MathNet 2009-part-02 — 175 problems (EN–VI)

If $a$ is an even positive integer and $A = a^n + a^{n-1} + \dots + a + 1$, $n \in \mathbb{N}^*$, is a perfect square, prove that $a$ is a multiple of $8$.

Let the triangle $ABC$ has barycenter $G$ and circumcenter $O$. The perpendicular bisectors of $GA$, $GB$ and $GC$ intersect at the points $A_1$, $B_1$, $C_1$. Prove that $O$ is the barycentre of the triangle $A_1B_1C_1$.

Find all triples of real numbers $(x, y, z)$ which are greater than $3$ and satisfy the equality:
$$
\frac{(x+2)^2}{y+z-2} + \frac{(y+4)^2}{z+x-4} + \frac{(z+6)^2}{x+y-6} = 36.
$$

In the plane are given $\nu$ different points such that any three of them are not collinear. We color these points red, green and black. In the sequel we consider all line segments with ends these $\nu$ points and we correspond to each of them an “algebraic value” according to the following rules:
1) If at least one of the ends of the line segment is black, then it has algebraic value 0.
2) If both ends of the line segment have the same colour, red or green then it has the algebraic value 1.
3) If ends of the line segment have different colour red or green, then it has the algebraic value -1.
Determine the least possible value of the sum of algebraic values of the all line segments.

Determine all integers $n \ge 1$ for which there exists $n$ real numbers $x_1, x_2, \dots, x_n$ in the closed interval $[-4, 2]$ such that the following three conditions are fulfilled:
- the sum of these real numbers is at least $n$;
- the sum of their squares is at most $4n$;
- the sum of their fourth powers is at least $34n$:

Let $\triangle ABC$ a triangle with $90^\circ \neq \hat{A} \neq 135^\circ$. Let $D$ and $E$ be external points to the triangle such that $DAB$ and $EAC$ are isosceles triangles with right angles in $D$ and $E$, respectively. Let $F = BE \cap CD$, and $M, N$ the midpoints of $BC, DE$, respectively.
Prove that, if three of the points $A, F, M, N$ are collinear, then the four points are collinear.

Decide whether the integers $1, 2, \ldots, 100$ can be arranged in the cells $C(i,j)$ of a $10 \times 10$ matrix (where $1 \le i, j \le 10$), such that the following conditions are satisfied:
(i) In every row, the entries add up to the same sum $S$.
(ii) In every column, the entries also add up to this sum $S$.
(iii) For every $k = 1, \dots, 10$ the ten entries $C(i,j)$ with $i-j \equiv k \pmod{10}$ add up to $S$.

Let $x, y, z$ be positive real numbers. Prove that
$$
\sum_{cyclic} \frac{xy}{xy + x^2 + y^2} \le \sum_{cyclic} \frac{x}{2x + z}.
$$

Suppose $\{a_n\}$ is a sequence in which all the terms are integers, and $a_2$ is odd. For any natural number $n$, $n(a_{n+1} - a_n + 3) = a_{n+1} + a_n + 3$. Furthermore, $a_{2009}$ is divisible by $2010$. Find the smallest integer $n$, $n \ge 2$, such that $a_n$ is divisible by $2010$.

There are $n$ points on the plane, no three of which are collinear. Each pair of points is joined by a red, yellow or green line. For any three points, the sides of the triangle they form consist of exactly two colours. Show that $n \lt 13$.

Let $\triangle ABC$ be a right-angled triangle with $\angle C = 90^\circ$. $CD$ is the altitude from $C$ to $AB$, with $D$ on $AB$. $\omega$ is the circumcircle of $\triangle BCD$. $\omega_1$ is a circle situated in $\triangle ACD$, which is tangent to the segments $AD$ and $AC$ at $M$ and $N$ respectively, and is also tangent to circle $\omega$.
(i) Show that $BD \cdot CN + BC \cdot DM = CD \cdot BM$.
(ii) Show that $BM = BC$.

Find all nonnegative integers $m$ and $n$ that satisfy the equation:
$$
107^{56}(m^2 - 1) - 2m + 3 = \binom{113^{114}}{n}.
$$
(If $n$ and $r$ are nonnegative integers satisfying $r \le n$, then $\binom{n}{r} = C_r^n = \frac{n!}{r!(n-r)!}$ and $\binom{n}{r} = 0$ if $r \gt n$.)

Let $\theta_1, \theta_2, \dots, \theta_{2008}$ be real numbers. Find the maximum value of
$$ \sin \theta_1 \cos \theta_2 + \sin \theta_2 \cos \theta_3 + \dots + \sin \theta_{2007} \cos \theta_{2008} + \sin \theta_{2008} \cos \theta_1. $$

Find the total number of solutions to the following system of equations:
$$
\begin{cases}
a^2 + bc \equiv a \pmod{37}, \\
b(a+d) \equiv b \pmod{37}, \\
c(a+d) \equiv c \pmod{37}, \\
bc + d^2 \equiv d \pmod{37}, \\
ad - bc \equiv 1 \pmod{37}.
\end{cases}
$$

In a school there are $2008$ students. Students are members of certain committees. A committee has at most $1004$ members and every two students join a common committee.

a. Determine the smallest possible number of committees in the school.

b. If it is further required that the union of any two committees consists of at most $1800$ students, will your answer in (a) still hold?

Show that the equation $y^{37} \equiv x^3 + 11 \pmod{p}$ is solvable for every prime $p$, where $p \le 100$.

Let $f : \mathbb{Z} \to \mathbb{Z}$ ($\mathbb{Z}$ is the set of integers) be such that $f(1) = 1$, $f(2) = 20$, $f(-4) = -4$ and $f(x+y) = f(x)+f(y)+axy(x+y)+bxy+c(x+y)+4$ for all $x, y \in \mathbb{Z}$, where $a, b$ and $c$ are certain constants.

a. Find a formula for $f(x)$, where $x$ is any integer.

b. If $f(x) \ge mx^2+(5m+1)x+4m$ for all nonnegative integers $x$, find the greatest possible value of $m$.

Let $a$, $b$, $c$ be positive real numbers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = a + b + c$. Prove that
$$
\frac{1}{(2a + b + c)^2} + \frac{1}{(2b + c + a)^2} + \frac{1}{(2c + a + b)^2} \leq \frac{3}{16}
$$

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that
$$
f(x-f(y))\gt y f(x)+x .
$$

Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all real $x, y$ the identity
$$
f(x f(x+y))=f(y f(x))+x^{2} .
$$