PROBLEMS AND SOLUTIONS FROM WINTER SCHOOLS OF MATHEMATICS IN VIETNAM

Prove that when A moves on major arc BC of circle (O) (such that 4ABC is still a acute-angle triangle and AB 6= AC), then the tangent line in (ADE) at D and E intersect at a fixed point.[r]

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TRAN NAM DUNG (Chief Author)

VO THANH DAT – HOANG DINH HIEU – LUONG VAN KHAI – NGUYEN DUY TUNG

DO THUY ANH – DO TRAN NGUYEN HUY – PHAM THI HONG NHUNG – NGO HOANG ANH

PROBLEMS AND SOLUTIONS

FROM WINTER SCHOOLS OF MATHEMATICS

IN VIETNAM

PREFACE

“T HE MORE PEOPLE YOU GO WITH , THE FURTHER YOU REACH ”

Before Vietnamese Mathematics Olympiad (VMO), students in Vietnam join in some Winter Schools of Mathematics, which are held in many areas of our country In these schools, many well-known Mathematics teachers try to help them review all the necessary materials used in VMO, emphasize the most important theories, and give out mock examinations to familiarize the students with solving the problems faster Besides, students also receive advice from the coaches, who participated in VMO in the past and now come back to convey their experiences to the following generations All these activities are to give competitors the best preparation for their most important Mathematics Contest in their student lives

With hope that students who want to participate in VMO next year have a useful document to review, and international students have a good view about how Vietnamese competitors study Math and prepare for VMO, we publish the book “Problems and Solutions from Winter Schools of Mathematics in Vietnam” In this attachment, we provide full solutions to most of the exercises, but for some of them we just give simple instructions so that readers can yourselves find out the solutions

We really appreciate Mr Luu Ba Thang (Ha Noi University of Education), Mr Tran Quoc Luat (Ha Tinh High School for the Gifted, Ha Tinh Province), Mr Vu Nguyen Duy (Huynh Man Dat High School for the Gifted, Kien Giang Province) for all their support We also would like

to thank our brother Mr Lê Phúc Lữ (FPT Software, Hồ Chí Minh City) for providing us the LaTeX template, our friends Đỗ Hà Ngọc Anh (Lê Quý Đôn High School for the Gifted, Bà Rịa – Vũng Tàu Province) for correcting all our grammar mistakes, and Lâm Quang Quỳnh Anh (VNU-HCM High School for the Gifted) for designing the cover page

This is the first time we have ever published a book in English, so we may have some mistakes in this document We really want to receive your comments, so that in other projects, we will not make any mistakes at all Please send your responses through email blogtoanhocchomoinguoi@gmail.com For more information and attachments, please visit our fanpage https://www.facebook.com/thcmn/ or our blog https://thcmn.wordpress.com/

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THCMN

Table of contents

1.1 Winter School of Vietnam Institute for Advanced Studying

Math 10

1.1.1 Day 1 10

1.1.2 Day 2 11

1.2 Northern Winter School 11

1.2.1 Day 1 11

1.2.2 Day 2 12

1.3 Northern Central Winter School 13

1.3.1 Day 1 13

1.3.2 Day 2 14

1.4 Southern Winter School 14

1.4.1 Day 1 14

1.4.2 Day 2 15

2 SOLUTIONS 17 2.1 Winter School of Vietnam Institute for Advanced Studying Math 18

2.2 Northern Winter School 27

2.3 Northern Central Winter School 35

2.4 Southern Winter School 43

7

1.1 Winter School of Vietnam Institute for

Ad-vanced Studying Math

Problem 3.Given two fixed point B,C lying on a circle (O) A variable point

A moves on that circle and always forms with B,C an acute-angled triangle.The interior bisector of dABCintersects line BC and (O) at D and E, respec-tively A point F lies on the segment BC such that FD = FE

1 Let H be the orthogonal projection of A on EF Prove that H alwayslies on a fixed circle

2 A circle (I) with center I is tangent to array AB, array AC, and line EF

at M, N, and P, respectively (Center I and point A belong to the samehalf-plane in accordance to line EF) Let Q be the point on MN suchthat PQ⊥EF Prove that line AQ always passes through a fixed point

Problem 4.At the 2016 Winter School, all teachers presented 100 Math

prob-lems (a hard problem is called “yam” ) for their students Whether a problem

is “yam” or not, there are exactly 20 students who can solve it To prepare

for the Ceremony, the School’s Committee will select qualified students forthe certificates In each particular way of selection, a problem is considered

"quacky" if it is a "yam" but 20 students who have solved it aren’t awarded the certificates, or it is not a "yam" but all 20 students who have solved it are

awarded the certificates Prove that there exists a selection so that among 100

given problems, there are no more than 4 "quacky" problems.

p2i + 1=

1

n2

Problem 6.Given an acute triangle ABC with incircle (I) is tangent to BC,CA, AB

at D, E, F, respectively Let H be the orthocenter of 4DEF and K is the foot

of the perpendicular from H to BC

1 Let J be the midpoint of EF Prove that AD intersects KJ at a pointlying on (T ), which is the circumcircle of triangle DHK

2 AD intersects (I) at the second point P Let M, N are the points lying on

DE, DF, respectively, such that dMPE = dNPF = 90o Prove that JD isthe radical axis of circle (T ) and the circumcircle of 4MNP

Problem 7.Find the smallest real constant c so that for all real numbers x, y, zthat x + y + z = 1, we have the following inequality:

x3+ y3+ z3− 1 ≤ c

x5+ y5+ z5− 1

...

H and circumcircle (O) CH intersects AB at D and intersects (O) the secondtime at AE Let F be the intersection of AC and BE, I be the intersection of Aand BC, J be the intersection of AB and. .. thecircumcircle of 4AUV Furthermore, I lies on the interior bisector ofangle A From Q is the midpoint of UV , J is the midpoint of KL

At the 2016 Winter School, all teachers presented 100 Math problems. .. major arc BC of circle (O) (such that4ABC is still a acute-angle triangle and AB 6= AC), then the tangentline in (ADE) at D and E intersect at a fixed point

2 Let M be the midpoint of BC Prove