toán học bắc trung nam giới thiệu đến các em học sinh bộ tài liệu dành cho học sinh giỏi

Denote (ω) as incircle of triangle ABE and it is tangent to AB, AE, BE respectively at P, F, K. We will say that a subset X of the set of cells of a board is malicious if every cycle on [r]

a b

a b c10

1 Selected problems from camps 9

2 Solution to tests of January camp 16

3 Solution to tests of March camp 24

4 Solution to tests of April camp 32

5 Solution to JBMO tests 40

6 Solution to IMO Team selection tests 51

7 Problems without solution 58

1

www.mawhiba.org.sa

This booklet is prepared bySultan Albarakati, Lê Phúc Lữ

with special thanks to the trainersFormer Olympiad StudentsAlzubair Habibullah, Alyazeed Basyoni, Shaden Alshammari,Omar Alrabiah, Majid Almarhoumi, Ali Alhaddad

Local TrainersTareq Salama, Safwat Altannani, Dr Abdulaziz Binobaid,

Waleed Aljabri, Adel Albarakati, Naif Alsalmi

Visitor Trainers Lukasz Bo ˙zyk, Tomasz Przybylowski, Dmytro Nomirovskii,

Dominik Burek, Ushangi Goginava, Smbat Gogyan,

Arsenii Nikolaiev, Lê Phúc Lữ, Melih Ucer, Abdulaziz Obeid

SAUDI ARABIAN IMO Booklet 2019

The Saudi Arabian team at IMO 2019

Asaad Mohammedsaleh Omar Habibullah

Thnaa Alhydary Marwan Alkhayat

Former Olympiad Students in the training team

Alzubair Habibullah Shaden Alshammari

Abdulaziz Al-Harthi

Team Training Administrators

Sultan Albarakati Fawzi Althukair Tarek Shehata

We also thanks to the helps of the people, teams during our camps

Organizers

Nada Altalhi, Saham AlHusseini, Akram El Ashy,

Hanan AlOtaibi, Mary Ann Callian, Nisha Mani, Venu Kas

Guest Executive Services Reservations

FC Helpdesk, Hanco Transport, Housing team,Business Transport, Tamimi KAUST team

Supervisions

Abdulrahman AlJedaani, Abdulrahman AlSaeed,

Abdulrahman bin Huzaim, Jaser AlShahrani, Khalid Hazazi,

Majed AlShayeb, Maryam AlSufyani, Naziha AlBarakati,

Noof AlNufaei, Seham Fatani, Sumayyah AlHaydary

SAUDI ARABIAN IMO Booklet 2019

This booklet contains the Team Selection Tests of the Saudi teams to the BalkanMathematics Olympiad, Balkan Junior Mathematics Olympiad, and the Interna-tional Mathematics Olympiad

The training was supported by the Ministry of Education, which commissionedMawhiba, the main establishment in Saudi Arabia that cares for the gifted students,

to do the task

We would like to express our gratitude to King Abdullah University of Science andTechnology KAUST for making its facilities on its beautiful campus available to usfor our training

The Saudi team had three main training camps during the academic year 2018-2019

In addition, the team had an intensive training period from March to the end ofJune 2019

During this academic year, the selected students participated in the following tests: The Asia Pacific Mathematics Olympiad, the European Girls MathematicsOlympiad in Ukraine, Balkan Mathematics Olympiad in Moldova and the JuniorBalkan mathematics Olympiad in Cyprus

con-It is our pleasure to share the selection tests problems with other IMO teams, hoping

it will contribute to future cooperation

Dr Fawzi A Al-ThukairLeader of the Saudi team in IMO 2019

نﺎﻘﻠﺒﻟاوﺔﻘﺑﺎﺴﻣنﺎﻘﻠﺒﻟاﻦﯿﺌﺷﺎﻨﻠﻟوﯿﻔﺼﺗتﺎ

دﺎﯿﺒﻤﻟوﻻا

ﻲﻟوﺪﻟاتﺎﯿﺿﺎﯾﺮﻠﻟ۲۰۱۹

ناﺐﯾرﺪﺗﻖﯾﺮﻔﻟانﺎﻛﻢﻋﺪﺑﻦﻣةرازوﻢﯿﻠﻌﺘﻟانوﺎﻌﺘﻟﺎﺑﻊﻣﺔﺴﺳﺆﻣﻚﻠﻤﻟاﺪﺒﻋﺰﯾﺰﻌﻟاوﮫﻟﺎﺟر

ﺔﺒھﻮﻤﻠﻟ

و

عاﺪﺑﻻا "

