de thu va loi giai HOMC 2006 - 2016

8/19/2019 de thu va loi giai HOMC 2006 - 2016

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Contents

1 Hanoi Open Mathematics Competition 3

1.1 Hanoi Open Mathematics Competition 2006 . . . 31.1.1 Junior Section . . . . . . . . . . . . . . . . 31.1.2 Senior Section . . . . . . . . . . . . . . . . 4

1.2 Hanoi Open Mathematics Competition 2007 . . . 61.2.1 Junior Section . . . . . . . . . . . . . . . . 6

1.2.2 Senior Section . . . . . . . . . . . . . . . . 81.3 Hanoi Open Mathematics Competition 2008 . . . 10

1 3 1 Junior Section 10


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Chapter 1

Hanoi Open MathematicsCompetition

1.1 Hanoi Open Mathematics Competition 2006

1.1.1 Junior Section

Question 1 . What is the last two digits of the number

(11 + 12 + 13 + · · ·+ 2006) 2?


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Question 5 . Suppose n is a positive integer and 3 arbitrarynumbers are choosen from the set {1, 2, 3, . . . , 3n +1 } with theirsum equal to 3n + 1.

What is the largest possible product of those 3 numbers?

Question 6 . The gure ABCDEF is a regular hexagon. Findall points M belonging to the hexagon such that

Area of triangle MAC = Area of triangle MCD.

Question 7 . On the circle ( O) of radius 15cm are given 2 pointsA, B . The altitude OH of the triangle OAB intersect ( O) at C .What is AC if AB = 16cm?

Question 8 . In ∆ ABC , P Q BC where P and Q are pointson AB and AC respectively. The lines P C and QB intersectat G. It is also given EF//BC , where G ∈ EF , E ∈ AB andF ∈ AC with P Q = a and EF = b. Find value of BC .


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(3 + 7 + 11 + · · ·+ 2007) 2?(A) 01; (B) 11; (C) 23; (D) 37; (E) None of the above.

Question 2 . What is largest positive integer n satisfying the

following inequality:n2006 < 72007 ?

(A) 7; (B) 8; (C) 9; (D) 10; (E) 11.Question 3 . Which of the following is a possible number of diagonals of a convex polygon?

(A) 02; (B) 21; (C) 32; (D) 54; (E) 63.Question 4 . Let m and n denote the number of digits in 2 2007


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Question 7 . Nine points, no three of which lie on the samestraight line, are located inside an equilateral triangle of side 4.Prove that some three of these points are vertices of a trianglewhose area is not greater than √ 3.Question 8 . Let a, b, c be positive integers. Prove that

(b + c−a)2(b + c)2 + a2

+ (c + a −b)2(c + a)2 + b2

+ (a + b−c)2(a + b)2 + c2 ≥

35

.

Question 9 . A triangle is said to be the Heron triangle if it

has integer sides and integer area. In a Heron triangle, the sidesa,b,c satisfy the equation b = a(a −c).

Prove that the triangle is isosceles.

Question 10 . Let a,b,c be positive real numbers such that1

bc

+ 1

ca

+ 1

ab ≥ 1. Prove that

a

bc

+ b

ca

+ c

ab ≥ 1.

Question 11 . How many possible values are there for the suma + b + c + d if a, b, c, d are positive integers and abcd = 2007.


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1.2.2 Senior Section


112 + 15 2 + 19 2 +

· · ·+ 20072

2?

(A) 01; (B) 21; (C) 31; (D) 41; (E) None of the above.

Question 2 . Which is largest positive integer n satisfying thefollowing inequality:

n2007

> (2007)n

.

(A) 1; (B) 2; (C) 3; (D) 4; (E) None of the above.

Question 3 . Find the number of different positive integertriples ( x,y,z ) satsfying the equations

x + y −z = 1 and x2 + y2 −z 2 = 1 .(A) 1; (B) 2; (C) 3; (D) 4; (E) None of the above


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Question 8 .Let ABC be an equilateral triangle. For a point M inside

∆ ABC , let D, E,F be the feet of the perpendiculars from M onto BC, CA, AB , respectively. Find the locus of all such pointsM for which ∠F DE is a right angle.

Question 9 . Let a1, a2, . . . , a 2007 be real numbers such that

a1+ a2+ · · ·+ a2007 ≥ (2007)2 and a21+ a22+ · · ·+ a22007 ≤ (2007)3−1.

