Found problems: 21
MathLinks Contest 3rd, 2
The sequence $\{x_n\}_{n\ge1}$ is defined by $x_1 = 7$, $x_{n+1} = 2x^2_n - 1$, for all positive integers $n$. Prove that for all positive integers $n$ the number $x_n$ cannot be divisible by $2003$.
MathLinks Contest 3rd, 3
On a $2004\times 2004$ chessboard we place $2004$ white knights$^1$ in the upper row, and $2004$ black ones in the lowest row. After a finite number of regular chess moves$^2$ , we get the opposite situation where the black ones are on the top and the white ones on the bottom lines.
In a [i]turn [/i] we make a move with each of the pieces of a color. If you know that each square except those on which the knights originally lie, must not be used more than once in this process, and that after each turn no $2$ knights of the same color can be attacking each other$^3$ , determine the number of ways in which this can be accomplished.
$^1$ also known as horses
$^2$ the knight can be moved either one square horizontally and two vertically or two squares horizontally and one vertically, in any direction on both horizontal and vertical lines
$^3$ a knight is attacking another knight, if in one chess move, the first one can be placed on the second one’s place
MathLinks Contest 3rd, 1
Find all functions$ f, g : (0,\infty) \to (0,\infty)$ such that for all $x > 0$ we have the relations:
$f(g(x)) = \frac{x}{xf(x) - 2}$ and $g(f(x)) = \frac{x}{xg(x) - 2}$
.
MathLinks Contest 3rd, 1
Let $P$ be the set of points in the Euclidean plane, and let $L$ be the set of lines in the same plane. Does there exist an one-to-one mapping (injective function) $f : L \to P$ such that for each $\ell \in L$ we have $f(\ell) \in \ell$?
MathLinks Contest 3rd, 1
For a triangle $ABC$ and a point $M$ inside the triangle we consider the lines $AM, BM,CM$ which intersect the sides $BC, CA, AB$ in $A_1, B_1, C_1$ respectively. Take $A', B', C'$ to be the intersection points between the lines $AA_1, BB_1, CC_1$ and $B_1C_1, C_1A_1, A_1B_1$ respectively.
a) Prove that the lines $BC', CB'$ and $AA'$ intersect in a point $A_2$;
b) Define similarly points $B_2, C_2$. Find the loci of $M$ such that the triangle $A_1B_1C_1$ is similar with the triangle $A_2B_2C_2$.
MathLinks Contest 3rd, 1
Let $a, b, c$ be positive reals. Prove that $$\sqrt{abc}(\sqrt{a} +\sqrt{b} +\sqrt{c}) + (a + b + c)^2 \ge 4 \sqrt{3abc(a + b + c)}.$$
MathLinks Contest 3rd, 3
An integer point of the usual Euclidean $3$-dimensional space is a point whose three coordinates are all integers. A set $S$ of integer points is called a [i]covered [/i] set if for all points $A, B$ in $S$ each integer point in the segment $[AB]$ is also in $S$.
Determine the maximum number of elements that a covered set can have if it does not contain $2004$ collinear points.
MathLinks Contest 3rd, 3
We say that a tetrahedron is [i]median [/i] if and only if for each vertex the plane that passes through the midpoints of the edges emerging from the vertex is tangent to the inscribed sphere. Also a tetrahedron is called [i]regular [/i] if all its faces are congruent. Prove that a tetrahedron is regular if and only if it is median.
MathLinks Contest 3rd, 3
Let $n \ge 3$ be an integer. Find the minimal value of the real number $k_n$ such that for all positive numbers $x_1, x_2, ..., x_n$ with product $1$, we have $$\frac{1}{\sqrt{1 + k_nx_1}}+\frac{1}{\sqrt{1 + k_nx_2}}+ ... + \frac{1}{\sqrt{1 + k_nx_n}} \le n - 1.$$
MathLinks Contest 3rd, 2
Let $a_1, a_2, ..., a_{2004}$ be integer numbers such that for all positive integers $n$ the number $A_n = a^n_1 + a^n_2 + ...+ a^n_{2004}$ is a perfect square. What is the minimal number of zeros within the $2004$ numbers?
