MathNet 2013

Prove that every polyhedron has two faces with the same number of vertices.

1) Let a,b be real numbers such that $a^n+b^n$ is an integer for n=1,2,3,4. Prove it is an integer for every $n\in\mathbb N^*$.

2) Is it true that if a,b are real and $a^n+b^n$ is an integer for n=1,2,3, then it is an integer for every $n\in\mathbb N^*$?

Find all pairs of positive rational numbers where the sum of the numbers in a pair is an integer and the sum of (multiplicative) inverses of the numbers in a pair is also an integer.

Gotham City is a large rectangle divided into smaller rectangular blocks by streets parallel to its sides. Assume there is at least one street and no street crosses the city from one side all the way to the other. Prove that some block does not touch the city's boundary.

Every positive rational number is colored either red or black, so that:
- x and x+1 have different colors;
- x and 1/x have the same color;
- 1 is red.

What color is $2012/2013$? (You are not asked to prove the existence of such a coloring.)

For each integer k≥2, let P(k) be its greatest prime divisor. Prove that infinitely many integers n satisfy $P(n)\lt P(n+1)\lt P(n+2)$.

The angles of a triangle are $22.5^\circ$, $45^\circ$ and $112.5^\circ$. Prove that inside this triangle there exists a point that is located on the median through one vertex, the angle bisector through another vertex and the altitude through the third vertex.

Let P and Q be real polynomials of degree n≥0. Suppose the coefficient of xⁿ in each is1 and P(P(x))=Q(Q(x)) for every real x. Prove P=Q.

A nine-digit number is called interesting if each digit from 1 to 9 appears exactly once, the digits 1 to 5 occur in order, but the digits 1 to 6 do not occur in order; for example 189236457.

How many interesting numbers are there?

The greatest common divisor of positive integers $a$, $b$, $c$ is $1$. It is known that $c$ divides $a + 2b$ and $a^2 - b^2$. Prove that $c$ also divides $a - b$.

Let $I$ be the incenter of triangle $ABC$. Let $R_A$, $R_B$ and $R_C$ be the radii of the circumcircles of triangles $BIC$, $CIA$ and $AIB$, respectively, and $R$ be the radius of the circumcircle of triangle $ABC$. Prove that $R_A + R_B + R_C \le 3R$.

Define a regular $n$-pointed star to be a union of $n$ line segments $P_{1}P_{2}$, $P_{2}P_{3}$, $\ldots$, $P_{n}P_{1}$ such that
- the points $P_{1}, P_{2}, \ldots, P_{n}$ are coplanar and no three of them are collinear;
- each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint;
- all of the angles at $P_{1}, P_{2}, \ldots, P_{n}$ are congruent;
- all of the $n$ line segments $P_{1}P_{2}, P_{2}P_{3}, \ldots, P_{n}P_{1}$ are congruent; and
- the path $P_{1}P_{2}\ldots P_{n}P_{1}$ turns counterclockwise at an angle less than $180^{\circ}$ at each vertex.
There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. Find all possible values of $n$ such that there are exactly 29 non-similar regular $n$-pointed stars.

Let I be the center of the incircle γ of triangle ABC and O its circumcenter. Let $\omega _A$ be the circle through B,C tangent to γ. Define $\omega _B$,$\omega _C$ similarly. Let A′ be the other intersection of $\omega _B$,$\omega _C$, and define B′,C′ similarly. Prove that AA′,BB′,CC′ concur at a point K collinear with O and I.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy
$$
\begin{aligned}
& f\left(\frac{\sqrt{3}}{3} x\right)=\sqrt{3} f(x)-\frac{2 \sqrt{3}}{3} x \\
& f(x) f(y)=f(x y)+f\left(\frac{x}{y}\right)
\end{aligned}
$$
for all $x, y \in \mathbb{R}$, with $y \neq 0$.

Find all values of $n$ for which there exists a convex cyclic non-regular polygon with $n$ vertices such that the measures of all its internal angles are equal.

a) Does there exist an integer $c$ and a polynomial $P(x)$ with integer coefficients for which $P(c) \neq c$, but $P(P(c)) = c$?

b) Does there exist an integer $c$ and a polynomial $P(x)$ with integer coefficients for which $P(c) \neq c$ and $P(P(c)) \neq c$, but $P(P(P(c))) = c$?

$ABC$ is a triangle, $H$ its orthocenter, $I$ its incenter, $O$ its circumcenter and $\omega$ its circumcircle. Line $CI$ intersects circle $\omega$ at point $D$ different from $C$. Assume that $AB = ID$ and $AH = OH$. Find the angles of triangle $ABC$.

Let $a$, $b$ and $c$ be positive real numbers such that $ab + bc + ca = 1$. Prove that
$$
\sqrt[4]{\frac{\sqrt{3}}{a} + 6\sqrt{3}b} + \sqrt[4]{\frac{\sqrt{3}}{b} + 6\sqrt{3}c} + \sqrt[4]{\frac{\sqrt{3}}{c} + 6\sqrt{3}a} \le \frac{1}{abc}.
$$
When does the equality hold?

Find the greatest and the least possible values of
$$\sqrt{4-a^2}+\sqrt{4-b^2}+\sqrt{4-c^2}$$
when a,b,c are strictly positive real numbers satisfying $a^2+b^2+c^2=6$.

Find all pairs of integers $(a, b)$ satisfying $(a+1)(b-1) = a^2 b^2$.