Prove that ak ∈ [2006; 2008] for all k ∈ {1, 2, . . . , 2007}.Question 10 . What is the smallest possible value of

x2 + 2 y2 −x −2y −xy?Question 11 . Find all polynomials P (x) satisfying the equation

(2x −1)P (x) = ( x −1)P (2x), ∀x.


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Question 15 . Let p = abcd be the 4-digit prime number. Provethat the equation

ax 3 + bx2 + cx + d = 0

has no rational roots.



Question 1 . How many integers from 1 to 2008 have the sumof their digits divisible by 5 ?

Question 2 . How many integers belong to ( a, 2008a), where a(a > 0) is given.

Question 3 . Find the coefficient of x in the expansion of

(1 + x)(1 2x)(1 + 3 x)(1 4x) (1 2008x)


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Find P (x2 + 1)?

Question 7 . The gure ABCDE is a convex pentagon. Findthe sum

∠DAC + ∠EBD + ∠ACE + ∠BDA + ∠CEB ?

Question 8 . The sides of a rhombus have length a and the areais S . What is the length of the shorter diagonal?

Question 9 . Let be given a right-angled triangle ABC with∠A = 900, AB = c, AC = b. Let E ∈ AC and F ∈ ABsuch that ∠AEF = ∠ABC and ∠AF E = ∠ACB . Denote byP ∈ BC and Q ∈ BC such that EP ⊥ BC and F Q ⊥ BC .Determine EP + EF + P Q?

Question 10 . Let a, b, c ∈ [1, 3] and satisfy the following con-ditionsmax{a b c} 2 a + b + c = 5


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has no solutions of positive integers x, y and z .

Question 4 . Prove that there exists an innite number of rela-tively prime pairs ( m, n ) of positive integers such that the equa-

tionx3 −nx + mn = 0

has three distint integer roots.

Question 5 . Find all polynomials P (x) of degree 1 such that

maxa≤x≤b

P (x) − mina≤x≤b P (x) = b−a, ∀a, b ∈R where a < b.

Question 6 . Let a, b, c ∈ [1, 3] and satisfy the following condi-tions

max

{a,b,c

}2, a + b + c = 5.

What is the smallest possible value of

2 2 2


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parallelogram. Let O be the intersection of BN and CP . FindM ∈ BC such that ∠P MO = ∠OM N .Question 10 . Let be given a right-angled triangle ABC with

∠A = 900, AB = c, AC = b. Let E ∈ AC and F ∈ ABsuch that ∠AEF = ∠ABC and ∠AF E = ∠ACB . Denote byP ∈ BC and Q ∈ BC such that EP ⊥ BC and F Q ⊥ BC .Determine EP + EF + F Q?



Question 1 . Let a,b,c be 3 distinct numbers from {1, 2, 3, 4, 5, 6}.Show that 7 divides abc + (7

−a)(7

−b)(7

−c).

Question 2 . Show that there is a natural number n such thath b d l i 2009


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Question 6 . Suppose that 4 real numbers a,b,c,d satisfy theconditions a2 + b2 = 4c2 + d2 = 4

ac + bd = 2Find the set of all possible values the number M = ab + cd cantake.

Question 7 . Let a, b, c, d be positive integers such that a + b+c + d = 99. Find the smallest and the greatest values of thefollowing product P = abcd.

Question 8 . Find all the pairs of the positive integers such thatthe product of the numbers of any pair plus the half of one of the numbers plus one third of the other number is three times

less than 1004 .

Question 9 . Let be given ∆ ABC with area (∆ ABC ) = 60cm 2.


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Question 1 . Let a,b,c be 3 distinct numbers from {1, 2, 3, 4, 5, 6}.Show that 7 divides abc + (7 −a)(7 −b)(7 −c).Question 2 . Show that there is a natural number n such thatthe number a = n! ends exacly in 2009 zeros.

Question 3 . Let a, b, c be positive integers with no commonfactor and satisfy the conditions

1a

+ 1b

= 1c

.

Prove that a + b is a square.

Question 4 . Suppose that a = 2b, where b = 210n +1 . Prove that

a is divisible by 23 for any positive integer n.

Question 5 . Prove that m7 − m is divisible by 42 for any


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Question 8 . Find all the pairs of the positive integers such thatthe product of the numbers of any pair plus the half of one of the numbers plus one third of the other number is three timesless than 1004 .

