Define a sequence $\langle f(n)\rangle^{\infty}_{n=1}$ of positive integers by $f(1) = 1$ and \[f(n) = \begin{cases} f(n-1) - n & \text{ if } f(n-1) > n;\\ f(n-1) + n & \text{ if } f(n-1) \leq n, \end{cases}\]for $n \geq 2.$ Let $S = \{n \in \mathbb{N} \;\mid\; f(n) = 1993\}.$ (i) Prove that $S$ is an infinite set. (ii) Find the least positive integer in $S.$ (iii) If all the elements of $S$ are written in ascending order as \[ n_1 < n_2 < n_3 < \ldots , \]show that \[ \lim_{i\rightarrow\infty} \frac{n_{i+1}}{n_i} = 3. \]

Show that there exists a finite set $A \subset \mathbb{R}^2$ such that for every $X \in A$ there are points $Y_1, Y_2, \ldots, Y_{1993}$ in $A$ such that the distance between $X$ and $Y_i$ is equal to 1, for every $i.$

Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.

Solve the following system of equations, in which $a$ is a given number satisfying $|a| > 1$: $\begin{matrix} x_{1}^2 = ax_2 + 1 \\ x_{2}^2 = ax_3 + 1 \\ \ldots \\ x_{999}^2 = ax_{1000} + 1 \\ x_{1000}^2 = ax_1 + 1 \\ \end{matrix}$

$a > 0$ and $b$, $c$ are integers such that $ac$ – $b^2$ is a square-free positive integer P. For exampleP could be $3*5$, but not $3^2*5$. Let $f(n)$ be the number of pairs of integers $d, e$ such that $ad^2 + 2bde + ce^2= n$. Show that$f(n)$ is finite and that $f(n) = f(P^{k}n)$ for every positive integer $k$. Original Statement: Let $a,b,c$ be given integers $a > 0,$ $ac-b^2 = P = P_1 \cdots P_n$ where $P_1 \cdots P_n$ are (distinct) prime numbers. Let $M(n)$ denote the number of pairs of integers $(x,y)$ for which \[ ax^2 + 2bxy + cy^2 = n. \]Prove that $M(n)$ is finite and $M(n) = M(P_k \cdot n)$ for every integer $k \geq 0.$ Note that the "$n$" in $P_N$ and the "$n$" in $M(n)$ do not have to be the same.

Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties: (i) $f(1) = 2$; (ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.

Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$

Let $c_1, \ldots, c_n \in \mathbb{R}$ with $n \geq 2$ such that \[ 0 \leq \sum^n_{i=1} c_i \leq n. \] Show that we can find integers $k_1, \ldots, k_n$ such that \[ \sum^n_{i=1} k_i = 0 \] and \[ 1-n \leq c_i + n \cdot k_i \leq n \] for every $i = 1, \ldots, n.$ Another formulation:Let $x_1, \ldots, x_n,$ with $n \geq 2$ be real numbers such that \[ |x_1 + \ldots + x_n| \leq n. \] Show that there exist integers $k_1, \ldots, k_n$ such that \[ |k_1 + \ldots + k_n| = 0. \] and \[ |x_i + 2 \cdot n \cdot k_i| \leq 2 \cdot n -1 \] for every $i = 1, \ldots, n.$ In order to prove this, denote $c_i = \frac{1+x_i}{2}$ for $i = 1, \ldots, n,$ etc.

