the diagonal determine by the remaining two vertices.)
1998 Hungary-Israel Binational
Let $ n$ be a positive integer. We consider the set $ P$ of all partitions of $ n$ into a sum of positive integers (the order is irrelevant). For every partition $ \alpha$, let $ a_{k}(\alpha)$ be the number of summands in $ \alpha$ that are equal to $ k, k = 1,2,...,n.$ Prove that

$ \sum_{\alpha\in P}\frac{1}{1^{a_{1}(\alpha)}a_{1}(\alpha)!\cdot 2^{a_{2}(\alpha)}a_{2}(\alpha)!...n^{a_{n}(\alpha)}a_{n}(\alpha)!}=1.$
1997 Hungary-Israel Binational
Is there an integer $ N$ such that $ \left(\sqrt{1997}-\sqrt{1996}\right)^{1998}=\sqrt{N}-\sqrt{N-1}$?
1997 Hungary-Israel Binational
Find all the real numbers $ \alpha$ satisfy the following property: for any positive integer $ n$ there exists an integer $ m$ such that $ \left |\alpha-\frac{m}{n}\right|<\frac{1}{3n}$.
1997 Hungary-Israel Binational
Let $ ABC$ be an acute angled triangle whose circumcenter is $ O$. The three diameters of the circumcircle that pass through $ A$, $ B$, and $ C$, meet the opposite sides $ BC$, $ CA$, and $ AB$ at the points $ A_1$, $ B_1$ and $ C_1$, respectively. The circumradius of $ ABC$ is of length $ 2P$, where $ P$ is a prime number. The lengths of $ OA_1$, $ OB_1$, $ OC_1$ are integers. What are the lengths of the sides of the triangle?
1997 Hungary-Israel Binational
Determine the number of distinct sequences of  letters of length 1997 which use each of the letters $A$, $B$, $C$ (and no others) an odd number of times.
1997 Hungary-Israel Binational
The three squares $ACC_{1}A''$, $ABB_{1}'A'$, $BCDE$ are constructed externally on the sides of a triangle $ABC$. Let $P$ be the center of the square $BCDE$. Prove that the lines $A'C$, $A''B$, $PA$ are concurrent.
1997 Hungary-Israel Binational
Can a closed disk can be decomposed into a union of two congruent parts having no common point?
1996 Hungary-Israel Binational
Find all integer sequences of the form $ x_i, 1 \le i \le 1997$, that satisfy $ \sum_{k=1}^{1997} 2^{k-1} x_{k}^{1997}=1996\prod_{k=1}^{1997}x_k$.
1996 Hungary-Israel Binational
$ n>2$ is an integer such that $ n^2$ can be represented as a difference of cubes of 2 consecutive positive integers.  Prove that $ n$ is a sum of 2 squares of positive integers, and that such $ n$ does exist.
1996 Hungary-Israel Binational
A given convex polyhedron has no vertex which belongs to exactly 3 edges. Prove that the number of faces of the polyhedron that are triangles, is at least 8.
1996 Hungary-Israel Binational
$ a_1, a_2, \cdots, a_n$ is a sequence of real numbers, and $ b_1, b_2, \cdots, b_n$ are real numbers that satisfy the condition $ 1 \ge b_1 \ge b_2 \ge \cdots \ge b_n \ge 0$. Prove that there exists a natural number $ k \le n$ that satisifes $ |a_1b_1 + a_2b_2 + \cdots + a_nb_n| \le |a_1 + a_2 + \cdots + a_k|$
1995 Hungary-Israel Binational
Let the sum of the first $ n$ primes be denoted by $ S_n$. Prove that for any positive integer $ n$, there exists a perfect square between $ S_n$ and $ S_{n+1}$.
1995 Hungary-Israel Binational
Let $ P_1$, $ P_2$, $ P_3$, $ P_4$ be five distinct points on a circle. The distance of $ P$ from the line $ P_iP_k$ is denoted by $ d_{ik}$. Prove that $ d_{12}d_{34} = d_{13}d_{24}$.
1995 Hungary-Israel Binational
The polynomial $ f(x)=ax^2+bx+c$ has real coefficients and satisfies $ \left|f(x)\right|\le 1$ for all $ x\in [0, 1]$. Find the maximal value of $ |a|+|b|+|c|$.
1995 Hungary-Israel Binational
Consider a convex polyhedron whose faces are triangles. Prove that it is possible to color its edges by either red or blue, in a way that the following property is satisfied: one can travel from any vertex to any other vertex while passing only along red edges, and can also do this while passing only along blue edges.
1994 Hungary-Israel Binational
Let $ m$ and $ n$ be two distinct positive integers. Prove that there exists a real number $ x$ such that $ \frac {1}{3}\le\{xn\}\le\frac {2}{3}$ and $ \frac {1}{3}\le\{xm\}\le\frac {2}{3}$. Here, for any real number $ y$, $ \{y\}$ denotes the fractional part of $ y$. For example $ \{3.1415\} = 0.1415$.
1994 Hungary-Israel Binational
Let $ a_1$, $ \ldots$, $ a_k$, $ a_{k+1}$, $ \ldots$, $ a_n$ be $ n$ positive numbers ($ k<n$). Suppose that the values of $ a_{k+1}$, $ a_{k+2}$, $ \ldots$, $ a_n$ are fixed. Choose the values of $ a_1$, $ a_2$, $ \ldots$, $ a_k$ that minimize the sum $ \sum_{i, j, i\neq j}\frac{a_i}{a_j}$
1994 Hungary-Israel Binational
Three given circles have the same radius and pass through a common point $ P$. Their other points of pairwise intersections are $ A$, $ B$, $ C$. We define triangle $ A'B'C'$, each of whose sides is tangent to two of the three circles. The three circles are contained in $ \triangle A'B'C'$. Prove that the area of $ \triangle A'B'C'$ is at least nine times the area of $ \triangle ABC$
1994 Hungary-Israel Binational
An $ n-m$ society is a group of $ n$ girls and $ m$ boys. Prove that there exists numbers $ n_0$ and $ m_0$ such that every $ n_0-m_0$ society contains a subgroup of five boys and five girls with the following property: either all of the boys know all of the girls or none of the boys knows none of the girls.
1993 Hungary-Israel Binational
Find all pairs of coprime natural numbers $a$ and $b$ such that the fraction $\frac{a}{b}$ is written in the decimal system as $b.a.$
1993 Hungary-Israel Binational
Determine all polynomials $f (x)$ with real coeffcients that satisfy

