MathDB

5

Part of 2018 China Team Selection Test

Problems(4)

Almost all n,i have k dividing nCi

Source: 2018 China TST Day 2 Q2

1/2/2018
Given a positive integer kk, call nn good if among (n0),(n1),(n2),...,(nn)\binom{n}{0},\binom{n}{1},\binom{n}{2},...,\binom{n}{n} at least 0.99n0.99n of them are divisible by kk. Show that exists some positive integer NN such that among 1,2,...,N1,2,...,N, there are at least 0.99N0.99N good numbers.
number theorybinomial coefficientsDivisibility
Sad Inequality

Source: 2018 China TST Day 4 Q2

1/21/2018
Given positive integers n,kn, k such that n≥4kn\ge 4k, find the minimal value λ=λ(n,k)\lambda=\lambda(n,k) such that for any positive reals a1,a2,…,ana_1,a_2,\ldots,a_n, we have ∑i=1naiai2+ai+12+⋯+ai+k2≤λ \sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}} \le \lambda Where an+i=ai,i=1,2,…,ka_{n+i}=a_i,i=1,2,\ldots,k
inequalities
Concurrent Lines Problem from a TST

Source: 2018 China TST 3 Day 2 Problem 5

3/27/2018
Let ABCABC be a triangle with ∠BAC>90∘\angle BAC > 90 ^{\circ}, and let OO be its circumcenter and ω\omega be its circumcircle. The tangent line of ω\omega at AA intersects the tangent line of ω\omega at BB and CC respectively at point PP and QQ. Let D,ED,E be the feet of the altitudes from P,QP,Q onto BCBC, respectively. F,GF,G are two points on PQ‾\overline{PQ} different from AA, so that A,F,B,EA,F,B,E and A,G,C,DA,G,C,D are both concyclic. Let M be the midpoint of DE‾\overline{DE}. Prove that DF,OM,EGDF,OM,EG are concurrent.
concurrentgeometrycircumcircle
The Modulus of a Polynomial are the Same

Source: 2018 China TST 4 Day 2 Problem 5

3/27/2018
Suppose the real number λ∈(0,1),\lambda \in \left( 0,1\right), and let nn be a positive integer. Prove that the modulus of all the roots of the polynomial f(x)=∑k=0n(nk)λk(n−k)xkf\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{k} are 1.1.
algebrapolynomial