MathNet 2015-part-02 — 191 problems (EN–VI)

Let $\mathbb{N} = \{1, 2, 3, \dots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n - 1)^2 \lt f(n)f(f(n)) \lt n^2 + n$ for every positive integer $n$.

Let $ABC$ be an acute-angled triangle with altitudes $AD$, $BE$, and $CF$. Let $H$ be the orthocentre, that is, the point where the altitudes meet. Prove that
$$
\frac{AB \cdot AC + BC \cdot BA + CA \cdot CB}{AH \cdot AD + BH \cdot BE + CH \cdot CF} \le 2.
$$

On a $(4n + 2) \times (4n + 2)$ square grid, a turtle can move between squares sharing a side. The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of $n$, what is the largest positive integer $k$ such that there must be a row or column that the turtle has entered at least $k$ distinct times?

Let $ABC$ be an acute-angled triangle with circumcenter $O$. Let $\Gamma$ be a circle with centre on the altitude from $A$ in $ABC$, passing through vertex $A$ and points $P$ and $Q$ on sides $AB$ and $AC$. Assume that $BP \cdot CQ = AP \cdot AQ$. Prove that $\Gamma$ is tangent to the circumcircle of triangle $BOC$.

Let $p$ be a prime number for which $\frac{p-1}{2}$ is also prime, and let $a$, $b$, $c$ be integers not divisible by $p$. Prove that there are at most $1+\sqrt{2 p}$ positive integers $n$ such that $n\lt p$ and $p$ divides $a^{n}+b^{n}+c^{n}$.

Determine positive integer $n$ such that the sum of his two smallest divisors is $6$ and the sum of his two largest divisors is $1122$.

Let $I$ be the incentre of the acute triangle $ABC$ and let $|AC| \gt |BC|$. The angle bisector and the altitude from vertex $C$ close an angle of $10^\circ$. If $\angle AIB = 120^\circ$, determine the angles of the triangle $ABC$. (Ilko Brnetić)

Does there exist a positive integer $n$ such that $n^2 + 2n + 2015$ is a perfect square?

The sum of squares of all solutions of the equation $x^4 + a x^2 + b = 0$ is $32$, and the product of all solutions of that equation is $4$. Determine $a$ and $b$. (Tamara Srnec)

Let $ABCD$ be a quadrilateral such that $|AB| = 6$, $|BC| = 9$, $|CD| = 18$ and $|AD| = 5$ hold. Determine the length of the diagonal $AC$ if it is known that it is a positive integer. (Andrea Aglić-Aljinović)

For any two positive real numbers $x$ and $y$ prove
$$
\log^2(xy) \ge \log(x^2)\log(y^2).
$$

Determine all values that the expression
$$
\frac{1 + \cos^2 x}{\sin^2 x} + \frac{1 + \sin^2 x}{\cos^2 x},
$$
can attain, where $x$ is a real number.

Determine the largest positive integer $n$ such that
$$
n + 5 \mid n^4 + 1395.
$$

Let $ABCD$ be a tetrahedron such that $\angle BAC = \angle CAD = \angle DAB = 90^\circ$, $|AD| = 2\sqrt{2}$ and $|AB| = |AC| = 3$ holds. Determine the radius of the inscribed sphere of the tetrahedron. (Mea Bombardelli)

There are ten white, and one red, blue, green, yellow and purple card. White cards are identical. On exactly one side of each card is the sign $X$. In how many ways is it possible to put the cards one on another such that no two cards face each other with the side having the sign $X$?

Ten chairs are arranged around a round table and marked with numbers $1$ to $10$ successively (in such a way that chairs $1$ and $10$ are also adjacent), and a knight is sitting in each chair. In the beginning, every knight has an even number of coins. Simultaneously, each knight gives half of his coins to his left neighbour, and the other half to his right neighbour. After that, the knight sitting in chair $1$ has $22$ coins, and each succeeding knight has two more coins, up until the knight in chair $10$ that has $40$ coins.

How many coins did the knight that ended up with $36$ coins have in the beginning? (Hong Kong)

Prove that there does not exist a positive integer $n$ such that $7^n - 1$ is divisible by $6^n - 1$.

The first $n$ positive integers are written on a board ($n \ge 3$). Ante repeats the following procedure: first he chooses two numbers on the board, and then he increases them both by the same arbitrary positive integer.
Determine all positive integers $n$ such that Ante can, by repeating this procedure, achieve that all numbers on the board are equal. (Ilko Brnetić)

Circles $k_1$ and $k_2$ intersect in points $A$ and $B$. Line $l$ intersects circle $k_1$ in points $C$ and $E$, and circle $k_2$ in points $D$ and $F$ in such a way that $D$ is between $C$ and $E$, and $E$ is between $D$ and $F$. Lines $CA$ and $BF$ intersect in point $G$, and lines $DA$ and $BE$ in point $H$.
Prove that $CF \parallel HG$.

Let $a$, $b$, $c$ and $d$ be four pairwise distinct real numbers. If $a$ and $b$ are solutions of the equation $x^2 - 10c x - 11d = 0$, and $c$ and $d$ are solutions of the equation $x^2 - 10a x - 11b = 0$, determine the sum $a + b + c + d$.