Indonesia Regional MOalso know as provincial level, is a qualifying round for National Math Olympiad Year 2018 Part APart B consists of 5 essay / proof problems, posted here Time: 90 minutes Rules$\bullet$ Write only the answers to the questions given. $\bullet$ Some questions can have more than one correct answer. You are asked to provide the most correct or exact answer to a question like this. Scores will only be given to the giver of the most correct or most exact answer. $\bullet$ Each question is worth 1 (one) point. $\bullet \bullet$ to be more exact: $\rhd$ in years 2002-08 time was 90' for part A and 120' for part B $\rhd$ since years 2009 time is 210' for part A and B totally $\rhd$ each problem in part A is 1 point, in part B is 7 points p1. The number of ordered pairs of integers $(a, b)$ so that $a^2 + b^2 = a + b$ is .. p2. Given a trapezoid $ABCD$, with $AD$ parallel to $BC$. It is known that $BD = 1$, $\angle DBA = 23^o$, and $\angle BDC = 46^o$. If the ratio $BC: AD = 9:5$ , then the length of the side $CD$ is ... p3. Suppose $a > 0$ and $0 < r_1 < r_2 < 1$ so that $a + ar_1 + ar_1^2 + ...$ and $a + ar_2 + ar_2^2+...$ are two infinite geometric series with the sum of $r_1$ and $r_2$, respectively. The value of $r_1 + r_2$ is .... p4. It is known that $S = \{10,11,12,...,N\}$. An element in $S$ is said to be trubus if the sum of its digits is the cube of a natural number. If $S$ has exactly $12$ trubus, then the largest possible value of $N$ is.... p5. The smallest natural number $n$ such that $\frac{(2n)!}{(n!)^2}$ to be completely divided by $30$ is .. p6. Given an isosceles triangle $ABC$ with $M$ midpoint of $BC$. Let $K$ be the centroid of triangle $ABM$. Point $N$ lies on the side $AC$ so that the area of the quadrilateral $KMCN$ is half of the area of triangle $ABC$. The value of $\frac{AN}{NC}$ is ... p7. In a box there are $n$ red marbles and $m$ blue marbles. Take $5$ marbles at a time. If the probability that $3$ red marbles and $2$ blue marbles are drawn is $\frac{25}{77}$ , then the smallest possible value of $m^2 + n^2$ is ... p8. Let P(x) be a non-constant polynomial with a non-negative integer coefficient that satisfies $P(10) = 2018$. Let $m$ and $M$ be the minimum and maximum possible values of $P(l)$, respectively). The value of $m + M$ is ... p9. A province consists of nine cities named $1, 2, 3, 4, 5, 6, 7, 8, 9$. From city $a$ there is a direct road to city $b$ if and only if $\overline{ab}$ and $\overline{ba}$ are two-digit numbers that are divisible by $3$. Two distinct cities $a_1$ and $a_n$ are said to be connected if there is a sequence of cities $a_1,a_2,...,a_{n-1},a_n$, so that there is a direct path from $a_i$ to $a_{i+ i}$ for each $i =1, ... , n -1$. The number of cities connected to city $4$ is ... p10. Given $37$ points as shown in the figure so that every two neighboring points are one unit apart. From each of the three distinct points a red triangle is drawn. The number of possible sidelengths of an equilateral red triangle is ... p11. A positive integer $k$ is taken at random with $k \le 2018$. The probability that $k^{1009}$ has a remainder of $2$ when divided by $2018$ is .... p12. Given the non-negative real numbers $a, b, c, d, e$ where $ab + bc + cd + de = 2018$. The minimum value of $a + b + c + d + e$ is ... p13. The number of subsets (including empty sets) of $X = \{1,2,3,... ,2017,2018\}$ which does not have two elements $x$ and $y$ so that $xy = 2018$ there are $m2^n$ with $m$ odd. The value of $m + n$ is ... p14. Let $S = \{ 1 ,2 ,..., n\}$. It is known that there are exactly $1001$ pairs $(a, b, c, d)$ with $a, b, c, d \in S$ and $a < b < c < d$ so that $a, b, c, d$ are arithmetic sequences. The value of $n$ is .... p15. The number of natural numbers $n$ such that $n^4 - 5n^3- 5n^2 + 4n + 10$ is a prime number is... p16. Point $M$ lies on the circumcircle of regular pentagon $ABCDE$. The greatest value that $\frac{MB + ME}{MA + MC + MD}$ might be is ... p17. For $x, y$ non-zero real numbers, the sum of the maximum and minimum values of $\frac{xy - 4y^2}{x^2 + 4y^2}$ is ... p18. An alien race has a unique language consisting of only two letters $X$ and $Z$. In this language, each word consists of at least one letter and no more than $11$ letters. For every two words, if the first and second words are written side by side then the result is not a word. For example, if $XXZ$ and $ZZZZX$ are words, then $XXZZZZZX$ is not a word. The maximum number of words in this language is ... 19. An acute triangle $ABC$ has integer sidelengths. It is known that $AC = BD$ where $D$ is a point on the line $BC$ such that $AD$ is perpendicular to $BC$. The smallest possible length of side $BC$ is ... p20. The largest natural number $n$ such that $$50\lfloor x \rfloor - \lfloor x \lfloor x \rfloor \rfloor = 100n - 27 \lceil x \rceil$$has a real solution $x$ is ...

p1. A number of $n$ students sit around a round table. It is known that there are as many male students as female students. If the number of pairs of $2$ people sitting next to each other is counted, it turns out that the ratio between adjacent pairs of the same sex and adjacent pairs of the opposite sex is $3:2$. Find the smallest possible $n$. p2. Let $a, b$, and $c$ be positive integers so that $$c=a+\frac{b}{a}-\frac{1}{b}.$$Prove that $c$ is the square of an integer. p3. Let $ \Gamma_1$ and $\Gamma_2$ be two different circles with the radius of same length and centers at points $O_1$ and $O_2$, respectively. Circles $\Gamma_1$ and $\Gamma_2$ are tangent at point $P$. The line $\ell$ passing through $O_1$ is tangent to $\Gamma_2$ at point $A$. The line $\ell$ intersects $\Gamma_1$ at point $X$ with $X$ between $A$ and $O_1$. Let $M$ be the midpoint of $AX$ and $Y$ the intersection of $PM$ and $\Gamma_2$ with $Y\ne P$. Prove that $XY$ is parallel to $O_1O_2$. p4. Let $a, b, c$ be positive real numbers with $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$. Prove that $$a+b+c+\frac{4}{1+(abc)^{2/3}}\ge 5$$ p5. On a chessboard measuring $200 \times 200$ square units are placed red or blue marbles so that each unit square has at most $ 1$ marble. Two marbles are said to be in a row if they are in the same row or column. It is known that for every red marble there are exactly $5$ blue marbles in a row and for every blue marble there are exactly $5$ red marbles in a row. Determine the maximum number of marbles possible on the chessboard.