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]
(1)Saud
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Arabia
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Mathematica
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Competition
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2019
SAMC 2019
Riyadh, June 2019 IMO Booklet
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(2)Table of contents
1 Selected problems from camps
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
(3)2
éKXđªË@ éJK.QªË@ éºÊỊỊÊË HAJAKQË@ HA ®K.AĨ
SAUDI ARABIAN
MATHEMATICAL COMPETITIONS 2019
Copyright @ Mawhiba 2018-2019 All rights reserved
The King Abdulaziz and His Companions Foundation for Giftedness and Creativity organization, Saudi Arabia
(4)3
This booklet is prepared by
Sultan Albarakati, Lê Phúc Lữ
with special thanks to the trainers
Former Olympiad Students
Alzubair Habibullah, Alyazeed Basyoni, Shaden Alshammari, Omar Alrabiah, Majid Almarhoumi, Ali Alhaddad
Local Trainers
Tareq 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
(5)4
The Saudi Arabian team at IMO 2019
Asaad Mohammedsaleh Omar Habibullah
Khalid Ajran Nawaf Alghamdi
Thnaa Alhydary Marwan Alkhayat
Former Olympiad Students in the training team
(6)5
General Supervisor of Competition Management
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
(7)Introduction
This booklet contains the Team Selection Tests of the Saudi teams to the Balkan Mathematics Olympiad, Balkan Junior Mathematics Olympiad, and the Interna-tional Mathematics Olympiad
The training was supported by the Ministry of Education, which commissioned Mawhiba, the main establishment in Saudi Arabia that cares for the gifted students, to the task
We would like to express our gratitude to King Abdullah University of Science and Technology KAUST for making its facilities on its beautiful campus available to us for 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 of June 2019
During this academic year, the selected students participated in the following con-tests: The Asia Pacific Mathematics Olympiad, the European Girls Mathematics Olympiad in Ukraine, Balkan Mathematics Olympiad in Moldova and the Junior Balkan mathematics Olympiad in Cyprus
It is our pleasure to share the selection tests problems with other IMO teams, hoping it will contribute to future cooperation