Question 9 .Given an acute-angled triangle ABC with area S ,let points A , B , C be located as follows: A is the point wherealtitude from A on BC meets the outwards facing semicirledrawn on BC as diameter. Points B , C are located similarly.

Evaluate the sum

T = (area ∆ BCA )2 + (area ∆ CAB )2 + (area ∆ ABC )2.

Question 10 . Prove that d2 +( a −b)2 < c 2, where d is diameterof the inscribed circle of ∆ ABC.Question 11 . Let A = {1, 2, . . . , 100} and B is a subset of A


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Question 10. Find the maximum value of

M = x

2x + y +

y2y + z

+ z

2z + x, x, y, z > 0.

1.5.2 Senior SectionQuestion 1 . The number of integers n ∈ [2000, 2010] such that22n + 2 n + 5 is divisible by 7 is

(A): 0; (B):1; (C): 2; (D): 3; (E) None of the above.

Question 2 . 5 last digits of the number 5 2010 are

(A): 65625; (B): 45625; (C): 25625; (D): 15625; (E) None of theabove.

Question 3 . How many real numbers a ∈ (1, 9) such that thecorresponding number a − 1a is an integer.(A): 0; (B):1; (C): 8; (D): 9; (E) None of the above.


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is an integer. Determine a, b, c, d.

Question 7 . Let P be the common point of 3 internal bisectorsof a given ABC. The line passing through P and perpendicular

to CP intersects AC and BC at M and N , respectively. If AP = 3cm, BP = 4cm, compute the value of

AM BN

?

Question 8 . If n and n3 +2 n2 +2 n +4 are both perfect squares,nd n.

Question 9. Let x, y be the positive integers such that 3 x2 +x = 4y2 + y. Prove that x −y is a perfect integer.Question 10. Find the maximum value of

M = x

2x + y +

y

2y + z +

z

2z + x, x, y, z > 0.


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3 (2011) 3 + 3 ×(2011) 2 + 4 ×2011 + 5?(A) 2010; (B) 2011; (C) 2012; (D) 2013; (E) None of theabove.

Question 4. Among the four statements on real numbers be-low, how many of them are correct?

“If a < b < 0 then a < b2”;“If 0 < a < b then a < b2”;“If a3 < b3 then a < b”;“If a2 < b2 then a < b”;“If |a| < |b| then a < b”.

(A) 0; (B) 1; (C) 2; (D) 3; (E) 4

Question 5. Let M = 7!

×8!

×9!

×10!

×11!

×12!. How many

factors of M are perfect squares?

Question 6. Find all positive integers ( m, n ) such that


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Question 9. Solve the equation

1 + x + x2 + x3 + · · ·+ x2011 = 0 .

Question 10. Consider a right-angle triangle ABC with A =90o, AB = c and AC = b. Let P ∈ AC and Q ∈ AB such that∠AP Q = ∠ABC and ∠AQP = ∠ACB. Calculate P Q + P E +QF, where E and F are the projections of P and Q onto BC ,respectively.

Question 11. Given a quadrilateral ABCD with AB = BC =3cm, CD = 4cm, DA = 8cm and ∠DAB + ∠ABC = 180o.Calculate the area of the quadrilateral.

Question 12. Suppose that a > 0, b > 0 and a + b 1.Determine the minimum value of

M = 1ab

+ 1

a2 + ab +

1ab + b2

+ 1

a2 + b2.


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(A) 2010; (B) 2011; (C) 2012; (D) 2013; (E) None of theabove.

Question 4. Prove that

1 + x + x2 + x3 + · · ·+ x2011 0for every x −1.Question 5. Let a, b, c be positive integers such that a + 2b +

3c = 100. Find the greatest value of M = abc.Question 6. Find all pairs ( x, y) of real numbers satisfying thesystem

x + y = 2x4 −y4 = 5x −3y

Question 7. How many positive integers a less than 100 suchthat 4 a2 + 3 a + 5 is divisible by 6.


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Question 10. Two bisectors BD and CE of the triangle ABC intersect at O. Suppose that BD.CE = 2BO.OC. Denote by H the point in BC such that OH ⊥ BC . Prove that AB.AC =2HB.HC.

