MathNet 2005 — Part 02

Find all numbers $n$ that can be expressed in the form $n = k + 2\lfloor\sqrt{k}\rfloor + 2$, where $k$ is a nonnegative integer.

Let $I$ be the incenter of a triangle $ABC$, and $D$ be the point of intersection of $AI$ with the circumcircle of $ABC$. Let $M$ be the midpoint of $AD$, and $E$ be the point of the segment $BD$ such that $IE$ is perpendicular to $BD$. If $IB + IE = \frac{AD}{2}$, $ME$ is parallel to $AB$, and the point $M$ is inside the segment $AI$, determine the measure of the angles of the triangle $ABC$.

If you want to color each positive entry with a color using the largest possible number of colors so that the following condition is met: If, in decimal notation, the number $B$ can be obtained from the number $A$, eliminating $A$ from the consecutive equal digits (aa) or By suppressing $A$ four consecutive digits that form two equal and consecutive pairs (abab), then $A$ and $B$ are of the same color. For example, 8, 833 and 22811 must be the same color and 72, 676772 and 173329898 must be the same color.
Determine the largest number of colors that can be used.

Let $n$ be a positive integer such that there are $k$ positive divisors of $n$, $k \gt 1$, whose sum is a prime number. Show that the product of those $k$ divisors is less than or equal to $n^{k-1}$.

We consider all finite sequences of positive terms less than or equal to $3$ and sum greater than $100$. For such a sequence $\alpha$ we consider a subsequence whose sum $S$ differs as little as possible from $100$, and we define the defect of $\alpha$ by $|S-100|$. Find the maximum value of the defect when $\alpha$ runs through all the sequences being considered.

Hallar the largest positive entero not divisible by $10$ which is a multiple of some numbers that are obtained by suppressing the consecutive digits of their decimal writing, none of them in the first or last position.

A flea jumps over whole points on the number line. In his first move he jumps from point $0$ and lands at point $1$. Then, if in one movement the flea jumped from point $a$ and landed on point $b$, in the next movement it jumps from point $b$ and falls on one of the points $b + (b - a) - 1$, $b + (b - a)$, $b + (b - a) + 1$.

Show that if the flea has fallen twice on the point $n$, for $n$ positive integer, then it must have made at least $t$ movements, where $t$ is the smallest integer greater than or equal to $2\sqrt{n}$.

In the trapezoid $ABCD$, the sum of the bases $AB$ and $CD$ is equal to the diagonal $BD$. Let $M$ be the midpoint of $BC$ and $E$ be the symmetrical point of $C$ with respect to the line $DM$. Prove that $\angle AEB = \angle ACD$.

Determine if there are positive integers $x$, $y$, $z$ such that the product $(x + y)(y + z)(z + x)$ is equal to:


a) $6767$


b) $7676$


c) $6776$

In each case, if the answer is affirmative, find all the ordered triples $(x, y, z)$ that satisfy the condition.

Around a circle a certain number of zeros and the same number of ones have been written. It is known that there are exactly $99$ triples of consecutive numbers containing two or three zeros. Determine the minimum number of triples of consecutive numbers that contain two or three ones.

Let $a$, $b$, $c$, $d$ be four different elements of the set $\{1, 2, 3, \ldots, 2005\}$, so that the sum of each three of them is a multiple of the fourth. Determine the largest value that $a + b + c + d$ can take.

Let $p \gt 3$ be a prime number. Yes
$$
\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{(p-1)^2} = \frac{n}{m}
$$
where the greatest common divisor of $n$ and $m$ is $1$, show that $p^3$ divides $n$.

Given two positive integers $a$ and $b$, $(a \nabla b)$ denotes the remainder obtained by dividing $a$ by $b$. This residue is one of the numbers $0, 1, \ldots, b-1$. Find all pairs of numbers $(a, p)$ such that $p$ is prime and it holds that
$$
(a \nabla p) + (a \nabla 2p) + (a \nabla 3p) + (a \nabla 4p) = a + p.
$$

Let $ABC$ be a triangle such that when constructing the squares $ABB_1A_2$, $BCC_1B_2$ and $CAA_1C_2$ outside the triangle, the points $A$, $B$ and $C$ remain inside the triangles $A_1B_1C_1$ and $A_2B_2C_2$. Show that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ have the same area.

Given a positive integer $n$, $2n$ aligned points in a plane are considered $A_1, A_2, \dots, A_{2n}$. Each point is colored blue or red using the following procedure: On the given plane, $n$ circles with end diameters $A_i$ and $A_j$ are drawn, disjointed two by two. Each $A_k$, $1 \le k \le 2n$, belongs to exactly one circle. The points are colored so that the two points on the same circle have the same color.

Determine how many different colorations of the $2n$ points can be obtained by varying the $n$ circumferences and distribution of the two colors.

Paladino has $n$ cards numbered $1$ to $n$ and divides them into two groups ($n \gt 1$). A division is perfect if at least one of the groups contains two cards such that the sum of the numbers on those cards is equal to the square of a natural number. What is the smallest value of $n$ for which all divisions of the $n$ cards into two groups are perfect?

Let $k$ be a positive integer. Show that for all $n \gt k$ the following is true:
There exist convex figures $F_1, \dots, F_n$ and $F$ such that no subset of $k$ figures chosen from $F_1, \dots, F_n$ completely covers $F$, but every subset of $k+1$ figures chosen from $F_1, \dots, F_n$ completely covers $F$.

Alan must choose a number with 37 digits different from 0 and write it on the pizza. Next, Beto can erase some digits of Alan's number (not all). Beto's objective is that the new number that falls on the pizza is multiple of 271. Decide if Alan can choose the number so that Beto finds it impossible to achieve his objective.

Find all numbers of the form $11\ldots1$ that have a multiple of the form $10\ldots01$.

Let $ABC$ be a triangle whose angles $\angle CAB$ and $\angle ABC$ are greater than $45^\circ$. $PQRS$ is a square such that $P$ and $Q$ are inside the side $AB$ in the order $APQB$, $R$ is inside the side $BC$, $S$ is inside the side $CA$. Let $M$ and $N$ be the feet of the perpendiculars drawn from $P$ to the side $CB$ and from $Q$ to the side $CA$, respectively. If $H$ is the intersection of $PM$ and $QN$, show that $CH$ is perpendicular to $AB$.