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Indonesia Regional MO 2015 Part B

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October 3, 2021
algebrageometrycombinatoricsnumber theorysystem of equations

Problem Statement

p1. Let X={1,2,3,4,5}X = \{1, 2, 3, 4, 5\}. Let F={A1,A2,A3,...,Am}F = \{A_1,A_2,A_3, ..., A_m\}, with Ai⊆XA_i\subseteq X and 22 members in AiA_i , for i=1,2,...,mi = 1,2,...,m. Determine the minimum mm so that for any B⊆XB\subseteq X, with B B having 33 members, there is a member of FF contained in B B. Prove your answer.
p2. Find all triples of real numbers (x,y,z)(x, y, z) that satisfy the system of equations:
(x+1)2=x+y+2(x + 1)^2 = x + y + 2 (y+1)2=y+z+2(y + 1)^2 = y + z + 2 (z+1)2=z+x+2(z + 1)^2 = z + x + 2
[url=https://artofproblemsolving.com/community/c6h2371594p19389035]p3. Given the isosceles triangle ABCABC, where AB=ACAB = AC. Let DD be a point in the segment BCBC so that BD=2DCBD = 2DC. Suppose also that point PP lies on the segment ADAD such that: ∠BAC=∠BPD\angle BAC = \angle BP D. Prove that ∠BAC=2∠DPC\angle BAC = 2\angle DP C.
p4. Suppose p1,p2,..,pnp_1, p_2,.., p_n arithmetic sequence with difference b>0b > 0 and pip_i prime for each i=1,2,...,ni = 1, 2, ..., n. a) If p1>np_1 > n, prove that every prime number pp with p≤np\le n, then pp divides bb . b) Give an example of an arithmetic sequence p1,p2,..,p10p_1, p_2,.., p_{10}, with positive difference and pip_i prime for i=1,2,...,10i = 1, 2, ..., 10.
p5. Given a set consisting of 2222 integers, A={±a1,±a2,...,±a11}A = \{\pm a_1, \pm a_2, ..., \pm a_{11}\}. Prove that there is a subset SS of AA which simultaneously has the following properties: a) For each i=1,2,...,11i = 1, 2, ..., 11 at most only one of the aia_i or −ai-a_i is a member of SS b) The sum of all the numbers in SS is divided by 20152015.
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