Question 11. Consider a right-angle triangle ABC with A =90o, AB = c and AC = b. Let P ∈ AC and Q ∈ AB such that∠AP Q = ∠ABC and ∠AQP = ∠ACB. Calculate P Q + P E +QF, where E and F are the projections of P and Q onto BC ,

respectively.Question 12. Suppose that |ax 2 + bx + c| |x2 −1| for all realnumbers x. Prove that |b2 −4ac| 4.1.7 Hanoi Open Mathematics Competition 2012


Q ti 1 A th t b ( b) Th


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Question 4 . A man travels from town A to town E throughtowns B, C and D with uniform speeds 3km/h, 2km/h, 6km/hand 3km/h on the horizontal, up slope, down slope and hori-zontal road, respectively. If the road between town A and townE can be classied as horizontal, up slope, down slope and hor-izontal and total length of each type of road is the same, whatis the average speed of his journey?

(A) 2km/h; (B) 2,5km/h; (C) 3km/h; (D) 3,5km/h; (E)

4km/h.


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Question 5 . How many different 4-digit even integers can beform from the elements of the set {1, 2, 3, 4, 5}.(A): 4; (B):5; (C): 8; (D): 9; (E) None of the above.

Question 6 . At 3:00 A.M. the temperature was 13 o below zero.By noon it had risen to 32 o. What is the average hourly increasein teparature?

Question 7 . Find all integers n such that 60 + 2 n − n2 is aperfect square.

Question 8 . Given a triangle ABC and 2 points K ∈ AB,N ∈ BC such that BK = 2AK , CN = 2BN and Q is thecommon point of AN and CK . Compute

S ∆ ABC S ∆ BC Q ·

Question 9 . Evaluate the integer part of the number

H = 1 + 20112 + 20112

20122 +

20112012·


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Question 14 . Let be given a triangle ABC with ∠A = 900

and the bisectrices of angles B and C meet at I . Suppose thatIH is perpendicular to BC (H belongs to BC ). If HB = 5cm,HC = 8cm, compute the area of

ABC .

Question 15 . Determine the greatest value of the sum M =xy + yz + zx, where x, y, z are real numbers satisfying the fol-lowing condition x2 + 2 y2 + 5 z 2 = 22 .


Question 1 . Let x = 6 + 2√ 5 + 6 −2√ 5√ 20 · The value of H = (1 + x5 −x7)2012

3 11

is

(A): 1; (B): 11; (C): 21; (D): 101; (E) None of the above.

Question 2 . Compare the numbers:P = 2α , Q = 3, T = 2β , where α = √ 2, β = 1 + 1√


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Question 5 . Let f (x) be a function such that f (x)+2 f x + 2010

x −1 =

4020 −x for all x = 1. Then the value of f (2012) is(A): 2010; (B):2011; (C): 2012; (D):2014; (E) None of theabove.

Question 6 . For every n = 2 , 3, . . . , we put

An = 1− 1

1 + 2 × 1− 1

1 + 2 + 3 ×···×1− 1

1 + 2 + 3 + · · ·+ n .Determine all positive integer n (n ≥ 2) such that

1An

is aninteger.

Question 7 . Prove that the number a = 1 . . . 12012

5 . . . 52011

6 is a

perfect square.


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Question 11. Suppose that the equation x3 + px2 + qx + r = 0has 3 real roots x1, x2, x3, where p, q, r are integer numbers. PutS n = xn1 + xn2 + xn3 , n = 1, 2, . . . Prove that S 2012 is an integer.

Question 12. In an isosceles triangle ABC with the base ABgiven a point M ∈ BC. Let O be the center of its circumscribedcircle and S be the center of the inscribed circle in ∆ ABC andSM AC. Prove that OM ⊥ BS.Question 13. A cube with sides of length 3cm is painted redand then cut into 3 ×3×3 = 27 cubes with sides of length 1cm.If a denotes the number of small cubes (of 1cm ×1cm×1cm) thatare not painted at all, b the number painted on one sides, c thenumber painted on two sides, and d the number painted on threesides, determine the value a

−b

−c + d?

Question 14 . Solve, in integers, the equation 16 x + 1 = ( x2 −y2)2


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Question 2 . How many natural numbers n are there so thatn2 + 2014 is a perfect square.


Question 3 . The largest integer not exceeding [( n +1) α]−[nα ],where n is a natural number, α =

√ 2013√ 2014, is:


Question 4 . Let A be an even number but not divisible by 10.The last two digits of A20 are:

(A): 46; (B): 56; (C): 66; (D): 76; (E): None of theabove.


