MathDB

3

Part of 2017 Korea Winter Program Practice Test

Problems(3)

Uniform number of distinct roots

Source: 2017 Korean Winter Program Practice Test 1 Day 1 #3

1/18/2017
Do there exist polynomials f(x)f(x), g(x)g(x) with real coefficients and a positive integer kk satisfying the following condition? (Here, the equation x2=0x^2 = 0 is considered to have 11 distinct real roots. The equation 0=00 = 0 has infinitely many distinct real roots.)
For any real numbers a,ba, b with (a,b)≠(0,0)(a,b) \neq (0,0), the number of distinct real roots of af(x)+bg(x)=0a f(x) + b g(x) = 0 is kk.
algebrapolynomial
Intersection satisfying a symmetric condition

Source: 2017 Korea Winter Program Practice Test 1 Day 2 #3

1/21/2017
Let △ABC\triangle ABC be a triangle with ∠A≠60∘\angle A \neq 60^\circ. Let IB,ICI_B, I_C be the B,CB, C-excenters of triangle ABCABC, let B′B^\prime be the reflection of BB with respect to ACAC, and let C′C^\prime be the reflection of CC with respect to ABAB. Let PP be the intersection of ICB′I_C B^\prime and IBC′I_B C^\prime. Denote by PA,PB,PCP_A, P_B, P_C the reflections of the point PP with respect to BC,CA,ABBC, CA, AB. Show that the three lines APA,BPB,CPCA P_A, B P_B, C P_C meet at a single point.
geometryTST
Inductive 0/1 sequence

Source: 2017 Korea Winter Program Practice Test 2 #7

8/14/2019
For a number consists of 00 and 11, one can perform the following operation: change all 11 into 100100, all 00 into 11. For all nonnegative integer nn, let AnA_n be the number obtained by performing the operation nn times on 11(starts with 100,10011,10011100100,…100,10011,10011100100,\dots), and ana_n be the nn-th digit(from the left side) of AnA_n. Prove or disprove that there exists a positive integer mm satisfies the following:
For every positive integer ll, there exists a positive integer k≤mk\le m satisfyingal+k+1=a1, al+k+2=a2, …, al+k+2017=a2017a_{l+k+1}=a_1,\ a_{l+k+2}=a_2,\ \dots,\ a_{l+k+2017}=a_{2017}
combinatorics