Let $a,b,c,d$ be four non-negative numbers satisfying \[ a+b+c+d=1. \] Prove the inequality \[ a \cdot b \cdot c + b \cdot c \cdot d + c \cdot d \cdot a + d \cdot a \cdot b \leq \frac{1}{27} + \frac{176}{27} \cdot a \cdot b \cdot c \cdot d. \]

a) Show that the set $ \mathbb{Q}^{ + }$ of all positive rationals can be partitioned into three disjoint subsets. $ A,B,C$ satisfying the following conditions: \[ BA = B; \& B^2 = C; \& BC = A; \] where $ HK$ stands for the set $ \{hk: h \in H, k \in K\}$ for any two subsets $ H, K$ of $ \mathbb{Q}^{ + }$ and $ H^2$ stands for $ HH.$ b) Show that all positive rational cubes are in $ A$ for such a partition of $ \mathbb{Q}^{ + }.$ c) Find such a partition $ \mathbb{Q}^{ + } = A \cup B \cup C$ with the property that for no positive integer $ n \leq 34,$ both $ n$ and $ n + 1$ are in $ A,$ that is, \[ \text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34. \]

Let $n,k \in \mathbb{Z}^{+}$ with $k \leq n$ and let $S$ be a set containing $n$ distinct real numbers. Let $T$ be a set of all real numbers of the form $x_1 + x_2 + \ldots + x_k$ where $x_1, x_2, \ldots, x_k$ are distinct elements of $S.$ Prove that $T$ contains at least $k(n-k)+1$ distinct elements.

Let $n > 1$ be an integer. In a circular arrangement of $n$ lamps $L_0, \ldots, L_{n-1},$ each of of which can either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \ldots .$ If $L_{j-1}$ ($j$ is taken mod $n$) is ON then $Step_j$ changes the state of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the state of any of the other lamps. If $L_{j-1}$ is OFF then $Step_j$ does not change anything at all. Show that: (i) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again, (ii) If $n$ has the form $2^k$ then all the lamps are ON after $n^2-1$ steps, (iii) If $n$ has the form $2^k + 1$ then all lamps are ON after $n^2 - n + 1$ steps.

Let $n \geq 2, n \in \mathbb{N}$ and $A_0 = (a_{01},a_{02}, \ldots, a_{0n})$ be any $n-$tuple of natural numbers, such that $0 \leq a_{0i} \leq i-1,$ for $i = 1, \ldots, n.$ $n-$tuples $A_1= (a_{11},a_{12}, \ldots, a_{1n}), A_2 = (a_{21},a_{22}, \ldots, a_{2n}), \ldots$ are defined by: $a_{i+1,j} = Card \{a_{i,l}| 1 \leq l \leq j-1, a_{i,l} \geq a_{i,j}\},$ for $i \in \mathbb{N}$ and $j = 1, \ldots, n.$ Prove that there exists $k \in \mathbb{N},$ such that $A_{k+2} = A_{k}.$

Let $S_n$ be the number of sequences $(a_1, a_2, \ldots, a_n),$ where $a_i \in \{0,1\},$ in which no six consecutive blocks are equal. Prove that $S_n \rightarrow \infty$ when $n \rightarrow \infty.$

Let $ABC$ be a triangle, and $I$ its incenter. Consider a circle which lies inside the circumcircle of triangle $ABC$ and touches it, and which also touches the sides $CA$ and $BC$ of triangle $ABC$ at the points $D$ and $E$, respectively. Show that the point $I$ is the midpoint of the segment $DE$.

A circle $S$ bisects a circle $S'$ if it cuts $S'$ at opposite ends of a diameter. $S_A$, $S_B$,$S_C$ are circles with distinct centers $A, B, C$ (respectively). Show that $A, B, C$ are collinear iff there is no unique circle $S$ which bisects each of $S_A$, $S_B$,$S_C$ . Show that if there is more than one circle $S$ which bisects each of $S_A$, $S_B$,$S_C$ , then all such circles pass through two fixed points. Find these points. Original Statement: A circle $S$ is said to cut a circle $\Sigma$ diametrically if and only if their common chord is a diameter of $\Sigma.$ Let $S_A, S_B, S_C$ be three circles with distinct centres $A,B,C$ respectively. Prove that $A,B,C$ are collinear if and only if there is no unique circle $S$ which cuts each of $S_A, S_B, S_C$ diametrically. Prove further that if there exists more than one circle $S$ which cuts each $S_A, S_B, S_C$ diametrically, then all such circles $S$ pass through two fixed points. Locate these points in relation to the circles $S_A, S_B, S_C.$