ﺔﺒھﻮﻣ "

رﺪﺠﺗوةرﺎﺷﻻاﻰﻟانوﺎﻌﺘﻟاومﺎﮭﺳﻻالﺎّﻌﻔﻟاﻦﻣﺔﻌﻣﺎﺟﻚﻠﻤﻟاﷲﺪﺒﻋمﻮﻠﻌﻠﻟو

،ﺔﯿﻨﻘﺘﻟاﺚﯿﺣتﺮﻓو

ﻨﻟ

ﺎ

ﻞﻛ

تﺎﻧﺎﻜﻣﻻاﻲﺘﻟا

ﺎﻨﺠﺘﺣاﺎﮭﻟﻲﻓﺐﯾرﺪﺘﻟاﻲﻓﺎﮭﻣﺮﺣﻲﻌﻣﺎﺠﻟاﻞﯿﻤﺠﻟا

ﻢﺗﺪﻘﻋﺔﺛﻼﺛتﺎﯿﻘﺘﻠﻣﺔﯿﺒﯾرﺪﺗلﻼﺧمﺎﻌﻟاﻲﺳارﺪﻟا۲۰۱۸-۲۰۱۹ ﺔﻓﺎﺿﻻﺎﺑﻰﻟا

ةﺮﺘﻓﺐﯾرﺪﺘﻟا

ﻒﺜﻜﻤﻟا

ﻲﺘﻟا

تأﺪﺑﻲﻓﺮﮭﺷسرﺎﻣ۲۰۱۹ﻰﻟاﺔﯾﺎﮭﻧﺮﮭﺷﻮﯿﻧﻮﯾ ﺎﻤﻛكرﺎﺷﺔﺒﻠﻄﻟانوﺰﯿﻤﺘﻤﻟاﻲﻓ

ﺪﯾﺪﻌﻟاﻦﻣ

تﺎﻘﺑﺎﺴﻤﻟا

ﺔﯿﻤﯿﻠﻗﻹاوﺎﮭﻨﻣ :دﺎﯿﺒﻤﻟواتﺎﯿﺿﺎﯾﺮﻟالوﺪﻟ

ﺎﯿﺳآوﻚﯿﻔﯿﺳﺎﺒﻟا ،دﺎﯿﺒﻤﻟواتﺎﺒﻟﺎﻄﻟالوﺪﻠﻟ

ﻞﻣﺄﻧنانﻮﻜﯾىﻮﺘﺤﻣاﺬھﺐﯿﺘﻜﻟا ًﺎﻣﺎﮭﺳإﺎﻨﻣﺔﯾﻮﻘﺘﻟﺮﺻاوانوﺎﻌﺘﻟاولدﺎﺒﺗتاﺮﺒﺨﻟاﺎﻨﻨﯿﺑ

د.يزﻮﻓﻦﺑﺪﻤﺣأﺮﯿﻛﺬﻟا

ﺲﯿﺋرﻖﯾﺮﻔﻟايدﻮﻌﺴﻟادﺎﯿﺒﻤﻟوﻼﻟ

ﻲﻟوﺪﻟا ۲۰۱۹

SAUDI ARABIAN IMO Booklet 2019

Selected problems from camps

Problem 2 Let P (x) be a polynomial of degree n ≥ 2 with rational coefficientssuch that P (x) has n pairwise different real roots forming an arithmetic progression.Prove that among the roots of P (x) there are two that are also the roots of somepolynomial of degree 2 with rational coefficients

Problem 3 Let ABCDEF be a convex hexagon satisfying AC = DF , CE = F Band EA = BD Prove that the lines connecting the midpoints of opposite sides ofthe hexagon ABCDEF intersect in one point

1.2 Test 2

Problem 4 Suppose that a, b, c, d are pairwise distinct positive integers such that

a + b = c + d = p for some odd prime p > 3 Prove that abcd is not a perfect square

Problem 5 There are 3 clubs A, B, C with non-empty members For any triplet

of members (a, b, c) with a ∈ A, b ∈ B, c ∈ C, two of them are friend and two ofthem are not friend (here the friend relationship is bidirectional) Prove that one ofthese statements must be true

1 There exist one student from A that knows all students from B

2 There exist one student from B that knows all students from C

3 There exist one student from C that knows all students from A

Problem 6 Let ABC be a triangle with A0, B0, C0 are midpoints of BC, CA, ABrespectively The circle (ωA) of center A has a big enough radius cuts B0C0 at

X1, X2 Define circles (ωB), (ωC) with Y1, Y2, Z1, Z2 similarly Suppose that thesecircles have the same radius, prove that X , X , Y , Y , Z , Z are concyclic

Problem 7 Let ABC be a triangle inscribed in a circle (ω) and I is the incenter.Denote D, E as the intersection of AI, BI with (ω) And DE cuts AC, BC at F, Grespectively Let P be a point such that P F k AD and P G k BE Suppose thatthe tangent lines of (ω) at A, B meet at K Prove that three lines AE, BD, KP areconcurrent or parallel.