MathLinks Contest 3rd, 1
In a soccer championship $2004$ teams are subscribed. Because of the extremely large number of teams the usual rules of the championship are modified as follows:
a) any two teams can play against one each other at most one game;
b) from any $4$ teams, $3$ of them play against one each other.
How many days are necessary to make such a championship, knowing that each team can play at most one game per day?
MathLinks Contest 3rd, 3
An integer $z$ is said to be a [i]friendly [/i] integer if $|z|$ is not the square of an integer. Determine all integers $n$ such that there exists an infinite number of triplets of distinct friendly integers $(a, b, c)$ such that $n = a+b+c$ and $abc$ is the square of an odd integer.
MathLinks Contest 3rd, 1
Let $S$ be a nonempty set of points of the plane. We say that $S$ determines the distance $d > 0$ if there are two points $A, B$ in $S$ such that $AB = d$.
Assuming that $S$ does not contain $8$ collinear points and that it determines not more than $91$ distances, prove that $S$ has less than $2004$ elements.
MathLinks Contest 3rd, 2
Find all functions $f : \{1, 2, ... , n,...\} \to Z$ with the following properties
(i) if $a, b$ are positive integers and $a | b$, then $f(a) \ge f(b)$;
(ii) if $a, b$ are positive integers then $f(ab) + f(a^2 + b^2) = f(a) + f(b)$.
MathLinks Contest 3rd, 3
Let $a$ and $b$ be different positive rational numbers such that the there exist an infinity of positive integers $n$ for which $a^n - b^n$ is an integer. Prove that $a$ and $b$ are also integers.
MathLinks Contest 3rd, 2
Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC, CA, AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all three semicircles has radius $t$. Prove that
$$\frac{s}{2} < t \le \frac{s}{2} + \left( 1 - \frac{\sqrt3}{2} \right)r.$$
MathLinks Contest 3rd, 2
Let n be a positive integer and let $a_1, a_2, ..., a_n, b_1, b_2, ... , b_n, c_2, c_3, ... , c_{2n}$ be $4n - 1$ positive real numbers such that $c^2_{i+j} \ge a_ib_j $, for all $1 \le i, j \le n$. Also let $m = \max_{2 \le i\le 2n} c_i$.
Prove that $$\left(\frac{m + c_2 + c_3 +... + c_{2n}}{2n} \right)^2 \ge \left(\frac{a_1+a_2 + ... +a_n}{n}\right)\left(\frac{
b_1 + b_2 + ...+ b_n}{n}\right)$$
MathLinks Contest 3rd, 1
Find all functions $f : (0, +\infty) \to (0, +\infty)$ which are increasing on $[1, +\infty)$ and for all positive reals $a, b, c$ they fulfill the following relation $f(ab)f(bc)f(ca)=f(a^2b^2c^2)+f(a^2)+f(b^2)+f(c^2)$.
MathLinks Contest 3rd, 3
Each point in the Euclidean space is colored with one of $n \ge 2$ colors, and each of the $n$ colors is used. Prove that one can find a triangle such that the color assigned to the orthocenter is different from all the colors assigned to the vertices of the triangle.
MathLinks Contest 3rd, 2
Prove that for all positive reals $a, b, c$ the following double inequality holds:
$$\frac{a+b+c}{3}\ge \sqrt[3]{\frac{(a+b)(b+c)(c+a)}{8}}\ge \frac{\sqrt{ab}+\sqrt{bc}\sqrt{ca}}{3}$$
MathLinks Contest 3rd, 2
Let $k \ge 1$ be an integer and $a_1, a_2, ... , a_k, b1, b_2, ..., b_k$ rational numbers with the property that for any irrational numbers $x_i >1$, $i = 1, 2, ..., k$, there exist the positive integers $n_1, n_2, ... , n_k, m_1, m_2, ..., m_k$ such that $$a_1\lfloor x^{n_1}_1\rfloor + a_2 \lfloor x^{n_2}_2\rfloor + ...+ a_k\lfloor x^{n_k}_k\rfloor=b_1\lfloor x^{m_1}_1\rfloor +2_1\lfloor x^{m_2}_2\rfloor+...+b_k\lfloor x^{m_k}_k\rfloor $$
Prove that $a_i = b_i$ for all $i = 1, 2, ... , k$.