(8)7 ﺔﻣﺪﻘﻣ يﻮﺤﯾ اﺬھ ﺐﯿﺘﻜﻟا ﻰﻠﻋ ﻞﺋﺎﺴﻣ تﺎﯿﻔﺼﺘﻟا ﺔﻘﺑﺎﺴﻤﻟ نﺎﻘﻠﺒﻟا و ﺔﻘﺑﺎﺴﻣ نﺎﻘﻠﺒﻟا ﻦﯿﺌﺷﺎﻨﻠﻟ و ﯿﻔﺼﺗ تﺎ دﺎﯿﺒﻤﻟوﻻا ﻲﻟوﺪﻟا تﺎﯿﺿﺎﯾﺮﻠﻟ ۲۰۱۹ نا ﺐﯾرﺪﺗ ﻖﯾﺮﻔﻟا نﺎﻛ ﻢﻋﺪﺑ ﻦﻣ ةرازو ﻢﯿﻠﻌﺘﻟا نوﺎﻌﺘﻟﺎﺑ ﻊﻣ ﺔﺴﺳﺆﻣ ﻚﻠﻤﻟا ﺪﺒﻋ ﺰﯾﺰﻌﻟا و ﮫﻟﺎﺟر ﺔﺒھﻮﻤﻠﻟ و عاﺪﺑﻻا " ﺔﺒھﻮﻣ " رﺪﺠﺗو ةرﺎﺷﻻا ﻰﻟا نوﺎﻌﺘﻟا و مﺎﮭﺳﻻا لﺎّﻌﻔﻟا ﻦﻣ ﺔﻌﻣﺎﺟ ﻚﻠﻤﻟا ﷲﺪﺒﻋ مﻮﻠﻌﻠﻟ و ،ﺔﯿﻨﻘﺘﻟا ﺚﯿﺣ تﺮﻓو ﻨﻟ ﺎ ﻞﻛ تﺎﻧﺎﻜﻣﻻا ﻲﺘﻟا ﺎﻨﺠﺘﺣا ﺎﮭﻟ ﻲﻓ ﺐﯾرﺪﺘﻟا ﻲﻓ ﺎﮭﻣﺮﺣ ﻲﻌﻣﺎﺠﻟا ﻞﯿﻤﺠﻟا ﻢﺗ ﺪﻘﻋ ﺔﺛﻼﺛ تﺎﯿﻘﺘﻠﻣ ﺔﯿﺒﯾرﺪﺗ لﻼﺧ مﺎﻌﻟا ﻲﺳارﺪﻟا ۲۰۱۸ -۲۰۱۹ ﺔﻓﺎﺿﻻﺎﺑ ﻰﻟا ةﺮﺘﻓ ﺐﯾرﺪﺘﻟا ﻒﺜﻜﻤﻟا ﻲﺘﻟا تأﺪﺑ ﻲﻓ ﺮﮭﺷ سرﺎﻣ ۲۰۱۹ ﻰﻟا ﺔﯾﺎﮭﻧ ﺮﮭﺷ ﻮﯿﻧﻮﯾ ﺎﻤﻛ كرﺎﺷ ﺔﺒﻠﻄﻟا نوﺰﯿﻤﺘﻤﻟا ﻲﻓ ﺪﯾﺪﻌﻟا ﻦﻣ تﺎﻘﺑﺎﺴﻤﻟا ﺔﯿﻤﯿﻠﻗﻹا و ﺎﮭﻨﻣ : دﺎﯿﺒﻤﻟوا تﺎﯿﺿﺎﯾﺮﻟا لوﺪﻟ ﺎﯿﺳآ و ﻚﯿﻔﯿﺳﺎﺒﻟا ، دﺎﯿﺒﻤﻟوا تﺎﺒﻟﺎﻄﻟا لوﺪﻠﻟ ﺔﯿﺑروﻷا ﻓ ﻲ ﺎﯿﻧاﺮﻛوا ، دﺎﯿﺒﻤﻟوا نﺎﻘﻠﺒﻟا ﻲﻓ ﺎﻓوﺪﻟﺎﻣ و دﺎﯿﺒﻤﻟوا ﻦﯿﺌﺷﺎﻨﻟا لوﺪﻟ نﺎﻘﻠﺒﻟا ﻲﻓ صﺮﺒﻗ ﻞﻣﺄﻧ نا نﻮﻜﯾ ىﻮﺘﺤﻣ اﺬھ ﺐﯿﺘﻜﻟا ًﺎﻣﺎﮭﺳإ ﺎﻨﻣ ﺔﯾﻮﻘﺘﻟ ﺮﺻاوا نوﺎﻌﺘﻟا و لدﺎﺒﺗ تاﺮﺒﺨﻟا ﺎﻨﻨﯿﺑ و لوﺪﻟا ﻟا ﺔﻛرﺎﺸﻤ ﻲﻓ دﺎﯿﺒﻤﻟوﻻا ﻲﻟوﺪﻟا د يزﻮﻓ ﻦﺑ ﺪﻤﺣأ ﺮﯿﻛﺬﻟا ﺲﯿﺋر ﻖﯾﺮﻔﻟا يدﻮﻌﺴﻟا دﺎﯿﺒﻤﻟوﻼﻟ ﻲﻟوﺪﻟا ۲۰۱۹
(9)Selected problems from camps
1 January camp
1.1 Test 1
Problem Suppose that x, y, z are non-zero real numbers such that
x = − y
z, y = − z
x, z = − x y
Find all possible values of T = x + y + z
Problem Let P (x) be a polynomial of degree n ≥ with rational coefficients such 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 some polynomial of degree with rational coefficients
Problem Let ABCDEF be a convex hexagon satisfying AC = DF , CE = F B and EA = BD Prove that the lines connecting the midpoints of opposite sides of the hexagon ABCDEF intersect in one point
1.2 Test 2
Problem Suppose that a, b, c, d are pairwise distinct positive integers such that a + b = c + d = p for some odd prime p > Prove that abcd is not a perfect square
Problem There are 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 of them are not friend (here the friend relationship is bidirectional) Prove that one of these 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
(10)Selected problems from camps
1.3 Test 1
Problem 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, G respectively Let P be a point such that P F k AD and P G k BE Suppose that the tangent lines of (ω) at A, B meet at K Prove that three lines AE, BD, KP are concurrent or parallel
Problem It is given a graph whose vertices are positive integers and an edge between numbers a and b exists if and only if
a + b + | a2+ b2+
Is this graph connected?