Problem 8 It is given a graph whose vertices are positive integers and an edgebetween numbers a and b exists if and only if

a + b + 1 | a2+ b2+ 1

Is this graph connected?

Problem 9 Define sequence of positive integers (an) as a1 = a and an+1 = a2

n+ 1for n ≥ 1 Prove that there is no index n for which

nY

Problem 1 Let p be an odd prime number

1 Show that p divides n2n+ 1 for infinitely many positive integers n

2 Find all n satisfy condition above when p = 3

Problem 2 Let I be the incenter of triangle ABC and J the excenter of the side

BC Let M be the midpoint of CB and N the midpoint of arc BAC of circle (ABC)

If T is the symmetric of the point N by the point A, prove that the quadrilateral

J M IT is cyclic

Problem 3 For n ≥ 3, it is given an 2n × 2n board with black and white squares

It is known that all border squares are black and no 2 × 2 subboard has all foursquares of the same color Prove that there exists a 2 × 2 subboard painted like achessboard, i.e with two opposite black corners and two opposite white corners

2.2 Test 2

Problem 4 There are n people with hats present at a party Each two of themgreeted each other exactly once and each greeting consisted of exchanging the hatsthat the two persons had at the moment Find all n ≥ 2 for which the order ofgreetings can be arranged in such a way that after all of them, each person has theirown hat back

SAUDI ARABIAN IMO Booklet 2019

10 Selected problems from camps

Problem 5 Let sequences of real numbers (xn) and (yn) satisfy x1 = y1 = 1 and

xn+1 = xn+ 2

xn+ 1 and yn+1 =

y2n+ 22yn for n = 1, 2, Prove that yn+1= x2 n holds for n = 0, 1, 2,

Problem 6 The triangle ABC (AB > BC) is inscribed in the circle Ω On thesides AB and BC, the points M and N are chosen, respectively, so that AM = CN.The lines M N and AC intersect at point K Let P be the center of the inscribedcircle of triangle AM K, and Q the center of the excircle of the triangle CN K tangent

to side CN Prove that the midpoint of the arc ABC of the circle Ω is equidistantfrom the P and Q

2.3 Test 3

Problem 7 Let 19 integer numbers are given Let Hamza writes on the paperthe greatest common divisor for each pair of numbers It occurs that the differencebetween the biggest and smallest numbers written on the paper is less than 180.Prove that not all numbers on the paper are different

Problem 8 Let ABCD is a trapezoid with ∠A = ∠B = 90◦ and let E is a pointlying on side CD Let the circle ω is inscribed to triangle ABE and tangents sides

AB, AE and BE at points P , F and K respectively Let KF intersects segments

BC and AD at points M and N respectively, as well as P M and P N intersect ω atpoints H and T respectively Prove that P H = P T

Problem 9 Let 300 students participate to the Olympiad Between each 3 ipants there is a pair that are not friends Hamza enumerates participants in someorder and denotes by xi the number of friends of i-th participant It occurs that

partic-{x1, x2, , x299, x300} = {1, 2, , N − 1, N }

Find the biggest possible value for N

3 April camp

3.1 Test 1

Problem 1 In a school there are 40 different clubs, each of them contains exactly

30 children For every i from 1 to 30 define ni as a number of children who attendexactly i clubs Prove that it is possible to organize 40 new clubs with 30 children

in each of them such, that the analogical numbers n1, n2, , n30 will be the samefor them

Problem 2 Let Pascal triangle be an equilateral triangular array of number, sists of 2019 rows and except for the numbers in the bottom row, each number isequal to the sum of two numbers immediately below it How many ways to assigneach of numbers a0, a1, , a2018 (from left to right) in the bottom row by 0 or 1such that the number S on the top is divisible by 1019

con-Problem 3 Find all functions f : R+ → R+ such that

f 3 (f (xy))2+ (xy)2
= (xf (y) + yf (x))2

for any x, y > 0

Problem 4 Let pairwise different positive integers a, b, c with gcd(a, b, c) = 1 aresuch that

a | (b − c)2, b | (c − a)2, c | (a − b)2.Prove, that there is no non-degenerate triangle with side lengths a, b and c

