MathNet 2016-part-03 — 184 problems (EN–VI)

Let $ABCD$ be a cyclic quadrilateral satisfying $AB = AD$ and $AB + BC = CD$.
Determine $\angle CDA$.

Find all $a \in \mathbb{R}$ for which there exists a function $f: \mathbb{R} \rightarrow \mathbb{R}$, such that
(i) $f(f(x))=f(x)+x$, for all $x \in \mathbb{R}$,
(ii) $f(f(x)-x)=f(x)+a x$, for all $x \in \mathbb{R}$.

King George has decided to connect the 1680 islands in his kingdom by bridges. Unfortunately the rebel movement will destroy two bridges after all the bridges have been built, but not two bridges from the same island.
What is the minimal number of bridges the King has to build in order to make sure that it is still possible to travel by bridges between any two of the 1680 islands after the rebel movement has destroyed two bridges?

Let $ABC$ be an acute triangle and let $H$ be its orthocenter. A point $G$ lies in the plane of the triangle such that $ABGH$ is a parallelogram. A point $I$ lies on the line $GH$ such that the line $AC$ bisects the segment $HI$. The line $AC$ intersects the circumcircle of triangle $GCI$ for a second time at the point $J$. Prove that $IJ = AH$.

Let a square scheme $2n \times 2n$, made of unit white squares be given. Allowed move is to change the color of three consecutive unit squares in a particular row or three consecutive unit squares in a particular column - unit square with white color goes to unit square with black color and vice versa.
Find all nonnegative integers, $n \ge 2$, for which with allowed moves the given square scheme can be colored like chess table.

Let $m$ and $n$ be positive integers such that $m\gt n$. Define $x_k = \frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by at least one odd prime.

Solve the equation $1 + x^2 + y^2 = \text{lcm}(x^2, y^2)$ in the set of natural numbers.

A magical square of dimensions $3 \times 3$ is a square with side $3$, consisting of $9$ unit squares, so that the real numbers written in the unit squares (one number in each unit square) satisfy the property: the sum of the numbers in the unit squares in any row is equal to the sum of the numbers in the unit squares in any column and is equal to the sum of the numbers in the unit squares in the two diagonals.
A rectangle of dimensions $m \times n$, $m \ge 3$, $n \ge 3$ is given, which consists of $mn$ unit squares. If in each unit square one number is written in such a way that each square of dimensions $3 \times 3$ is magical, then how many different numbers can be used at most to fill the rectangle?

Solve the equation $xyz + yzt + xzt + xyt = xyzt + 3$ in the set of natural numbers.

A segment $AB$ and its midpoint $K$ are given. An arbitrary point $C$, different from $K$, is chosen on the perpendicular to $AB$ through $K$. Let $N$ be the intersection of $AC$ and the line passing through $B$ and the midpoint of the segment $CK$. Let $U$ be the intersection of $AB$ with the line that passes through $C$ and the midpoint $L$ of the segment $BN$. Prove that the ratio of the areas of the triangles $CNL$ and $BUL$ doesn't depend on the choice of point $C$.

Let $n \ge 3$ and $a_1, a_2, \ldots, a_n$ be positive real numbers for which $\frac{1}{1+a_1^4} + \frac{1}{1+a_2^4} + \ldots + \frac{1}{1+a_n^4} = 1$ holds. Prove the inequality $a_1 a_2 \ldots a_n \ge (n-1)^{n/4}$.

A trapezoid $ABCD$ ($AB \parallel CD$, $AB \gt CD$) is circumscribed. The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.

Find all the triples of integers $(a,b,c)$ such that the number
$$
N = \frac{(a-b)(b-c)(c-a)}{2} + 2
$$
is a power of $2016$.
(A power of $2016$ is an integer of the form $2016^n$, where $n$ is a non-negative integer).

A $5 \times 5$ table is called *regular* if each of its cells contains one of four pairwise distinct real numbers, such that each of them occurs exactly once in every $2 \times 2$ subtable. The sum of all numbers of a *regular table* is called the *total sum* of the table. With any four numbers, one constructs all possible regular tables, computes their total sums and counts the distinct outcomes. Determine the maximum possible count.

Solve the equation
$$
x_1^4 + x_2^4 + \dots + x_{14}^4 = 2016^3 - 1
$$
in the set of integers.

Let $ABCD$ be a parallelogram and let $E$, $F$, $G$ and $H$ be the midpoints of the sides $AB$, $BC$, $CD$ and $DA$, respectively. If $BH \cap AC = I$, $BD \cap EC = J$, $AC \cap DF = K$ and $AG \cap BD = L$, then prove that the quadrilateral $IJKL$ is a parallelogram.

Let $x$, $y$, $z$ be positive real numbers. Prove that
$$
\sqrt{\frac{xy}{x^2 + y^2 + 2z^2}} + \sqrt{\frac{yz}{y^2 + z^2 + 2x^2}} + \sqrt{\frac{zx}{z^2 + x^2 + 2y^2}} \le \frac{3}{2}.
$$

Solve the equation $x + y^2 + (\text{gcd}(x, y))^2 = x y \cdot \text{gcd}(x, y)$ in the set of natural numbers.

Find the value of $\cot \left(\cot^{-1} 2 + \cot^{-1} 3 + \cot^{-1} 4 + \cot^{-1} 5\right)$.

Find the minimum value of $x^{2}+4 y^{2}-2 x$, where $x$ and $y$ are real numbers that satisfy $2 x+8 y=3$.