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Art of Problem Solving

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MathLinks Contest 1st

3

February - May 2003

Round 1

1

In a country there are $n$ major cities, $n \ge 4$ , connected by railroads, such that each city is directly connected to each other city. Each railroad company in that country has but only one train, which connects a series of cities, at least two, such that the train doesn’t pass through the same city twice in one shift. The companies divided the market such that any two cities are directly $^1$ connected only by one company. Prove that among any $n+1$ companies, there are two which have no common train station or there are two cities that are connected by two trains belonging to two of these $n+1$ companies.

$^1$ directly connected means that they are connected by a railroad, without no other station between them

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2

Prove that for all positive integers $a, b, c$ the following inequality holds:
$\frac{a + b}{a + c}+\frac{b + c}{b + a}+\frac{c + a}{c + b} \le \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$

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3

Let $x_0 = 1$ and $x_1 = 2003$ and define the sequence $(x_n)_{n \ge 0}$ by: $x_{n+1} =\frac{x^2_n + 1}{x_{n-1}}$ , $\forall n \ge 1$
Prove that for every $n \ge 2$ the denominator of the fraction $x_n$ , when $x_n$ is expressed in lowest terms is a power of $2003$ .

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Round 2

1

Let $A$ be a finite set of positive integers. Prove that there exists a finite set $B$ of positive integers such that $A \subset B$ and $\prod_{x \in B} x =\sum_{x \in B}x^2$

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2

In a triangle $\vartriangle ABC$ , $\angle B = 2\angle C$ . Let $P$ and $Q$ be points on the perpendicular bisector of segment $BC$ such that rays $AP$ and $AQ$ trisect $\angle A$ . Prove that $PQ$ is smaller than $AB$ if and only if $\angle B$ is obtuse.

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3

Prove that in any acute triangle with sides $a, b, c$ circumscribed in a circle of radius $R$ the following inequality holds:
$\frac{\sqrt2}{4} <\frac{Rp}{2aR + bc} <\frac{1}{2}$ where $p$ represents the semi-perimeter of the triangle.

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Round 3

1

A pack of $2003$ circus flees are deployed by their circus trainer, named Gogu, on a sufficiently large table, in such a way that they are not all lying on the same line. He now wants to get his Ph.D. in fleas training, so Gogu has thought the fleas the following trick: we chooses two fleas and tells one of them to jump over the other one, such that any flea jumps as far as twice the initial distance between him and the flea over which he is jumping. The Ph.D. circus committee has but only one task to Gogu: he has to make the his flees gather around on the table such that every flea represents a vertex of a convex polygon. Can Gogu get his Ph.D., no matter of how the fleas were deployed?

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2

Let a be a non-zero integer, and $n \ge 3$ another integer. Prove that the following polynomial is irreducible in the ring of integer polynomials (i.e. it cannot be written as a product of two non-constant integer polynomials):
$f(x) = x^n + ax^{n-1} + ax^{n-2} +... + ax -1$

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3

Let $(A_i)_{i\ge 1}$ be sequence of sets of two integer numbers, such that no integer is contained in more than one $A_i$ and for every $A_i$ the sum of its elements is $i$ . Prove that there are infinitely many values of $k$ for which one of the elements of $A_k$ is greater than $13k/7$ .

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Round 4

1

Given are $4004$ distinct points, which lie in the interior of a convex polygon of area $1$ .
Prove that there exists a convex polygon of area $\frac{1}{2003}$ , included in the given polygon, such that it does not contain any of the given points in its interior.

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2

Let $f$ be a polynomial with real coefficients such that for each positive integer n the equation $f(x) = n$ has at least one rational solution. Find $f$ .

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3

Find the triangle of the least area which can cover any triangle with sides not exceeding $1$ .

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Round 5

1

In a triangle $ABC$ , $\angle B = 70^o$ , $\angle C = 50^o$ . A point $M$ is taken on the side $AB$ such that $\angle MCB = 40^o$ , and a point $N$ is taken on the side $AC$ such that $\angle NBC = 50^o$ . Find $\angle NMC$ .

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2

Let $m$ be the greatest number such that for any set of complex numbers having the sum of all modulus of all the elements $1$ , there exists a subset having the modulus of the sum of the elements in the subset greater than $m$ . Prove that $\frac14 \le m \le \frac12.$
(Optional Task for 3p) Find a smaller value for the RHS.

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3

Prove that if the positive reals $a, b, c$ have sum $1$ then the following inequality holds
$(ab)^{ \frac54} + (bc)^{\frac54} + (ca)^{\frac54} < \frac14 .$

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Round 6

1

Let $a, m$ be two positive integers, $a \ne 10^k$ , for all non-negative integers $k$ and $d_1, d_2, ... , d_m$ random decimal $^1$ digits with $d_1 > 0$ . Prove that there exists some positive integer $n$ for which the representation in the decimal base of the number $a^n$ begins with the digits $d_1, d_2, ... , d_m$ in this order.

$^1$ lesser or equal with $9$

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2

Given is a triangle $ABC$ and on its sides the triangles $ABM, BCN$ and $CAP$ are build such that $\angle AMB = 150^o$ , $AM = MB$ , $\angle CAP = \angle CBN = 30^o$ , $\angle ACP = \angle BCN = 45^o$ . Prove that the triangle $MNP$ is an equilateral triangle.

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3

Consider $(f_n)_{n\ge 0}$ the Fibonacci sequence, defined by $f_0 = 0$ , $f_1 = 1$ , $f_{n+1} = f_n + f_{n-1}$ for all positive integers $n$ . Solve the following equation in positive integers $nf_nf_{n+1} = (f_{n+2} - 1)^2.$ .

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Round 7

1

Prove that for every positive numbers $x, y, z$ the following inequality holds:
$\sqrt{4x^2 + 4x(y + z) + (y - z)^2} <\sqrt{4y^2 + 4y(z + x) + (z - x)^2}+\sqrt{4z^2 + 4z(x + y) + (x - y)^2}.$

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2

Consider the circles $\omega$ , $\omega_1$ , $\omega_2$ , where $\omega_1$ , $\omega_2$ pass through the center $O$ of $\omega$ . The circle $\omega$ cuts $\omega_1$ at $A, E$ and $\omega_2$ at $C, D$ . The circles $\omega_1$ and $\omega_2$ intersect at $O$ and $M$ . If A $D$ cuts $CE$ at $B$ and if $MN \perp BO$ , ( $N \in BO$ ) prove that $2MN^2 \le BM \cdot MO$ .

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3

For a set $S$ , let $|S|$ denote the number of elements in $S$ . Let $A$ be a set of positive integers with $|A| = 2001$ . Prove that there exists a set $B$ such that all of the following conditions are fulfilled:
a) $B \subseteq A$ ;
b) $|B| \ge 668$ ;
c) for any $x, y \in B$ we have $x + y \notin B$ .

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