$$f (x^{2}-2x) = f^{2}(x-2)$$

for all $x.$
1993 Hungary-Israel Binational
Distinct points $A, B , C, D, E$ are given in this order on a semicircle with radius $1$. Prove that

$$AB^{2}+BC^{2}+CD^{2}+DE^{2}+AB \cdot BC \cdot CD+BC \cdot CD \cdot DE < 4.$$
1993 Hungary-Israel Binational
Find the largest possible number of rooks that can be placed on a $3n \times 3n$ chessboard so that each rook is attacked by at most one rook.
1993 Hungary-Israel Binational
In the questions below: $G$ is a ﬁnite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group.



Suppose $k \geq 2$ is an integer such that for all $x, y \in G$ and $i \in \{k-1, k, k+1\}$ the relation $(xy)^{i}= x^{i}y^{i}$ holds. Show that $G$ is Abelian.
1993 Hungary-Israel Binational
In the questions below: $G$ is a ﬁnite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group.



Suppose that $n \geq 1$ is such that the mapping $x \mapsto x^{n}$ from $G$ to itself is an isomorphism. Prove that for each $a \in G, a^{n-1}\in Z (G).$
1993 Hungary-Israel Binational
In the questions below: $G$ is a ﬁnite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group.



Show that every element of $S_{n}$ is a product of $2$-cycles.
1993 Hungary-Israel Binational
In the questions below: $G$ is a ﬁnite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group.



Let $H \leq G$ and $a, b \in G.$ Prove that $|aH \cap Hb|$ is either zero or a divisor of $|H |.$
1993 Hungary-Israel Binational
In the questions below: $G$ is a ﬁnite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group.



Let $H \leq G, |H | = 3.$ What can be said about $|N_{G}(H ) : C_{G}(H )|$?
1993 Hungary-Israel Binational
In the questions below: $G$ is a ﬁnite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group.



Let $a, b \in G.$ Suppose that $ab^{2}= b^{3}a$ and $ba^{2}= a^{3}b.$ Prove that $a = b = 1.$
1993 Hungary-Israel Binational
In the questions below: $G$ is a ﬁnite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group.



Assume $|G'| = 2$. Prove that $|G : G'|$ is even.
1992 Hungary-Israel Binational
Prove that if $c$ is a positive number distinct from $1$ and $n$ a positive integer, then

$$n^{2}\leq \frac{c^{n}+c^{-n}-2}{c+c^{-1}-2}. $$
1992 Hungary-Israel Binational
A set $S$ consists of $1992$ positive integers among whose units digits all $10$ digits occur. Show that there is such a set $S$ having no nonempty subset $S_{1}$ whose sum of elements is divisible by $2000$.
1992 Hungary-Israel Binational
We are given $100$ strictly increasing sequences of positive integers: $A_{i}= (a_{1}^{(i)}, a_{2}^{(i)},...), i = 1, 2,..., 100$. For $1 \leq r, s \leq 100$ we deﬁne the following quantities: $f_{r}(u)=$ the number of elements of $A_{r}$ not exceeding $n$; $f_{r,s}(u) =$ the number of elements of $A_{r}\cap A_{s}$ not exceeding $n$. Suppose that $f_{r}(n) \geq\frac{1}{2}n$ for all $r$ and $n$. Prove that there exists a pair of indices $(r, s)$ with $r \not = s$ such that $f_{r,s}(n) \geq\frac{8n}{33}$ for at least ﬁve distinct $n-s$ with $1 \leq n < 19920.$
1992 Hungary-Israel Binational
We are given a convex pentagon $ABCDE$ in the coordinate plane such that $A$, $B$, $C$, $D$, $E$ are lattice points. Let $Q$ denote the convex pentagon bounded by the five diagonals of the pentagon $ABCDE$ (so that the vertices of $Q$ are the interior points of intersection of diagonals of the pentagon $ABCDE$). Prove that there exists a lattice point inside of $Q$ or on the boundary of $Q$.
1992 Hungary-Israel Binational
We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$

$$F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n}, $$

where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof.



Prove that $1+L_{2^{j}}\equiv 0 \pmod{2^{j+1}}$ for $j \geq 0$.
1992 Hungary-Israel Binational
We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ $$F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},$$ where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof.



Prove that $$\sum_{k=1}^{n}[\alpha^{k}F_{k}+\frac{1}{2}]=F_{2n+1}\; \forall n>1.$$