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prove that the equation f (x) = 2x2 −1 has two real roots.Question 13 . Solve the system of equations

1

x +

1

y =

1

63x

+ 2y

= 56

Question 14 . Solve the system of equations

x3 + y = x2 + 12y3 + z = 2y2 + 13z 3 + x = 3z 2 + 1

Question 15 . Denote by Q and N∗ the set of all rational and

positive integer numbers, respectively. Suppose that ax + b

x ∈Q

for every x ∈ N∗. Prove that there exist integers A, B,C suchthat

+ b A + B


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(A): 12012

; (B): 12013

; (C): 12014

; (D): 12015

; (E): Noneof the above.

Question 3 . What is the largest integer not exceeding 8 x3 +

6x −1, where x = 12 3

2 + √ 5 + 3

2 −√ 5 ?(A): 1; (B): 2; (C): 3; (D): 4; (E) None of the above.Question 4 . Let x0 = [α ], x1 = [2α ] − [α ], x2 = [3α ] − [2α ],x4 = [5α ]−[4α], x5 = [6α] −[5α], ..., where α =

√ 2013√ 2014. Thevalue of x9 is

(A): 2; (B): 3; (C): 4; (D): 5; (E): None of the above.

Question 5 . The number n is called a composite number if itcan be written in the form n = a × b, where a, b are positiveintegers greater than 1.


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∆ ACD such that AMN is an equilateral triangle. Determine BMC.

Question 8 . Let ABCDE be a convex pentagon and

area of ∆ ABC = area of ∆ BCD = area of ∆ CDE

= area of ∆ DEA = area of ∆ EAB.

Given that area of ∆ ABCDE = 2 . Evaluate the area of area of ∆ ABC.

Question 9 . A given polynomial P (t) = t3 + at 2 + bt + c has 3distinct real roots. If the equation ( x2 + x + 2013) 3 + a(x2 + x +2013)2 + b(x2 + x + 2013) + c = 0 has no real roots, prove that

P (2013) > 164

.

Question 10 . Consider the set of all rectangles with a givenarea S Find the largest value of


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Hanoi Open Mathematical Competition 2016Senior Section

Saturday, 12 March 2016 08h30-11h30

Question 1. How many are there 10-digit numbers composed from the digits 1,

2, 3 only and in which, two neighbouring digits differ by 1.

(A): 48 (B): 64 (C): 72 (D): 128 (E): None of the above.

Question 2. Given an array of numbers A = (672 , 673, 674, . . . , 2016) on ta-ble. Three arbitrary numbers a,b,c ∈ A are step by step replaced by number13

min(a,b,c ). After 672 times, on the table there is only one number m, such that

(A): 0 < m < 1 (B): m = 1 (C): 1 < m < 2 (D): m = 2 (E): None of theabove.

Question 3. Given two positive numbers a, b such that the condition a3 + b3 =a 5 + b5 , then the greatest value of M = a 2 + b2 −ab is(A):

1

4

(B): 1

2

(C): 2 (D): 1 (E): None of the above.

Question 4. In Zoo, a monkey becomes lucky if he eats three different fruits.What is the largest number of monkeys one can make lucky having 20 oranges, 30


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Question 9. Let rational numbers

a, b, c satisfy the conditions

a + b + c = a 2 + b2 + c2 ∈Z .

Prove that there exist two relative prime numbers m, n such that abc = m 2

n 3 .

Question 10. Given natural numbers a, b such that 2015 a 2 + a = 2016b2 + b. Prove

that √ a −b is a natural number.Question 11. Let I be the incenter of triangle ABC and ω be its circumcircle. Letthe line AI intersect ω at point D = A. Let F and E be points on side BC andarc BDC respectively such that ∠BAF = ∠CAE <

12∠BAC . Let X be the second

point of intersection of line E I with ω and T be the point of intersection of segmentDX with line AF . Prove that TF.AD = ID.AT .

Question 12. Let A be point inside the acute angle xOy. An arbitrary circle ωpasses through O, A ; intersecting Ox and Oy at the second intersection B and C,respectively. Let M be the midpoint of BC. Prove that M is always on a xed line(when ω changes, but always goes through O and A).

