Question bank Math Olympiad Vault 1985

$a$, $b$, $c$, $d$ are integers. Show that $x^2 + a x + b = y^2 + c y + d$ has infinitely many integer solutions iff $a^2 - 4b = c^2 - 4d$.

$A$, $B$ are real numbers. Find a necessary and sufficient condition for $A x + B [x] = A y + B [y]$ to have no solutions except $x = y$.

$a$, $b$, $c$, $d$ are integers with $ad \neq bc$. Show that $\frac{1}{(ax+b)(cx+d)}$ can be written in the form $\frac{r}{ax+b} + \frac{s}{cx+d}$. Find the sum
$$
\frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 10} + \dots + \frac{1}{2998 \cdot 3001}
$$

A convex quadrilateral is inscribed in a circle of radius $1$. Show that its perimeter minus the sum of its two diagonals lies between $0$ and $2$.

Prove or disprove that there exists an integer which is doubled when the initial digit is transferred to the end.

Let $P_{1}$ and $P_{2}$ be regular polygons of $1985$ sides and perimeters $x$ and $y$ respectively. Each side of $P_{1}$ is tangent to a given circle of circumference $c$ and this circle passes through each vertex of $P_{2}$. Prove $x + y \geq 2c$. (You may assume that $\tan \theta \geq \theta$ for $0 \leq \theta \lt \frac{\pi}{2}$.)

Prove that $2^{n-1}$ divides $n!$ if and only if $n=2^{k-1}$ for some positive integer $k$.

The lengths of the sides of a triangle are $6$, $8$ and $10$ units. Prove that there is exactly one straight line which simultaneously bisects the area and perimeter of the triangle.

Let $1 \lt x_{1} \lt 2$ and, for $n = 1, 2, \ldots$, define $x_{n+1} = 1 + x_{n} - \frac{1}{2} x_{n}^{2}$. Prove that, for $n \geq 3$, $\left| x_{n} - \sqrt{2} \right| \lt 2^{-n}$.

For each real number $x_{1}$, the sequence $x_{1}, x_{2}, \ldots, x_{n}, \ldots$ is constructed by making $$
x_{n+1} = x_{n}\left(x_{n} + \frac{1}{n}\right) \text{ for each } n \geq 1
$$ show that there is exactly one value of $x_{1}$ for which $0 \lt x_{n} \lt x_{n+1} \lt 1$ for each $n$.

Non-aligned rays are considered $O x, O y$. by the point $A \in O x$ Pairs of lines are drawn $r_{1}, r_{2}$, anti-parallel to the angle $x O y$; be $M, N$ the intersections of $r_{1}$ with $O y$ and of $r_{2}$ with $O x$, respectively. be $P$ The point of intersection of the angles of the angles $A M y, A N y$. Find the geometric place of $P$ when varying $A$.

Let $\mathbb{Z}$ be the set of integers and $\mathbb{Z} \times \mathbb{Z}$ the set of ordered pairs of integers. The sum of these pairs is defined by $$
(a, b)+\left(a', b'\right)=\left(a+a', b+b'\right)
$$ being $(-a,-b)$ the opposite of $(a, b)$. Study if there is a subset $E$ of $\mathbb{Z} \times \mathbb{Z}$ that meets the following conditions:
a) The sum of two pairs of $E$ is also from $E$.
b) The $(0,0)$ pair belongs to $E$.
c) If $(a, b)$ is not $(0,0)$, then either $(a, b)$ belongs to $E$, or $(-a,-b)$ belongs to $E$, but not both.

We will say that a square matrix is of a constant sum if the sum of the elements of each row, of each column, and of each diagonal, are equal values. Similarly, a square matrix is of constant product if the products of the elements of each row, each column and each diagonal are equal. Determine the square matrices of order $3$ over $\mathbb{R}$ which are both sum and constant product.

Let $n$ and $k$ be two prime natural numbers with each other, with $0 \lt k \lt n$. Each number of the set $\mathcal{M} = \{1, 2, \ldots, n-1\}$ is colored either blue or white. It is known that 1) for each $i \in \mathcal{M}$, the elements $i$ and $n-i$ have the same color. 2) For each $i \in \mathcal{M}$, $i \neq k$, the elements $i$ and $|i-k|$ have the same color. Prove that all elements of $\mathcal{M}$ have the same color.

Let us consider three natural numbers $a, b, c$ such that the ratio $$
\frac{a+b+c}{a b c}
$$ is the inverse of an integer and positive $k$ number. It is requested to demonstrate:
a) $a^{3}+b^{3}+c^{3}$ is not prime.
b) For each $k \in \mathbb{N}$ there are triples of natural $a, b, c$ that meet the conditions.

Let $\mathcal{M}$ be a set of 1985 different positive integers, none of which has prime divisors greater than 26. Demonstrate that $\mathcal{M}$ contains at least a subset of four different elements, the product of which is the Fourth power of an integer.

A center circumference $O$ passes through the vertices $A$ and $C$ of a triangle $A B C$ and cuts the segments again $A B$ and $B C$ at different points $K$ and $N$, respectively. Circumferences circumscribed to triangles $A B C$ and $K B N$ are cut exactly at two different points $B$ and $M$. Prove that the angle $\widehat{O M B}$ is a right angle.

Find the equation of the circumference that passes through the affixes of the solutions of the equation $$
z^{3}+(-1+i) z^{2}+(1-i) z+i=0
$$

A circle has the center on the $AB$ side of the $ABCD$ quadrilateral. The other three sides are tangent to the circle. Prove that $AD + BC = AB$.

Given the equation $x^{5}-p x-1=0$, study the value of $p$ So that there are two solutions of the equation, $x_{1}, x_{2}$, which at the same time are solutions of $x^{2}-a x+b=0$, with $a, b$ whole.