MathNet 2015-part-05 — 191 problems (EN–VI)

Four families have two children each and all eight children are born after the year 1989. All four youngest siblings are born in the same year, and the sum of the digits of the year is equal to the product of the non zero digits. The differences of the ages between the siblings of each family is a perfect square. Find the birth years of the four eldest siblings in each family, given that their ages are all different.

Find the smallest positive integer which has exactly $2015$ positive divisors.

A positive integer $n \ge 2$ is lucky if $n^2$ can be represented as a sum of $n$ consecutive positive integers. Prove that:

a) the number $7$ is lucky,

b) the number $10$ is not lucky,

c) the product of any two lucky numbers is a lucky number.

Eight numbers, all zero, are written on a blackboard. A *move* consists in randomly selecting four of them, $a$, $b$, $c$, $d$, and replacing them by $a+3$, $b+3$, $c+2$ and $d+1$, respectively.

a) What is the smallest number of moves after which on the blackboard can appear eight consecutive numbers?

b) Is there a sequence of moves after which all the numbers on the blackboard are equal to $2015$?

Find all positive integers that have exactly $8$ positive divisors, among which three are primes of the form $a$, $\overline{bc}$ and $\overline{cb}$, given that $a + \overline{bc} + \overline{cb}$ is a perfect square and $a, b, c$ are digits, with $b \lt c$.

a) Find all positive integers $a$ for which
$$
\frac{1}{4} \lt \frac{1}{a+1} + \frac{1}{a+2} + \frac{1}{a+3} \lt \frac{1}{3}.
$$

b) Prove that for any integer $p \ge 2$ there exist $p$ consecutive positive integers $a_1, a_2, \dots, a_p$ such that
$$
\frac{1}{p+1} \lt \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_p} \lt \frac{1}{p}.
$$

Let $x$, $y$, $n$ be positive integers such that
$$
n = \frac{x^2 - 1}{2} = \frac{y^2 - 1}{3}.
$$

Prove that $n = y^2 - x^2$ and $20$ divides $n$.

Given a triangle $ABC$ with $m(\angle A) = 90^\circ$, $AC \lt AB$, consider on the rays $BA$ and $AC$ points $E$ and $D$ respectively, such that $A \in (BE)$, $C \in (AD)$, $AE = AC$ and $AD = AB$. Denote by $M$ and $N$ the midpoints of $[BC]$ and $[DE]$ respectively, and let $\{R\} = EC \cap BD$. Show that $MN = RA$.

Determine $n \in \mathbb{N}$ such that the numbers $n+8$, $2n+1$, $4n+1$ are all perfect cubes.

Consider a triangle $ABC$, such that $\angle B = 90^\circ$. Denote by $I$ the in-center and let $F$, $D$ and $E$ be the points where the incircle touches sides $[AB]$, $[BC]$, and $[AC]$ respectively. If $CI \cap EF = \{M\}$ and $DM \cap AB = \{N\}$, show that:

a) $AI = ND$;

b) $FM = \frac{EI \cdot EM}{EC}$.

Find the distinct primes $p, q, r$ and $s$ satisfying
$$
1 - \frac{1}{p} - \frac{1}{q} - \frac{1}{r} - \frac{1}{s} = \frac{1}{pqrs}.
$$

Consider a triangular pyramidal frustum $ABCA'B'C'$. Points $D \in (AA')$, $E \in (BB')$ and $F \in (CC')$ are such that the planes $(AEF)$ and $(DB'C')$ are parallel. Prove that the planes $(A'EF)$ and $(DBC)$ are also parallel.

Denote by $p(a)$ the first digit of the natural number $a$. Show that each of the sets
$$
A = \{n \in \mathbb{N} \mid p(5^n) - p(2^n) \gt 0\}, \quad B = \{n \in \mathbb{N} \mid p(5^n) - p(2^n) \lt 0\}
$$
has infinitely many elements.

In a regular tetrahedron $ABCD$ consider planes that are parallel to its faces such that each edge is divided into 6 equal segments. These planes determine, on the edges of the tetrahedron, on its faces and in its interior a set consisting of 80 points of intersection. Denote this set by $V$.
Find the maximum number of elements of a subset $W$ of the set $V \cup \{A, B, C, D\}$, having the property: any three points of $W$ are not collinear and the plane generated by them is neither parallel with any of the faces of the tetrahedron $ABCD$, nor contains one of his faces.
Mihai Monea

Prove that the set $\{\sqrt{1}, \sqrt{2}, \sqrt{3}, \dots, \sqrt{2015}\}$ does not contain a non-constant arithmetic sequence of length 45.

A quadratic function $f$ sends any interval $I$ of length 1 to an interval $f(I)$ of length at least 1.
Prove that for any interval $J$ of length 2, the length of the interval $f(J)$ is at least 4.

Let $P$ be a point in the interior of the triangle $ABC$. The lines $AP$, $BP$, $CP$ intersect $BC$, $AC$, $AB$ at $A_1$, $B_1$, $C_1$, respectively. Given that
$$
s(PBA_1) + s(PCB_1) + s(PAC_1) = \frac{1}{2}s(ABC),
$$
(where $s(XYZ)$ denotes the area of $XYZ$) prove that $P$ lies on one of the medians of triangle $ABC$.

Let $a, b, c, d \ge 0$ be real numbers such that $a + b + c + d = 1$. Prove that
$$
\sqrt{a + \frac{(b-c)^2}{6} + \frac{(c-d)^2}{6} + \frac{(d-b)^2}{6}} + \sqrt{b} + \sqrt{c} + \sqrt{d} \le 2.
$$

Find all triples $(a, b, c)$ of non-zero complex numbers with equal absolute values, for which
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{a} + 1 = 0.
$$

Let $k \ge 1$, be a positive integer, let $p_1, p_2, \dots, p_k$ be distinct primes and denote $n = p_1p_2\dots p_k$. For a function $f : \{1, 2, \dots, n\} \to \{1, 2, \dots, n\}$, define $p(f) = f(1)f(2)\dots f(n)$.

a) Determine the number of functions $f$ such that $p(f)$ divides $n$.

b) For $n=6$, determine the number of functions $f$ such that $p(f)$ divides $36$.