Question 13. Find all triples ( a,b,c ) of real numbers such that

|2a + b

| ≥4 and

|ax2 + bx + c| ≤1 ∀x∈[−1, 1].

Question 14 Let f (x) = x 2 + px + q where p q are integers Prove that there is


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Digit 1 2 3 41 12 121 12122 21 123 12323 23 212 2121

32 232 2123321 2321323 2323

32123232

3 4 6 8

.

We can see that a number ending by 2 in previous column generates 2 numbersfor next column (we can add 1 or 3 at the end), but a number ending by 1 or 3generate 1 number for next column (we can add only 2 at the end). From this, wecan make a table.

The rst row is number of digits, the second row is the number of k-digit numberssatisfying the condition and ending with 1, 3, the third row is the number of k-digitnumbers satisfying the condition and ending with 2.

1 2 3 4 5 6 7 8 9 102 2 4 4 8 8 16 16 32 321 2 2 4 4 8 8 16 16 32

.

Question 2. (A).

Note that


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We haveab(a 2 −b

2 )2 ≥0⇔2a3 b3 ≤ab

5 + a 5 b⇔(a3 + b3 )2 ≤ (a + b)(a

5 + b5 ). (1)

Combining a3 + b3 = a 5 + b5 and (1), we nd

a 3 + b3 ≤a + b⇔a2 + b2 −ab ≤1.

The equality holds if a = 1 , b = 1 .

Question 4. (D).

First we leave tangerines on a side. We have 20 + 30 + 40 = 90 fruites. As wefeed the happy monkey is not more than one tangerine, each monkey eats fruits of these 90 at least 2.

Hence, the monkeys are not more than 90/2 = 45. We will show how you canbring happiness to 45 monkeys:

5 monkeys eat: orange, banana, tangerine;15 monkeys eat: orange, peach, tangerine;25 monkeys eat peach, banana, tangerine.At all 45 lucky monkeys - and left ve unused tangerines!

Question 5. (E).

We have 3x 2 + x = 4y2 + y

⇔(x

−y)(3x + 3 y + 1) = y2 .

We prove that ( x −y; 3x + 3 y + 1) = 1 .Indeed, if d = ( x −y; 3x + 3 y + 1) then y2 is divisible by d2 and y is divisible byd; x is divisible by d, i.e. 1 is divisible by d, i.e. d = 1.


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Type 2: Three vertices lie in distinct horizontal lines.We have 3 ×3 ×3 triangles of these type. But we should remove degeneratedtriangles from them. There are 5 of those (3 vertical lines and two diagonals). So,

we have 27 - 5 = 22 triangles of this type.Total, we have 54 + 22 = 76 triangles.For those students who know about C kn this problem can be also solved as C

39 −8where 8 is the number of degenerated triangles.

Question 8. Let abc, where a ,b, c ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9}, a = 0 and abc =(a + b + c)3 .Note that 100 ≤ (a + b + c)

3

≤ 999 and 3√ 100 ≤ a + b + c ≤

3√ 999. Hence5 ≤a + b + c ≤9.If a + b+ c = 5 then abc = ( a + b + c)3 =5 3 = 125 and a + b+ c = 8 (not suitable).

If a + b+ c = 6 then abc = ( a + b + c)3 =6 3 = 216 and a + b+ c = 9 (not suitable).If a + b + c = 7 then abc = ( a + b + c)3 =7 3 = 343 and a + b + c = 10 (not

suitable).If a + b + c = 8 then abc = ( a + b + c)3 =8 3 = 512 and a + b + c = 8 (suitable).If a + b + c = 9 then abc = ( a + b + c)3 =9 3 = 729 and a + b + c = 18 (not

suitable).Conclusion: abc = 512.

Question 9. Put a + b + c = a 2 + b2 + c2 = t.We have 3 (a 2 + b2 + c2 ) ≥ (a + b + c)

2 , then t∈[0; 3].

Since t∈Z then t∈ {0;1;2;3}.If t = 0 then a = b = c = 0 and abc = 0 = 0

1.

If t = 3 then


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Hencey = rq (q

− p)

x = rp ( p −q )z = rpq ⇒abc = − [ pq ( p −q )]

2

( p2 + q 2 − pq )3 .

