Indonesia Regional MOalso know as provincial level, is a qualifying round for National Math Olympiad Year 2019 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. In the bag there are $7$ red balls and $8$ white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ... p2. Given a regular hexagon with a side length of $1$ unit. The area of the hexagon is ... p3. It is known that $r, s$ and $1$ are the roots of the cubic equation $x^3 - 2x + c = 0$. The value of $(r-s)^2$ is ... p4. The number of pairs of natural numbers $(m, n)$ so that $GCD(n,m) = 2$ and $LCM(m,n) = 1000$ is ... p5. A data with four real numbers $2n-4$, $2n-6$, $n^2-8$, $3n^2-6$ has an average of $0$ and a median of $9/2$. The largest number of such data is ... p6. Suppose $a, b, c, d$ are integers greater than $2019$ which are four consecutive quarters of an arithmetic row with $a <b <c <d$. If $a$ and $d$ are squares of two consecutive natural numbers, then the smallest value of $c-b$ is ... p7. Given a triangle $ABC$, with $AB = 6$, $AC = 8$ and $BC = 10$. The points $D$ and $E$ lies on the line segment $BC$. with $BD = 2$ and $CE = 4$. The measure of the angle $\angle DAE$ is ... p8. Sequqnce of real numbers $a_1,a_2,a_3,...$ meet $\frac{na_1+(n-1)a_2+...+2a_{n-1}+a_n}{n^2}=1$ for each natural number $n$. The value of $a_1a_2a_3...a_{2019}$ is .... p9. The number of ways to select four numbers from $\{1,2,3, ..., 15\}$ provided that the difference of any two numbers at least $3$ is ... p10. Pairs of natural numbers $(m , n)$ which satisfies $$m^2n+mn^2 +m^2+2mn = 2018m + 2019n + 2019$$are as many as ... p11. Given a triangle $ABC$ with $\angle ABC =135^o$ and $BC> AB$. Point $D$ lies on the side $BC$ so that $AB=CD$. Suppose $F$ is a point on the side extension $AB$ so that $DF$ is perpendicular to $AB$. The point $E$ lies on the ray $DF$ such that $DE> DF$ and $\angle ACE = 45^o$. The large angle $\angle AEC$ is ... p12. The set of $S$ consists of $n$ integers with the following properties: For every three different members of $S$ there are two of them whose sum is a member of $S$. The largest value of $n$ is .... p13. The minimum value of $\frac{a^2+2b^2+\sqrt2}{\sqrt{ab}}$ with $a, b$ positive reals is .... p14. The polynomial P satisfies the equation $P (x^2) = x^{2019} (x+ 1) P (x)$ with $P (1/2)= -1$ is .... p15. Look at a chessboard measuring $19 \times 19$ square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of $k$ coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of $k$ so that the game never ends for any initial square selection is ....
2019 Indonesia Regional
Part A
Part B
Problem 1. Given a cube $ABCD.EFGH$ with an edge with length 4 units and $P$ be the midpoint/center of side $EFGH$. If $M$ is the midpoint of $PH$, determine the length of segment $AM$. Problem 2. Find all reals $k$ such that the system of equations \begin{align*} a^2 + ab &= kb^2 \\ b^2 + bc &= kc^2 \\ c^2 + ca &= ka^2 \end{align*}have (a) real positive solution(s) $a, b, c$. Problem 3. Each cell of a checkerboard with size $m \times n$ is colored with either black or white, such that: (a) The number of black and white cells in each row are the same. (b) If a row intersects a column at some black cell, then said row and column contain the same number of black tiles. (c) If a row intersects a column at some white cell, then said row and column contain the same number of white tiles. Determine all possible values of $m$ and $n$ so that the coloring above can be done. Problem 4. Determine all nonnegative integers $k$ such that we can always find a noninteger positive real $x$ which satisfies \[ \lfloor x + k \rfloor^{\lfloor x + k \rfloor} = \lceil x \rceil^{\lfloor x \rfloor} + \lfloor x \rfloor^{\lceil x \rceil}. \] Problem 5. On a triangle $ABC$ where $AC > BC$, with $O$ as its circumcenter. Let $M$ be the circumcircle of $ABC$ such that $CM$ is the bisector of $\angle{ACB}$/ Let $\Gamma$ be the circle with diameter $CM$. The bisectors of $BOC$ and $AOC$ intersect $\Gamma$ respectively at points $P$ and $Q$. If $K$ is the midpoint of $CM$, prove that points $P, Q, O, K$ are concyclic.