MathNet 2008-part-02 — 165 problems (EN–VI)

Show that the product of three consecutive integers is the sum of four integer cubes.

Let $a$, $b$ and $c$ be the sides of a triangle and $p = a + b + c$. Show that

a. $\frac{a}{b+c-a} + \frac{b}{c+a-b} + \frac{c}{a+b-c} \ge 3$;

b. $a^k + b^k \gt \frac{c^k}{2^{k-1}}$ for $k = 2, 3, 4, \dots$;

c. $a^k + b^k + c^k \ge \frac{p^k}{3^{k-1}}$ for $k = 2, 3, 4, \dots$.

Let $H$ be an arbitrary point on the altitude $CP$ of an acute triangle $ABC$. The lines $AH$ and $BH$ intersect $BC$ and $AC$ in $M$ and $N$ respectively. Show that $\angle NPC = \angle MPC$.

Let $x_1$ and $y_1$ be positive integers satisfying $x_1 + y_1 = 2^r$, where $r$ is a positive integer. If $x_1 \lt y_1$, let $x_2 = 2x_1$ and $y_2 = 2^r - x_2$. On the other hand, if $x_1 \gt y_1$, then we let $y_2 = 2y_1$ and $x_2 = 2^r - y_2$. Apply the same procedure to $(x_2, y_2)$, and so on. Show that the procedure will end in finitely many steps, i.e. there exists a $k$ such that $x_k = y_k$. (For example, let $(x_1, y_1) = (7, 57)$, with $x_1 + y_1 = 2^6$. Then $(x_2, y_2) = (14, 50)$, $(x_3, y_3) = (28, 36)$, $(x_4, y_4) = (56, 8)$, $(x_5, y_5) = (48, 16)$, and finally $(x_6, y_6) = (32, 32)$.)

For each permutation $x_1, x_2, \dots, x_{10}$ of $1, 2, \dots, 10$, compute
$$
|2x_1 - 3x_2| + |2x_2 - 3x_3| + |2x_3 - 3x_4| + \dots + |2x_{10} - 3x_1|.
$$
Let $S$ be the maximum possible value of this sum. Find the number of permutations attaining $S$.

Let $p$ be a prime number and $f(x)$ be a polynomial with rational coefficients and is of degree $p$. Furthermore (i) $f(m)$ is an integer for any integer $m$ and (ii) $p$ divides $f(p+1) - f(1)$. Find all such polynomials.

Let $S \subseteq \mathbb{R}$ be a set of real numbers. We say that a pair $(f, g)$ of functions from $S$ into $S$ is a Spanish Couple on $S$, if they satisfy the following conditions:
(i) Both functions are strictly increasing, i.e. $f(x) \lt f(y)$ and $g(x) \lt g(y)$ for all $x, y \in S$ with $x \lt y$;
(ii) The inequality $f(g(g(x))) \lt g(f(x))$ holds for all $x \in S$.
Decide whether there exists a Spanish Couple

a. on the set $S = \mathbb{N}$ of positive integers;
b. on the set $S = \{ a - 1 / b : a, b \in \mathbb{N} \}$.

For an integer $m$, denote by $t(m)$ the unique number in $\{1,2,3\}$ such that $m+t(m)$ is a multiple of $3$. A function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $f(-1)=0$, $f(0)=1$, $f(1)=-1$ and
$$
f\left(2^{n}+m\right)=f\left(2^{n}-t(m)\right)-f(m) \quad \text{ for all integers } m, n \geq 0 \text{ with } 2^{n} \gt m.
$$

Prove that $f(3p) \geq 0$ holds for all integers $p \geq 0$.

Let $a$, $b$, $c$, $d$ be positive real numbers such that
$$
a b c d = 1 \quad \text{ and } \quad a + b + c + d \gt \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}
$$
Prove that
$$
a + b + c + d \lt \frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{a}{d}
$$

Let $f: \mathbb{R} \rightarrow \mathbb{N}$ be a function which satisfies
$$
f\left(x+\frac{1}{f(y)}\right)=f\left(y+\frac{1}{f(x)}\right) \quad \text{ for all } x, y \in \mathbb{R}
$$
Prove that there is a positive integer which is not a value of $f$.

