MathNet 1993 — Part 01

Let $P_{1}, P_{2}, \ldots, P_{1993}=P_{0}$ be distinct points in the $xy$-plane with the following properties:
(i) both coordinates of $P_{i}$ are integers, for $i=1,2, \ldots, 1993$;
(ii) there is no point other than $P_{i}$ and $P_{i+1}$ on the line segment joining $P_{i}$ with $P_{i+1}$ whose coordinates are both integers, for $i=0,1, \ldots, 1992$.
Prove that for some $i, 0 \leq i \leq 1992$, there exists a point $Q$ with coordinates ($q_{x}, q_{y}$) on the line segment joining $P_{i}$ with $P_{i+1}$ such that both $2q_{x}$ and $2q_{y}$ are odd integers.

Find the total number of different integer values the function
$$
f(x) = [x] + [2x] + \left[\frac{5x}{3}\right] + [3x] + [4x]
$$
takes for real numbers $x$ with $0 \leq x \leq 100$.

Note: $[t]$ is the largest integer that does not exceed $t$.

Determine all positive integers $n$ for which the equation
$$
x^{n}+(2+x)^{n}+(2-x)^{n}=0
$$
has an integer as a solution.

Let $ABCD$ be a quadrilateral such that all sides have equal length and angle $\angle ABC$ is $60$ degrees. Let $\ell$ be a line passing through $D$ and not intersecting the quadrilateral (except at $D$). Let $E$ and $F$ be the points of intersection of $\ell$ with $AB$ and $BC$ respectively. Let $M$ be the point of intersection of $CE$ and $AF$.
Prove that $CA^{2} = CM \times CE$.

Let
$$
f(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{0} \quad \text{and} \quad g(x) = c_{n+1} x^{n+1} + c_{n} x^{n} + \cdots + c_{0}
$$
be non-zero polynomials with real coefficients such that $g(x) = (x + r) f(x)$ for some real number $r$. If $a = \max \left(\left|a_{n}\right|, \ldots, \left|a_{0}\right|\right)$ and $c = \max \left(\left|c_{n+1}\right|, \ldots, \left|c_{0}\right|\right)$, prove that $\frac{a}{c} \leq n+1$.

A real number with absolute value less than $1$ is written in each cell of an $n \times n$ array, so that the sum of the numbers in each $2 \times 2$ square is zero. Show that for $n$ odd the sum of all the numbers is less than $n$.

Find a real-valued function $f(x)$ on the non-negative reals such that $f(0) = 0$, and $f(2x + 1) = 3f(x) + 5$ for all $x$.

The sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = 8$, $a_2 = 18$, $a_{n+2} = a_{n+1} \cdot a_n$.
Find all terms which are perfect squares.

Let $P_1P_2...P_n$ be a polygon inscribed in a circle and $Q$ a point outside the plane of the circle. Let $\beta_i$ be the plane perpendicular to $QP_i$ and passing through $P_i$, $i = 1, 2, ..., n$. Prove that the intersection of all planes $\beta_i$ is a point.

$ABCD$ is a convex quadrilateral with $\angle BAC = 30^\circ$, $\angle CAD = 20^\circ$, $\angle ABD = 50^\circ$, $\angle DBC = 30^\circ$. If the diagonals intersect at $P$, show that $PC = PD$.

Given a circle and its center $O$, a point $A$ inside the circle and a distance $h$, construct a triangle $BAC$ with $\angle A = 90^\circ$, $B$ and $C$ on the circle and the altitude from $A$ with length $h$.

On each face of two dice some positive integer is written. The two dice are thrown and the numbers on the top faces are added. Determine whether one can select the integers on the faces so that the possible sums are $2,3,4,5,6,7,8,9,10,11,12,13$, all equally likely?

Let $Q$ be a unit cube. We say a tetrahedron is "good" if all its edges are equal and all its vertices lie on the boundary of $Q$. Find all possible volumes of "good" tetrahedra.

Let's consider three pairwise non-parallel straight lines in the plane. Three points are moving along these lines with different non-zero velocities, one on each line (we consider the movement as having taken place for infinite time and continuing infinitely in the future). Is it possible to determine these straight lines, the velocities of each moving point and their positions at some "zero" moment in such a way that the points never were, are or will be collinear?

Suppose two functions $f(x)$ and $g(x)$ are defined for all $x$ such that $2 \lt x \lt 4$ and satisfy $2 \lt f(x) \lt 4$, $2 \lt g(x) \lt 4$, $f(g(x)) = g(f(x)) = x$ and $f(x) \cdot g(x) = x^{2}$ for all such values of $x$. Prove that $f(3) = g(3)$.

A convex quadrangle $A B C D$ is inscribed in a circle with the centre $O$. The angles $\angle A O B, \angle B O C, \angle C O D$ and $\angle D O A$, taken in some order, are of the same size as the angles of quadrangle $A B C D$. Prove that $A B C D$ is a square.

A square is divided into 16 equal squares, obtaining the set of 25 different vertices. What is the least number of vertices one must remove from this set, so that no 4 points of the remaining set are the vertices of any square with sides parallel to the sides of the initial square?

Do there exist positive integers $a \gt b \gt 1$ such that for each positive integer $k$ there exists a positive integer $n$ for which $a n + b$ is a $k$th power of a positive integer?

Let $a_{1}, a_{2}, \ldots, a_{n}$ and $b_{1}, b_{2}, \ldots, b_{n}$ be two finite sequences consisting of $2n$ different real numbers. Rearranging each of the sequences in the increasing order we obtain $a_{1}^{\prime}, a_{2}^{\prime}, \ldots, a_{n}^{\prime}$ and $b_{1}^{\prime}, b_{2}^{\prime}, \ldots, b_{n}^{\prime}$. Prove that
$$
\max_{1 \leq i \leq n} \left| a_{i} - b_{i} \right| \geq \max_{1 \leq i \leq n} \left| a_{i}^{\prime} - b_{i}^{\prime} \right|.
$$

Prove that for any odd positive integer $n$, $n^{12} - n^{8} - n^{4} + 1$ is divisible by $2^{9}$.