We prove that ( pq ( p −q ); p2 + q 2 − pq ) = 1 .Suppose that s = ( pq ( p −q ) ; p2 + q 2 − pq ) ; s > 1 then s| pq ( p −q ) .Case 1. Let s| p. Since s|( p2 + q 2 − pq ) then s|q and s = 1 (not suitable).Case 2. Let s|q. Similarly, we nd s = 1 (not suitable).Case 3. If s

|( p

−q ) then s

|( p

−q )2

−( p2 + q 2

− pq )

⇒

s

| pq

⇒

s

| p

s|q (not suitable).

If t = 2 then a + b + c = a 2 + b2 + c2 = 2 .We reduce it to the case where t = 1 , which was to be proved.

Question 10. From equality

2015a 2 + a = 2016b2 + b, (1)

we nd a ≥b.If a = b then from (1) we have a = b = 0 and √ a −b = 0 .If a > b, we write (1) asb2 = 2015( a 2 −b

2 ) + ( a −b)⇔b2 = ( a −b)(2015a + 2015b + 1) . (2)

Let (a, b) = d then a = md ; b = nd, where (m, n ) = 1 . Since a > b then m > n ;and put m

−n = t > 0.

Let ( t, n ) = u then n is divisible by u; t is divisible by u and m is divisible by u.That follows u = 1 and then ( t, n ) = 1 .

Putting b = nd ; a −b = td in (2), we nd


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Since CI is bisector of ∠ACL , we get ILAI

= CLAC

. Furthermore, ∠DCL =

∠DCB = ∠DAB = ∠CAD = 12∠BAC. Hence, the triangles DCL and DCA are

similar. Therefore, CLAC

= DC AD

.Finally, we have ∠DIC = ∠IAC + ∠ICA = ∠ICL + ∠LCD = ∠ICD . It

follows DIC is a isosceles triangle at D . Hence DC AD

= IDAD

.

Summarizing all these equalities, we get T F AT

= ILAI

= CLAC

= DC AD

= IDAD ⇒

T F

AT =

ID

AD ⇒TF.AD = ID.AT as desired.

Question 12. Let (Ox ), (Oy ) be circles passing throught O, A and tangent to


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Since (Ox ) is tangent to Ox, ∠ADC = ∠AOB. Since OBAC is cyclic, ∠ABO =

∠ACD. So triangles AOB, ADC are similar. ThereforeABAC

=OBDC

(1)

Similarly, ABE ACO, soBE CO

=ABAC

(2)From (1) and (2), we deduce that

OB

CD

= BE

OC ⇒

OB

BE

= CD

OC Hence

OE BE

= ODOC ⇒

ON BE

= OP OC ⇒

ON NB

= ON

BE −NO =

OP OC −OP

= OP CP

It follows, if NP intersects BC at M, then MBMC ·

P C P O ·

NONB

= 1 (by Menelaus’

Theorem in triangle OBC ) conclusionMBMC = 1 , it follows N P passes through M is

midpoint of BC.

Question 13. From the assumptions, we have |f (±1)| ≤1, |f (0)| ≤1 and

f (1) = a + b + cf (−1) = a −b + cf (0) = c

⇔

a = 12

[f (1) + f (−1)] −f (0)b = 1

2[f (1) −f (−1)]

c = f (0)


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Indeed, we have

f [f (x) + x] = [f (x) + x]2 + p[f (x) + x] + q = f 2 (x) + 2 f (x).x + x 2 + pf (x) + px + q

= f (x)[f (x) + 2 x + p] + x 2 + px + q = f (x)[f (x) + 2 x + p] + f (x)

= f (x)[f (x) + 2 x + p + 1] = f (x)[x 2 + px + q + 2 x + p + 1]

= f (x)[(x + 1) 2 + p(x + 1) + q ] = f (x)f (x + 1) ,

which proves (1).Putting m := f (2015) + 2015 gives

f (m ) = f [f (2015) + 2015] = f (2015)f (2015 + 1) = f (2015)f (2016),

as desired.

Question 15. We have

T −2(18ab+9 ca+29 bc) = (5 a−3b)2 +(4 a−3c)

2 +(4 b−5c)2 +( a−3b+3 c)

2

≥0, ∀a,b,c∈R .That follows T ≥2. The equality occures if and only if

5a −3b = 04a

−3c = 0

4b −5c = 0a −3b + 3 c = 0

⇔

5a −3b = 04a −3c = 04b −5c = 018ab + 9 ca + 29 bc = 1


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de thu va loi giai HOMC 2006 - 2016