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Indonesian Junior MO 2019 (Nationals), Day 1

Source:

November 11, 2021
functiongeometry3D geometryprobabilityindonesia juniorsIJNAMO

Problem Statement

Actually, this is an MO I participated in :) but it's really hard to get problems from this year if you don't know some people.
P1. Let ff be a function satisfying f(x+1)+f(x−1)=2f(x)f(x + 1) + f(x - 1) = \sqrt{2} f(x), for all reals xx. If f(x−1)=af(x - 1) = a and f(x)=bf(x) = b, determine the value of f(x+4)f(x + 4). We found out that this is the modified version of a problem from LMNAS UGM 2008, Senior High School Level, on its First Round. This is also the same with Arthur Engel's "Problem Solving Strategies" Book, Example Problem E2.
P2. The sequence of "Sanga" numbers is formed by the following procedure. i. Pick a positive integer nn. ii. The first term of the sequence (U1)(U_1) is 9n9n. iii. For k≥2k \geq 2, Uk=Uk−1−17U_k = U_{k-1} - 17. Sanga[r][r] is the "Sanga" sequence whose smallest positive term is rr. As an example, for n=3n = 3, the "Sanga" sequence which is formed is 27,10,−7,−24,−41,….27, 10, -7, -24, -41, \ldots. Since the smallest positive term of such sequence is 1010, for n=3n = 3, the sequence formed is called Sanga[10][10]. For n≤100n \leq 100, determine the sum of all nn which makes the sequence Sanga[4][4].
P3. The cube ABCD.EFGHABCD.EFGH has an edge length of 6 cm. Point RR is on the extension of line (segment) EHEH with EH:ER=1:2EH : ER = 1 : 2, such that triangle AFRAFR cuts edge GHGH at point PP and cuts edge DHDH at QQ. Determine the area of the region bounded by the quadrilateral AFPQAFPQ.
[url=https://artofproblemsolving.com/community/q1h2395046p19649729]P4. Ten skydivers are planning to form a circle formation when they are in the air by holding hands with both adjacent skydivers. If each person has 2 choices for the colour of his/her uniform to be worn, that is, red or white, determine the number of different colour formations that can be constructed.
P5. After pressing the start button, a game machine works according to the following procedure. i. It picks 7 numbers randomly from 1 to 9 (these numbers are integers, not stated but corrected) without showing it on screen. ii. It shows the product of the seven chosen numbes on screen. iii. It shows a calculator menu (it does not function as a calculator) on screen and asks the player whether the sum of the seven chosen numbers is odd or even. iv. Shows the seven chosen numbers and their sum and products. v. Releases a prize if the guess of the player was correct or shows the message "Try again" on screen if the guess by the player was incorrect. (Although the player is not allowed to guess with those numbers, and the machine's procedures are started all over again.) Kiki says that this game is really easy since the probability of winning is greater than 9090%. Explain, whether you agree with Kiki.
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