Problem Define sequence of positive integers (an) as a1 = a and an+1 = a2n+ for n ≥ Prove that there is no index n for which
n Y
k=1
a2k+ ak+
is a perfect square
2 March camp - BMO TST
2.1 Test 1
Problem Let p be an odd prime number
1 Show that p divides n2n+ for infinitely many positive integers n. Find all n satisfy condition above when p =
Problem 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 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 × subboard has all four squares of the same color Prove that there exists a × subboard painted like a chessboard, i.e with two opposite black corners and two opposite white corners
2.2 Test 2
Problem There are n people with hats present at a party Each two of them greeted each other exactly once and each greeting consisted of exchanging the hats that the two persons had at the moment Find all n ≥ for which the order of greetings can be arranged in such a way that after all of them, each person has their own hat back
(11)10 Selected problems from camps
Problem Let sequences of real numbers (xn) and (yn) satisfy x1 = y1 = and
xn+1 =
xn+ xn+
and yn+1 =
y2n+ 2yn
for n = 1, 2,
Prove that yn+1= x2n holds for n = 0, 1, 2,
Problem The triangle ABC (AB > BC) is inscribed in the circle Ω On the sides 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 inscribed circle 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 equidistant from the P and Q
2.3 Test 3
Problem Let 19 integer numbers are given Let Hamza writes on the paper the greatest common divisor for each pair of numbers It occurs that the difference between the biggest and smallest numbers written on the paper is less than 180 Prove that not all numbers on the paper are different
Problem Let ABCD is a trapezoid with ∠A = ∠B = 90◦ and let E is a point lying 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 at points H and T respectively Prove that P H = P T
Problem Let 300 students participate to the Olympiad Between each partic-ipants there is a pair that are not friends Hamza enumerates particpartic-ipants in some order and denotes by xi the number of friends of i-th participant It occurs that
{x1, x2, , x299, x300} = {1, 2, , N − 1, N } Find the biggest possible value for N
3 April camp
3.1 Test 1
Problem In a school there are 40 different clubs, each of them contains exactly 30 children For every i from to 30 define ni as a number of children who attend exactly 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 same for them
Problem Let Pascal triangle be an equilateral triangular array of number, con-sists of 2019 rows and except for the numbers in the bottom row, each number is equal to the sum of two numbers immediately below it How many ways to assign each of numbers a0, a1, , a2018 (from left to right) in the bottom row by or such that the number S on the top is divisible by 1019
Problem Find all functions f : R+ → R+ such that
(12)Selected problems from camps
3.2 Test 2
Problem Let pairwise different positive integers a, b, c with gcd(a, b, c) = are such 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 Let be given a positive integer n > Find all polynomials P (x) non constant, with real coefficients such that
P (x)P (x2) P (xn) = Pxn(n+1)2
for all x ∈ R
Problem Let ABC be an acute, non isosceles triangle with O, H are circumcenter and orthocenter, respectively Prove that the nine-point circles of AHO, BHO, CHO has two common points
3.3 Test 3
Problem Let P (x) be a monic polynomial of degree 100 with 100 distinct non-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 polynomials A(x), B(x) such that A(x)B(x) = P (x) and A(x) = B(x) has no root in (−1; 1)
Problem Let ABC be a triangle, the circle having BC as diameter cuts AB, AC 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 All of the numbers 1, 2, 3, , 1000000 are initially colored black On each move it is possible to choose the number x (among the colored numbers) and change the color of x and of all of the numbers that are not co-prime with x (black into white, white into black) Is it possible to color all of the numbers white?
4 JBMO TST
4.1 Test 1
Problem Find the smallest integer m for which there are positive integers n > k > satisfying the equation
11 | {z }
n
= 11 | {z }
k
·m
Problem Chess horse attacks fields in distance √5 Let several horses are put on the board 12 × 12 such, that every square of size × contains at least one horse Find the maximal possible number of cells that are not under attack (horse doesn’t attack it’s own cell)
(13)12 Selected problems from camps
Problem How many integers n satisfy to the following conditions?
i) 219 ≤ n ≤ 2019,
ii) there exist x, y ∈ Z such that ≤ x < n < y and y is divisible by all integers from to n, except two numbers x and x +
Problem 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 the triangle 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 the projections 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 In square ABCD with side point E lies on BC and F lies on CD such that∠EAB = 20◦, ∠EAF = 45◦ Find the length of altitude AH of 4AEF
Problem Prove the inequality for non-negative a, b, c
a√3a2+ 6b2+ b√3b2+ 6c2+ c√3c2+ 6a2 ≥ (a + b + c)2.
Problem Find all primes p such that there exist integers m and n satisfying p = m2+ n2 and p | m3+ n3+ 8mn.
Problem An 11 × 11 square is partitioned into 121 smaller × squares, 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 of the rectangle we can obtain regardless of the positions of the black squares? It is allowed to cut the rectangle along the grid lines
4.3 Test 3
Problem Determine the maximal number of disjoint crosses (5 squares) which can be put inside × chessboard such that sides of a cross are parallel to sides of the chessboard
Problem Find all pairs of positive integers (m, n) such that
(14)Selected problems from camps
Problem Let ABC be an acute, non isosceles triangle Take two points D, E inside this triangle such that
∠DAB = ∠DCB, ∠DAC = ∠DBC; ∠EAB = ∠EBC, ∠EAC = ∠ECB
Prove that triangle ADE is right
Problem 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
m X
i=1 ai−
n X
i=m+1
...
Problem There are 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 of them are not friend (here the friend... members in B Consider a new member x to make the maximum number of members increased by It is easy to see that if x ∈ A or x ∈ C, then the conditions are still true (since the members in B remains...
Solution We will prove the statement by induction on the maximum number of members in clubs A, B, C
For n = 1, each club has exactly one member and the statement is obviously true