Indonesia Regional MOalso know as provincial level, is a qualifying round for National Math Olympiad Year 2016 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. Suppose $a, b, c$ are three natural numbers that satisfy $2^a + 2^b + 2^c = 100$. The value of $a + b + c$ is ... . p2. A function f has the property $f(65x+1) = x^2-x+1$ for all real numbers $x$. The value of $f (2016)$ is ... . p3. Three different numbers $a, b, c$ will be chosen one by one at random from $1, 2, 3, 4,..., 10$ with respect to the order. The probability that $ab+c$ is even is ... p4. Point $P$ is a point on the convex quadrilateral $ABCD$ with $PA = 2$, $PB = 3$, $PC = 5$, and $PD = 6$. The maximum area of the quadrilateral $ABCD$ is... p5. If $0 < x <\frac{\pi}{2}$ and $4 \tan x + 9 \cot x \le 12$, then the possible values of $\sin x$ are ... . p6. For every natural number $n$, let $s(n)$ represent the result of the sum of the digits of n in decimal writing. For example, $s(2016) = 2+0+1+6 = 9$. The result of the sum of all natural numbers n such that $n + s(n) = 2016$ is ... p7. Among $30$ students, $15$ students liked athletics, $17$ students liked basketball, and $17$ students love chess. Students who enjoy athletics and basketball as much as students who love basketball and chess. A total of $8$ students enjoy athletics and chess. Students who like basketball and chess are twice as much as students who love happy all three. Meanwhile, $4$ students were not happy with any of the three. From the $30$ students, three students were chosen randomly. Probability of each each student who is selected only likes chess or basketball is ... p8. Given a cube $ABCD:EFGH$ with side length $5$. Points $I$ and $J$ any point on $BF$ with $IJ = 1$. Any point $K$ and $L$ on $CG$ with $KL = 2$. Ants move from $A$ to $H$ on the path $AIJKLH$. The shortest path is... p9. The number of triple prime numbers $(p, q, r)$ that satisfies $15p+7pq+qr = pqr$ is ... p10. If $x^2 + xy + 8x =-9$ and $4y^2 + 3xy + 16y =-7$, then the value of $x + 2y$ maybe is ... p11. The lengths of the sides of a triangular pyramid are all integers. Its five ribs are $14$ long each; $20$, $40$, $52$, and $70$. The number of possible lengths of the sixth side is... p12. A chess player plays a minimum of once every day for seven day with a total of $m$ matches. Maximum value of $m$ such that there are two or more consecutive days with a total of four matches is ... p13. Mr. Adi's house has a broken water meter, that cannot show numbers $3$ and $9$. For example, the number shown on the meter after $22$ is $24$ and also the number shown after $28$ is $40$. For example, in one month, Pak Adi's water meter shows $478$ m$^3$. The actual loss of Mr. Adi because the broken meter is ... m$^3$. p14. The result of the sum of all the real numbers $x$ that fulfil $\lfloor 8x-1008 \rfloor + \lfloor x \rfloor = 2016$ is ... . p15. Suppose $a_1, a_2, ..., a_{120}$ are $120$ permutations of the word MEDAN which is sort alphabetically like in a dictionary, for example $a_1 = ADEMN$, $a_2 = ADENM$, $a_3 = ADMEN$, and so on. The result of the sum of all $k$ indexes until the letter $A$ is the third letter in the permutation $a_k$ is ... p16. Suppose $ABCDE$ is a regular pentagon with area $2$. The points $P, Q, R, S, T$ are the intersection of the diagonals of the pentagon $ABCDE$ such that $PQRST$ is a regular pentagon. If the area of $PQRST$ is written in the form $a-\sqrt{b}$ with $a$ and $b$ natural numbers, then the value of $a + b$ is ... p17. Triangle $ABC$ has a circumcircle of radius $1$. If the two medians of triangle ABC each have a length of $1$, then the perimeter of the triangle $ABC$ is .... p18. Sequence $x_0, x_1, x_2, ... , x_n$ is defined as $x_0 = 10$, $_x1 = 5$, and $x_{k+1} = x_{k-1}-\frac{1}{x_k}$ for $k = 1, 2, 3, ..., n-1$ and we get $x_n = 0$. The value of $n$ is ... p19. In a soccer tournament involving $n$ teams, each team plays against the other team exactly once. In one game, $3$ points will be awarded to the winning team and $0$ points to the losing team, while $ 1$ point is given to each team when the match ends in a draw. After the match ends, only one team gets the most points and only that team gets the number of wins at least. The smallest value of $n$ so that this is possible ... p20. The sequence of non-negative numbers $a_1, a_2, a_3, ... $ is defined with $a_1 = 1001$ and $an+2 = |a_{n+1}-a_n|$ for $n\ge 1$. If it is known that a_2 < 1001 and $a_{2016} = 1$, so the number of possible values of $a_2$ is...
2016 Indonesia Regional
Part A
Part B
p1. Let $a$ and $ b$ be different positive real numbers so that $a +\sqrt{ab}$ and $b +\sqrt{ab}$ are rational numbers. Prove that $a$ and $ b$ are rational numbers. p2. Find the number of ordered pairs of natural numbers $(a, b, c, d)$ that satisfy $$ab + bc + cd + da = 2016.$$ p3. For natural numbers $k$, we say a rectangle of size $1 \times k$ or $k\times 1$ as strips. A rectangle of size $2016 \times n$ is cut into strips of all different sizes . Find the largest natural number $n\le 2016$ so we can do that. Note: $1\times k$ and $k \times 1$ strips are considered the same size. p4. Let $PA$ and $PB$ be the tangent of a circle $\omega$ from a point $P$ outside the circle. Let $M$ be any point on $AP$ and $N$ is the midpoint of segment $AB$. $MN$ cuts $\omega$ at $C$ such that $N$ is between $M$ and $C$. Suppose $PC$ cuts $\omega$ at $D$ and $ND$ cuts $PB$ at $Q$. Prove $MQ$ is parallel to $AB$. p5. Given a triple of different natural numbers $(x_0, y_0, z_0)$ that satisfy $x_0 + y_0 + z_0 = 2016$. Every $i$-hour, with $i\ge 1$, a new triple is formed $$(x_i,y_i, z_i) = (y_{i-1} + z_{i-1} x_{i-1}, z_{i-1} + x_{i-1}-y_{i-1}, x_{i-1} + y_{i-1}- z_{i-1})$$Find the smallest natural number $n$ so that at the $n$-th hour at least one of the found $x_n$, $y_n$, or $z_n$ are negative numbers.