Problem 5 Let be given a positive integer n > 1 Find all polynomials P (x) nonconstant, with real coefficients such that

3.3 Test 3

Problem 7 Let P (x) be a monic polynomial of degree 100 with 100 distinct integer real roots Suppose that each of polynomials P (2x2− 4x) and P (4x − 2x2)has exactly 130 distinct real roots Prove that there exist non constant polynomialsA(x), B(x) such that A(x)B(x) = P (x) and A(x) = B(x) has no root in (−1; 1).Problem 8 Let ABC be a triangle, the circle having BC as diameter cuts AB, AC

non-at F, E respectively Let P a point on this circle Let C0, B0 be the projections of

P upon the sides AB, AC respectively Let H be the orthocenter of the triangle

AB0C0 Show that ∠EHF = 90◦

Problem 9 All of the numbers 1, 2, 3, , 1000000 are initially colored black Oneach move it is possible to choose the number x (among the colored numbers) andchange the color of x and of all of the numbers that are not co-prime with x (blackinto white, white into black) Is it possible to color all of the numbers white?

4 JBMO TST

4.1 Test 1

Problem 1 Find the smallest integer m for which there are positive integers n >

k > 1 satisfying the equation

11 1

| {z }n

= 11 1

| {z }k

·m

Problem 2 Chess horse attacks fields in distance √

5 Let several horses are put

on the board 12 × 12 such, that every square of size 2 × 2 contains at least one horse.Find the maximal possible number of cells that are not under attack (horse doesn’tattack it’s own cell)

SAUDI ARABIAN IMO Booklet 2019

12 Selected problems from camps

Problem 3 How many integers n satisfy to the following conditions?

i) 219 ≤ n ≤ 2019,

ii) there exist x, y ∈ Z such that 1 ≤ x < n < y and y is divisible by all integersfrom 1 to n, except two numbers x and x + 1

Problem 4 Let AD be the altitude of the right angled triangle ABC with ∠A =

90◦ Let DE be the altitude of the triangle ADB and DZ be the altitude of thetriangle ADC respectively Let N is chosen on the line AB such that CN is parallel

to EZ Let A0 be the symmetric of A with respect to the line EZ and I, K theprojections of A0into AB and AC respectively Prove that∠NA0T = ∠ADT , where

T is the intersection point of IK and DE

4.2 Test 2

Problem 1 In square ABCD with side 1 point E lies on BC and F lies on CDsuch that∠EAB = 20◦, ∠EAF = 45◦ Find the length of altitude AH of 4AEF.Problem 2 Prove the inequality for non-negative a, b, c

Problem 4 An 11 × 11 square is partitioned into 121 smaller 1 × 1 squares, 4

of which are painted black, the rest being white We cut a fully white rectangle(possibly a square) out of the big 11 × 11 square What is the maximal area ofthe rectangle we can obtain regardless of the positions of the black squares? It isallowed to cut the rectangle along the grid lines

4.3 Test 3

Problem 1 Determine the maximal number of disjoint crosses (5 squares) whichcan be put inside 8 × 8 chessboard such that sides of a cross are parallel to sides ofthe chessboard

Problem 2 Find all pairs of positive integers (m, n) such that

125 · 2n− 3m = 271

inside this triangle such that

∠DAB = ∠DCB, ∠DAC = ∠DBC;

∠EAB = ∠EBC, ∠EAC = ∠ECB

Prove that triangle ADE is right

Problem 4 Let n be a positive integer and let a1, a2, , an be any real numbers.Prove that there exists m, k ∈ {1, 2, , n} such that

mX

i=1

ai−

nX

i=m+1

ai

... all members in B Consider

a new member x to make the maximum number of members increased by It iseasy to see that if x ∈ A or x ∈ C, then the conditions are still true (since themembers...

Solution We will prove the statement by induction on the maximum number ofmembers in clubs A, B, C

For n = 1, each club has exactly one member and the statement is obviously true.Suppose... E, then T P = T Q.Remark Note that point T is the Miquel point of the completed quadrilateral

AM N C.BK In this problem, we have the following lemma (trillium theorem):Let L be the midpoint