Mathematics

# Hong Kong Team Selection Test 1

#### Morning

I gone out on morning and bought myself a notebook, that’s all. Just do something random. The selection test begins at two. I went to Sha Tin at one. Just memorizing some stupid formulas about complex number, and turn out useless.

#### Problems

here you go

1. In $\delta ABC$, let AD be the angle bisector of angle BAC, with D on BC. The perpendiculars from B to AD intersect the circumcircle of $\delta ABD$ at B and E. Prove that E,A and the circumcenter O of $ABC$ are colinear.
2. In a committee there are n members. Each pair of members are either friends or enemies. Each committee member has exactly three enemies. It is also known that for each committee member, an enemy of his friend is automatically his own enemy. Find all possible values of n.
3. Let f(x) be a monic cubic polynomial with f(0)=-64, and all roots of f(x) are non-negative real numbers. Find the largest possible value of f(-1).
4. Consider a number with 2016 digits formed by 1,2,3 and 4. Find the number of those number containing an even number of digit 1.
5. Find the digit after the decimal point of the number $\frac{1}{1009}+\frac{1}{1010}+...+\frac{1}{2016}$.
6. Given two infinite sequence ${a_n}$ and ${b_n}$ of real numbers satisfying $a_n+b_n=\frac{a_n+b_n}{2}$ and $a_{n+1}b_{n+1}=a_nb_n$ for all $n>1$. Suppose $b_{2016}=1$ and $a_1>0$. Find all possible value of $a_1$.

#### Battlefield and solutions

First, according to a student, it is the easiest, but the teacher said no… And I would choose the worse one. I finished two problems, and remain four made some nontrivial progress… Secondly, I have a big complaint is although there are 6 problems, none of one is a Number theory! As I have a strong strength (maybe) of Number theory…

First the geometry one, looking at the teacher draft is very easy, though I made nothing, I am so suck at geometry, I tried complex bash and realized that I forgot those formulas. And I am lazy now… There are several solved that one.

Question two and three are solved and written within an hour, 2 problems are my goal. And the thing is according to a student, I fell a trap… Problem 3 is just stupid polynomial mixed a trivial inequality, Problem 2 are solved using some equivalence relation, oh my god that one I am little proud of myself… I spotted the equivalence relation behind, friends are related,and I use the partition of set, and boom easily done.

Problem four is just confusing, first the original problem is not like the above, the problem is actually stated wrongly, and later clarified (very late actually). My (mostly one) attempted by just writing out the formula, and don’t know how to compute it, the teacher said it can see by smaller cases, oh holy chloe I am stupid… However the official solution is attained by induction, make 2016 arbitary, actually I thought if I focus more, I may finished without regret.

Problem 5 is hardest one, the teacher solution is bash, Mine is like the official solution, try bonding the sum, I use AM-HM perfectly (because the answer is 6), but I use I really stupid trivial bond attain a upperbond 9, I try to improve but no inequality appear in my head… While the official solution is being smart.

Problem 6 is taking time, my thought is to construct two sequence, one is sum and one is product, and find recurrence of them, at this moment I am same to the teacher’s one, and alas, I failed due to some careless mistake I suppose. My friend Chan Ho spot the sum of root and product of root, and able to bond by AM-GM.

#### Regret

They are not much of my regret, at least I have a way to proceed studying, First I must studied the chapter 1-3 of the geometry book of evan chen. And able to use complex bash and barycentric coordinates easily. I think I should attain the goal when phrase 2 end. The another goal is to study combinatorics, the resource is likely too hard for me. And the time, are way too less for me. Also next time I must buy some snacks and more drink for me. And more focus, please?

#### Night

Got a bad night, of course except typing this blog, I am now very scared I not always think time flies too much, the summer holiday is too short, three weeks left! Are you serious? Too much math and too much homework…

I am sorry that I can’t attain 1000 words.