Prove that for any four positive real numbers $a$, $b$, $c$, $d$ the inequality
$$
\frac{(a-b)(a-c)}{a+b+c}+\frac{(b-c)(b-d)}{b+c+d}+\frac{(c-d)(c-a)}{c+d+a}+\frac{(d-a)(d-b)}{d+a+b} \geq 0
$$
holds. Determine all cases of equality.

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary.
Find the largest $n$ for which there exist $n$ boxes $B_{1}, \ldots, B_{n}$ such that $B_{i}$ and $B_{j}$ intersect if and only if $i \not \equiv j \pm 1(\bmod n)$.

For every positive integer $n$ determine the number of permutations $(a_{1}, a_{2}, \ldots, a_{n})$ of the set $\{1,2, \ldots, n\}$ with the following property:
$$
2(a_{1}+a_{2}+\cdots+a_{k}) \quad \text{is divisible by} \ k \quad \text{for} \ k=1,2, \ldots, n.
$$

In the coordinate plane consider the set $S$ of all points with integer coordinates. For a positive integer $k$, two distinct points $A, B \in S$ will be called $k$-friends if there is a point $C \in S$ such that the area of the triangle $A B C$ is equal to $k$. A set $T \subset S$ will be called a $k$-clique if every two points in $T$ are $k$-friends. Find the least positive integer $k$ for which there exists a $k$-clique with more than 200 elements.

Let $S=\{x_{1}, x_{2}, \ldots, x_{k+\ell}\}$ be a $(k+\ell)$-element set of real numbers contained in the interval $[0,1]$; $k$ and $\ell$ are positive integers. A $k$-element subset $A \subset S$ is called nice if
$$
\left|\frac{1}{k} \sum_{x_{i} \in A} x_{i}-\frac{1}{\ell} \sum_{x_{j} \in S \backslash A} x_{j}\right| \leq \frac{k+\ell}{2 k \ell} .
$$
Prove that the number of nice subsets is at least $\frac{2}{k+\ell}\binom{k+\ell}{k}$.

Given trapezoid $A B C D$ with parallel sides $A B$ and $C D$, assume that there exist points $E$ on line $B C$ outside segment $B C$, and $F$ inside segment $A D$, such that $\angle D A E=\angle C B F$. Denote by $I$ the point of intersection of $C D$ and $E F$, and by $J$ the point of intersection of $A B$ and $E F$. Let $K$ be the midpoint of segment $E F$; assume it does not lie on line $A B$.
Prove that $I$ belongs to the circumcircle of $A B K$ if and only if $K$ belongs to the circumcircle of $C D J$.

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be points in $ABCD$ such that $PQDA$ and $QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $E$ on the line segment $PQ$ such that $\angle PAE = \angle QDE$ and $\angle PBE = \angle QCE$. Show that the quadrilateral $ABCD$ is cyclic.

Let $k$ and $n$ be integers with $0 \leq k \leq n-2$. Consider a set $L$ of $n$ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $I$ the set of intersection points of lines in $L$. Let $O$ be a point in the plane not lying on any line of $L$.
A point $X \in I$ is colored red if the open line segment $OX$ intersects at most $k$ lines in $L$. Prove that $I$ contains at least $\frac{1}{2}(k+1)(k+2)$ red points.

There is given a convex quadrilateral $A B C D$. Prove that there exists a point $P$ inside the quadrilateral such that
$$
\angle P A B+\angle P D C=\angle P B C+\angle P A D=\angle P C D+\angle P B A=\angle P D A+\angle P C B=90^{\circ} \tag{1}
$$
if and only if the diagonals $A C$ and $B D$ are perpendicular.

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a, b, c$ are integers (not necessarily positive) satisfying the equations
$$
a^{n}+p b=b^{n}+p c=c^{n}+p a,
$$
then $a=b=c$.