Let triangle $ABC$ be such that its circumradius is $R = 1.$ Let $r$ be the inradius of $ABC$ and let $p$ be the inradius of the orthic triangle $A'B'C'$ of triangle $ABC.$ Prove that \[ p \leq 1 - \frac{1}{3} \cdot (1+r)^2. \] Click to reveal hidden texttypo corrected

Given a triangle $ABC$, let $D$ and $E$ be points on the side $BC$ such that $\angle BAD = \angle CAE$. If $M$ and $N$ are, respectively, the points of tangency of the incircles of the triangles $ABD$ and $ACE$ with the line $BC$, then show that \[\frac{1}{MB}+\frac{1}{MD}= \frac{1}{NC}+\frac{1}{NE}. \]

On an infinite chessboard, a solitaire game is played as follows: at the start, we have $n^2$ pieces occupying a square of side $n.$ The only allowed move is to jump over an occupied square to an unoccupied one, and the piece which has been jumped over is removed. For which $n$ can the game end with only one piece remaining on the board?

For three points $A,B,C$ in the plane, we define $m(ABC)$ to be the smallest length of the three heights of the triangle $ABC$, where in the case $A$, $B$, $C$ are collinear, we set $m(ABC) = 0$. Let $A$, $B$, $C$ be given points in the plane. Prove that for any point $X$ in the plane, \[ m(ABC) \leq m(ABX) + m(AXC) + m(XBC). \]

Let $A$, $B$, $C$, $D$ be four points in the plane, with $C$ and $D$ on the same side of the line $AB$, such that $AC \cdot BD = AD \cdot BC$ and $\angle ADB = 90^{\circ}+\angle ACB$. Find the ratio \[\frac{AB \cdot CD}{AC \cdot BD}, \] and prove that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles $ACD$ and $BCD$ at the point $C$ are perpendicular.)

The vertices $D,E,F$ of an equilateral triangle lie on the sides $BC,CA,AB$ respectively of a triangle $ABC.$ If $a,b,c$ are the respective lengths of these sides, and $S$ the area of $ABC,$ prove that \[ DE \geq \frac{2 \cdot \sqrt{2} \cdot S}{\sqrt{a^2 + b^2 + c^2 + 4 \cdot \sqrt{3} \cdot S}}. \]

A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$ a.) Show that every prime number $n$ has property $P.$ b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$

Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1\mid a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.

Show that for any finite set $S$ of distinct positive integers, we can find a set $T \supseteq S$ such that every member of $T$ divides the sum of all the members of $T$. Original Statement: A finite set of (distinct) positive integers is called a DS-set if each of the integers divides the sum of them all. Prove that every finite set of positive integers is a subset of some DS-set.

Let $S$ be the set of all pairs $(m,n)$ of relatively prime positive integers $m,n$ with $n$ even and $m < n.$ For $s = (m,n) \in S$ write $n = 2^k \cdot n_o$ where $k, n_0$ are positive integers with $n_0$ odd and define \[ f(s) = (n_0, m + n - n_0). \] Prove that $f$ is a function from $S$ to $S$ and that for each $s = (m,n) \in S,$ there exists a positive integer $t \leq \frac{m+n+1}{4}$ such that \[ f^t(s) = s, \] where \[ f^t(s) = \underbrace{ (f \circ f \circ \cdots \circ f) }_{t \text{ times}}(s). \] If $m+n$ is a prime number which does not divide $2^k - 1$ for $k = 1,2, \ldots, m+n-2,$ prove that the smallest value $t$ which satisfies the above conditions is $\left [\frac{m+n+1}{4} \right ]$ where $\left[ x \right]$ denotes the greatest integer $\leq x.$