MathNet 2010 — Part 06

In the triangle $ABC$ with $AC \gt AB$, $D$ is the foot of the perpendicular from $A$ onto $BC$ and $E$ is the foot of the perpendicular from $D$ onto $AC$. Let $F$ be the point on the line $DE$ such that $EF \cdot DC = BD \cdot DE$. Prove that $AF$ is perpendicular to $BF$.

In an isosceles triangle $ABC$, $AB = AC$. Let $E$ be a point on $AC$ and extend $AB$ to a point $D$ such that $BD = EC$. Let $F$ be the intersection of $BC$ and $DE$ and let the circumcircle of $ABC$ intersect the circumcircle of $BDF$ at $G$. Prove that $DE$ is perpendicular to $FG$.

Suppose that $a_1, \dots, a_{15}$ are prime numbers forming an arithmetic progression with common difference $d \gt 0$. If $a_1 \gt 15$, prove that $d \gt 30,000$.

Positive real numbers $a$, $b$, $c$, $d$ satisfy:
$$
a(c^2 - 1) = b(b^2 + c^2) \text{ and } d \le 1.
$$
Prove that
$$
d(a\sqrt{1-d^2} + b^2\sqrt{1+d^2}) \le \frac{(a+b)c}{2}.
$$

Let $N = 2010! + 1$. Prove that


a) $N$ is not divisible by $4021$;


b) $N$ is not divisible by $2027$, $2029$, $2039$;


c) $N$ has a prime divisor greater than $2050$.

In countries of the Silk Road there are finite number of cities and some of them are connected by directed roads (one pair of cities can have several roads, moreover they can have an opposite direction). It is known that every two paths by these roads form the city $A$ to the city $B$ use at least one same road. Prove that some of the roads is common part for all the paths from $A$ to $B$ ($A$ and $B$ are two given cities).

In convex quadrilateral $ABCD$
$$
\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^\circ \text{ and } AD = BC.
$$
Prove that segments $DB$, $CA$ and $DC$ form the sides of the right triangle.

At an acute angle $\triangle ABC$ point $M$ is the center of the page $BC$, and points $D, E$ and $F$ are the bases of the heights from the vertex $A, B$ and $C$, respectively. Let it be $H$ orthocenter $\triangle ABC$, $S$ center longer $AH$, whereas $G$ section longer $FE$ and $AH$. If it is $N$ the point of intersection of the center of gravity of the length $AM$ and described circles $\triangle BCH$, prove that it is $\measuredangle HMA = \measuredangle GNS$.

Let $a_{0}$ and $a_{n}$ be different divisors of the natural number $m\gt 1$, and let the sequence of natural numbers $a_{0}, a_{1}, a_{2}, \ldots, a_{n}$ satisfy
$$
a_{i+1}=\left|a_{i} \pm a_{i-1}\right| \quad \text{ for } 0\lt i\lt n
$$
If NZD is $\left(a_{0}, \ldots, a_{n}\right)=1$, prove that there is a term in the sequence that is less than $\sqrt{m}$.

An $n \times n$ table contains the numbers $1,2, \ldots, n^{2}$, each exactly once. Call it a Niš table if the product of the $n$ entries in any $n$ cells, no two in the same row or column, has the same remainder modulo $n^{2}+1$. Does such a table exist for:

a. $n=8$;

b. $n=10$?

(Such a selection of $n$ cells is called scattered.)

Let $A$ be an infinite subset of the set of natural numbers. Determine all natural numbers $n$ such that for each $a \in A$
$$
a^{n}+a^{n-1}+\ldots+a^{1}+1 \mid a^{n!}+a^{(n-1)!}+\ldots+a^{1!}+1
$$

Let $O$ be the center of the circumscribed circle $\triangle ABC$. The line through $O$ intersects the sides $CA$ and $CB$ at points $D$ and $E$, respectively, and the inscribed circle $\triangle ABO$ at point $P$ inside the triangle (different from $O$). Point $Q$ on page $AB$ is such that $\frac{AQ}{QB} = \frac{DP}{PE}$. Prove that $\angle APQ = 2 \cdot \angle CAP$. (Dusan Djukic)

Some of the $n$ cities are connected by airline routes (all routes are two-way). There is exactly $m$ line. Let $d_{i}$ be the number of lines departing from the city $i$, for $i=1,2, \ldots, n$. If $1 \leqslant d_{i} \leqslant 2010$, for each $i=1,2, \ldots, n$, prove that it holds
$$
\sum_{i=1}^{n} d_{i}^{2} \leqslant 4022 m-2010 n
$$
Determine all $n$ for which equality can be reached.

For which natural numbers $n$ is there a multiple of 11 that has a sum of digits equal to $n$?

Find all prime numbers $p$, $q$ and $r$ such that $p \gt q \gt r$ and the numbers $p-q$, $p-r$ and $q-r$ are also prime.

A teacher gives Matej four pieces of paper, each containing a non-zero digit. Matej arranges them in a line thus forming a four-digit number. He then exchanges two of the pieces (without flipping or rotating them in the process) to form another four-digit number. Can he always do this in such a way that the two numbers he obtains are not relatively prime regardless of the digits that were written on the pieces of paper?

Let $\triangle ABC$ be an acute triangle with $|AB| \gt |AC|$. Let $D$ be a point on the side $AB$, such that the angles $\angle ACD$ and $\angle CBD$ are equal. Let $E$ denote the midpoint of $BD$, and let $S$ be the circumcentre of the triangle $BCD$. Prove that the points $A, E, S$ and $C$ lie on the same circle.

Find all integers $n$, such that the equation $x^2 + nx + n + 5 = 0$ has only integer solutions.

For what positive integers $n \ge 3$ does there exist a convex $n$-gon which can be divided into finitely many parallelograms?

Let $H$ be the orthocentre of the acute triangle $ABC$ and let $D$ be a point inside the triangle $ABH$. A line that passes through the point $D$ and is parallel to the line $AH$ intersects the segments $BC$ and $AB$ at $K$ and $L$. A line that passes through the point $D$ and is parallel to the line $BH$ intersects the segments $AC$ and $AB$ at $M$ and $N$. Prove that $C, D$ and $H$ lie on the same line if $K, L, M$ and $N$ lie on the same circle.